UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICOPOSGRADO EN CIENCIAS FÍSICAS

INSTITUTO DE CIENCIAS NUCLEARES

SUPERESPACIOS DEFORMADOS Y TEORÍAS DE CAMPOS SUPERSIMÉTRICAS NO CONMUTATIVAS

TESISQUE PARA OPTAR POR EL GRADO DE:

DOCTORA EN CIENCIAS (FÍSICAS)

PRESENTA:DALIA BERENICE CERVANTES CABRERA

TUTORA PRINCIPALDRA. MARÍA ANTONIA LLEDÓ BARRENA

UNIVERSIDAD DE VALENCIA MIEMBROS DEL COMITÉ TUTOR

DR. MIGUEL SOCOLOVSKY VAJOVSKYINSTITUTO DE CIENCIAS NUCLEARES

DR. ZBIGNIEW ANTONI OZIEWICZ KAWASSFACULTAD DE ESTUDIOS SUPERIORES

MÉXICO, D. F. FEBRERO 2014

i

A mi madre...

Libertango

Mi libertad me ama y todo el ser le entrego.

Mi libertad destranca la cárcel de mis huesos.

Mi libertad se ofende si soy feliz con miedo.

Mi libertad desnuda me hace el amor perfecto.

Mi libertad me insiste con lo que no me atrevo.

Mi libertad me quiere con lo que llevo puesto.

Mi libertad me absuelve si alguna vez la pierdo

por cosas de la vida que a comprender no acierto.

Mi libertad no cuenta los años que yo tengo,

pastora inclaudicable de mis eternos sueños.

Mi libertad me deja y soy un pobre espectro,

mi libertad me llama y en trajes de alas vuelvo.

Mi libertad comprende que yo me sienta preso

de los errores míos sin arrepentimiento.

Mi libertad quisieran el astro sin asueto

y el átomo cautivo, ser libre ½qué misterio!

Ser libre. Ya en su vientre mi madre me decía

�ser libre no se compra ni es dádiva o favor�.

Yo vivo del hermoso secreto de esta orgía:

si polvo fui y al polvo iré, soy polvo de alegría

y en leche de alma preño mi libertad en �or.

De niño la adoré, deseándola crecí,

mi libertad, mujer de tiempo y luz,

la quiero hasta el dolor y hasta la soledad.

Mi libertad me sueña con mis amados muertos,

mi libertad adora a los que en vida quiero.

Mi libertad me dice, de cuando en vez, por dentro,

que somos tan felices como deseamos serlo.

Mi libertad conoce al que mató y al cuervo

que ahoga y atormenta la libertad del bueno.

Mi libertad se infarta de hipócritas y necios,

mi libertad trasnocha con santos y bohemios.

Mi libertad es tango de par en par abierto

y es blues y es cueca y choro, danzón y romancero.

Mi libertad es tango, juglar de pueblo en pueblo,

y es murga y sinfonía y es coro en blanco y negro

Mi libertad es tango que baila en diez mil puertos

y es rock, malambo y salmo y es ópera y �amenco.

Mi libertango es libre, poeta y callejero,

tan viejo como el mundo, tan simple como un credo.

De niño la adoré, deseándola crecí,

mi libertad, mujer de tiempo y luz,

la quiero hasta el dolor y hasta la soledad.

Música: Astor Piazzolla

Letra: Horacio Ferrer

(1990)

iv

Agradecimientos

Quiero agradecer a mi asesora la Dra. María Antonia por su paciencia al dirigirme eneste trabajo de investigación, animándome y ayudándome a aprender una nueva rama delas matemáticas. Así como darme la oportunidad de participar con su grupo de trabajo enValencia. Al Prof. Oziewicz por siempre creer y alentar con sus discusiones académicas,recomendaciones, consejos y apoyo mi carrera cientí�ca. Al Prof. Adolfo por su enormepaciencia, su permanente solidaridad académica y humana. A Miguel Socolovsky profesory amigo por iniciarme en dos de mis grandes pasiones, matemáticas en la física de altasenergías y el baile. A los sinodales por sus observaciones y sugerencias las que hanenriquecido este trabajo. Principalmente deseo dar las gracias a mi familia y amigos enlos que siempre he encontrado apoyo y amor.Finalmente agradezco al Consejo Nacional de Ciencia y Tecnología (CONACYT) y alPrograma de Apoyo a Proyectos de Investigación e Inovación Tecnológica (PAPPIT) porsus apoyos brindados en la realización de parte de mis estudios de doctorado.

vi

Contents

Contents vii

Summay 2

Resumen 3

Introduction 5

Preliminaries 91.1 Basic algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Classic Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 A�ne varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2 Projective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4 Sheaf theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.5 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5.1 A�ne schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.2 Projective schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.6 Functor of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.7 Coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.8 Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.9 Supergeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.9.1 Supervectorial spaces and superalgebras . . . . . . . . . . . . . . 311.9.2 Modules for superalgebras . . . . . . . . . . . . . . . . . . . . . . 351.9.3 Hopf superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 361.9.4 Superspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.9.5 Projective supergeometry . . . . . . . . . . . . . . . . . . . . . . 401.9.6 The functor of points of projective superspace . . . . . . . . . . . 421.9.7 The functor of points of the Grassmannian superscheme . . . . . 431.9.8 Supergroup functors . . . . . . . . . . . . . . . . . . . . . . . . . 451.9.9 Actions of Supergroups . . . . . . . . . . . . . . . . . . . . . . . . 46

Chiral Superspaces 492.1 Real and chiral super�elds in Minkowski superspace . . . . . . . . . . . . 49

2.1.1 Scalar super�elds . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

vii

CONTENTS

2.1.2 Action of the Lorentz group SO(1,3) . . . . . . . . . . . . . . . . 502.1.3 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.1.4 Shifted coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 532.1.5 Supersymmetric theories . . . . . . . . . . . . . . . . . . . . . . . 54

2.2 The super Grassmannian variety . . . . . . . . . . . . . . . . . . . . . . . 552.2.1 Plücker embedding of the Grassmannian variety and the big cell . 562.2.2 The Poincaré group plus dilatations and the big cell . . . . . . . . 572.2.3 The Plücker embedding for the super Grassmannian . . . . . . . . 582.2.4 The superstraightening algorithm . . . . . . . . . . . . . . . . . . 612.2.5 The conformal and Poincaré supergroups and the big cell . . . . . 63

2.3 Quantum super Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . 662.3.1 Quantum supergroups . . . . . . . . . . . . . . . . . . . . . . . . 662.3.2 Presentation of the quantum super Grassmannian Grq . . . . . . . 682.3.3 Grq as a homogeneous quantum space . . . . . . . . . . . . . . . . 71

2.4 Quantum deformation of the big cell inside the super Grassmannian . . . 732.4.1 Super setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.4.2 Quantum setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Quantum Twistors 793.1 Poisson algebra and �-product . . . . . . . . . . . . . . . . . . . . . . . . 793.2 Algebraic star product on Minkowski space . . . . . . . . . . . . . . . . . 813.3 Di�erential star product on the big cell . . . . . . . . . . . . . . . . . . . 82

3.3.1 Explicit computation up to order 2 . . . . . . . . . . . . . . . . . 823.3.2 Di�erentiability at arbitrary order . . . . . . . . . . . . . . . . . . 86

3.4 Poincaré coaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.4.1 The coaction as a di�erential operator . . . . . . . . . . . . . . . 90

3.5 The real forms: the Euclidean and Minkowskian signatures . . . . . . . . 933.5.1 The real forms in the classical case . . . . . . . . . . . . . . . . . 943.5.2 The real forms in the quantum case . . . . . . . . . . . . . . . . . 96

3.6 The deformed quadratic invariant . . . . . . . . . . . . . . . . . . . . . . 97

Conclusions 99

References 101

Appendix 1 107

Appendix 2 109B.1 A basis for the Poincaré quantum group . . . . . . . . . . . . . . . . . . 109

B.1.1 Generators and relations for the Poincaré quantum group . . . . . 109B.1.2 The Diamond Lemma . . . . . . . . . . . . . . . . . . . . . . . . 111B.1.3 A basis for the Poincaré quantum group . . . . . . . . . . . . . . 112

1

CONTENTS

Summary

In this Thesis we give a quantum deformation of the chiral super Minkowski space infour complex dimensions, as the big cell inside a quantum super Grassmannian. Thequantization is performed in such way that the actions of the Poincaré and conformalquantum supergroups on the quantum Minkowski and quantum conformal superspacesare preserved.

As a starting point, we beging with the classical case, the big cell beging a dense opensubset that can be identi�ed with the complexi�ed Minkowski space inside the Grass-mannian Gp2, 4q recognized as the space of 2-complex planes in the complex vectorialspace C4. This Grassmannian is also an homogeneous space, and it is an embeddingin to the proyective space P5. The image of this embedding of Gp2, 4q de�ne a Plückerrelation. The big cell is invariant under the Poincaré group time dilatations. All thesefacts are extend to the supergroups category, using the tool of functors of points overlocal algebras for each superspace. Moreover the Grassmannian super ring is given interms of generators and Plücker relations, and the superspace of the big cell is associatedwith the superalgebra of chiral super�elds.

For the quantization, we substitute the supergroups by quantum supergroups followingManin's work, and the corresponding homogeneous spaces by quantum homogeneousspaces. We obtain the deformed Grassmannian super variety in terms of generators andquantum super Plücker relations and the corresponding coaction of the Poincaré quantumsupergroup on the quantum big cell identi�ed with the quantum chiral super Minkowskispace. Taking the limit, we recover the classic case. This deformation corresponds to astar product in the ring of formal power series in the indeterminate q over C. We givethe explicit formula for this star product on polynomials on the complexi�ed Minkowskispace, and we proof that the star product can be extended to act on smooth functions asa di�erential star product. We obtain a quadratic Poisson bracket of deformation. Alsowe get the coaction of the quantum Poincaré supergroup plus dilatations in terms of thisstar product, which is di�erential too.

To complete the construction of the non commutative Minkowski and Euclidian spaces,we give the adequate real forms.Finally a deformed quadratic invariant under the Poincaré quantum supergroup is given.

2

CONTENTS

Resumen

En esta Tesis, se da una deformación cuántica del superespacio quiral de Minkowski encuatro dimensiones complejas como la celda mayor dentro de una super Grassmannianacuántica. La cuantización se realiza de tal forma que las acciones de los supergrupos cuán-ticos de Poincaré y conformal en los superespacios cuánticos de Minkowski y conformalse preservan.

Para ello se comienza con el caso clásico, la celda mayor que es un subconjunto abiertodenso puede ser identi�cada con el espacio compleji�cado de Minkowski dentro de laGrassmanniana Gp2, 4q que corresponde con el espacio de 2-planos complejos en el espaciovectorial complejo C4. También esta Grassmanniana es un espacio homogéneo y estáencajado en el espacio proyectivo P5. La imagen de este encaje en Gp2, 4q de�ne unarelación de Plücker o cuádrica de Klein. La celda mayor es invariante bajo el grupo dePoincaré más dilataciones.

Todos los resultados anteriores se generaliza en la categoría de supergrupos, usando laherramienta de funtores de puntos sobre álgebras locales. Además el super anillo Grass-manniano es dado en términos de generadores y super relaciones de Plücker. El superes-pacio que corresponde a la celda mayor es asociado con la superálgebra de supercamposquirales.

Para la cuantización, sustituimos los supergrupos por supergrupos cuánticos siguiendoel trabajo de Manin y los correspondientes espacios homogéneos por espacios homogé-neos cuánticos. Se obteniene la super variedad Grassmanniana deformada en términosde generadores y relaciones super cuánticas de Plücker así como la correspondiente coac-ción del supergrupo de Poincaré más dilataciones sobre la celda mayor identi�cada conel superespacio quiral cuántico de Minkowski. Tomando el límite se recupera el casoclásico. A esta deformación le corresponde un producto estrella en el anillo de seriesde potencias formales en la indeterminada q sobre los complejos. Damos una formúlaexplícita de este producto estrella entre monomios en el espacio de Minkowski compleji�-cado, demostrando que puede ser extendida a la acción en funciones diferenciables comoun producto estrella diferencial. Obteniendo un corchete de Poisson cuadrático para ladeformación. También se consigue escribir la coacción del grupo cuántico de Poincarémás dilataciones en términos del producto estrella la cual es también diferencial.

Para completar la construcción de los espacios no comutativos de Minkowski y Euclideo,encontramos adecuadas formas reales. Finalmente se presenta un invariante cuadráticobajo la coacción del supergrupo cuántico de Poincaré.

3

CONTENTS

4

CONTENTS

Introduction

In this Thesis we investigate some aspects of supersymmetric �eld theories by means ofalgebraic and geometric techniques. It can be divided in three related parts:

The �rst part of this Thesis is developed in Chapter 1.The principal objective of this chapter is to provide the mathematical structure used inthe followings chapters.

The second part of this Thesis is treat in Chapter 2.The main intentions of this part are described below,

1. To produce a non commutative version of the Minkowski and conformal superspaces.

2. To show the quantum super Grassmannian as a homogeneous quantum space.

3. To realize the quantized super Minkowski space as the big cell of the quantumconformal space, as it happens in the non quantum case.

4. To show the action of the corresponding quantum symmetry groups over quantumMinkowski superspace.

The third part is developed in Chapter 3.In this chapter our goals are detailed next,

1. To de�ne a non-commutative star product for the conformal complexi�cation ofMinkowski space.

2. To show that the action of the star product on polynomials can be reproduced bya bidi�erential operator.

3. To de�ne a coaction of the Poincaré group plus dilations on Minkowski space com-patible with the star product.

4. To show that this coaction can be reproduced by a di�erential operator up to someorder in the quantization parameter.

5. To complete the construction of the non commutative Minkowski and Euclidianspaces, giving adequate real forms.

6. To found a deformed quadratic invariant under the Poincaré group plus dilations.

In the history of physics, deformations of mathematical structures have been used inseveral occasions. For example, when Galilean transformations between inertial framesof reference were seen not to describe adequately the physical world, a deformation ofthe group law was introduced to solve this problem. The Lorentz group is a deformationof the Galilei group in terms of the parameter 1{c. From the mathematical point of view

5

CONTENTS

it is not di�cult to imagine the deformation inside the category of groups, but from thephysical point of view it has enormous consequences. In this deformation scheme, the oldstructure is seen as a limit or contraction when the parameter takes a preferred value.

In the quantum theory the word quantization means changing (or deforming) the algebraof observables (usually functions over a phase space) to a non commutative one (usuallyoperators over a Hilbert space).

In our investigation we apply the theory of quantum deformation to superspaces givingrise to noncommutative superspace.

Starting from the classical setting, the Minkowski space in four dimensions is just R4

with the pseudoeuclidean metric diagp1,�1,�1,�1q. The Poincaré group Pp1, 3q is thegroup that preserves such metric, while the conformal group is the group that preservesthe metric up to a global factor. It is in fact the group SOp2, 4q and it acts non linearlyon a compacti�cation of the Minkowski space, obtained by adjoining to it not just apoint at in�nity, but the closure of a cone [23]. This compacti�cation turns out to be theGrassmannian manifold Gp2, 4q, that is, the space of 2-planes inside a four dimensionalvector space and the Poincaré group together with the dilations is precisely the subgroupof SOp2, 4q consisting of the elements that leave the Minkowski space invariant.The rest of the conformal transformations may send a point in the Minkowski space toa point at in�nity. The Minkowski space sits inside the Grassmannian Gp2, 4q as its bigcell, which is a dense open set inside it.We will refer to the Grassmannian Gp2, 4q as the conformal space.

The relation between the Poincaré and conformal group on one side and the Minkowskiand conformal space on the other side, are well known, see for example Ref. [1]. Abrief but complete review can also be found in Ref. [24], which we have followed veryclosely in spirit and notation. The analysis in there starts by considering the spin groupof SOp2, 4q, that is, SUp2, 2q, which contains the group SLp2,CqR, which is the spin groupof SOp1, 3q. In general it is more natural to work in the complexi�ed spaces (SLp4,Cqand SLp2,Cq � SLp2,Cq respectively) and to look at the end for the particular real formassociated to Minkowskian signature.

This approach is very useful when extending the results to the Minkowski and confor-mal superspaces (see [23]), since the action on spinors is explicit in the formalism. Thefascinating subject of supergeometry emerges here as a very natural framework. Superge-ometry extends standard algebraic and di�erential geometry in a less dramatic way thannon commutative geometry [25]. The category of algebras considered in supergeometryare non commutative, but their non commutativity a�ects only to some generators thatanticommute. These are the odd generators.

The functor of points is another tool, that one borrows from standard algebraic geometryand extends to the super setting. For a standard algebraic variety, the geometric pointsare the morphisms from the coordinate ring of the variety to the ground �eld. If oneconsiders the morphisms from the coordinate ring to another commutative ring, say R,one has the R-points of the algebraic variety. For a supervariety one takes R to be

6

CONTENTS

a commutative superalgebra, and the R-points still make sense. The geometric pointsturn out to give the points of a standard variety called the reduced variety, and theodd variables disappear. Only by considering morphisms to superalgebras instead thanto commutative algebras or just the ground �eld one can recover the role of the oddvariables, and with it all the information needed to reconstruct the supervariety.

We then see that via the functor of points the true nature of the odd variables appearsin a beautiful, perfectly consistent framework [24, 26, 27], which is very close to the wayin which physicists have been thinking and talking about superspaces and supergroups.The terminology and perhaps the level of rigor becomes more sophisticated, but a closerlook reveals many old concepts that have been used implicitly by physicists are at thecore of the mathematical formulation.

The next step is to produce a non commutative version of the Minkowski and conformalsuperspaces. In order to do this, we need to substitute the commutative superalgebrasby noncommutative ones, but in that step the geometric intuition that we had retainedin supergeometry with the functor of points is lost. We have then to rely on the algebraiccounterpart of the geometric objects and try to generalize them to the non commutativesetting. Non commutative geometry [25] is certainly the most complete framework to doso, and ultimately it will be connected to the quantization of space and superspace.

The approach followed in the �rst part of this Thesis is:

We substitute the supergroups by quantum supergroups and the corresponding homoge-neous spaces by quantum homogeneous spaces. This was the approach followed in Refs.[7, 8, 9] for the non super case. We are then able to preserve also the realization of thequantized super Minkowski space as the big cell (appropriately de�ned) of the quantumconformal space.

The problem of quantizing the Minkowski superspace has appeared in many places in thephysics literature. We mention some references, although our list is not exhaustive. We�nd a �rst step as early as in Refs. [28], [29] and in deformations (mostly with constantPoisson bracket) inspired in string theory [30, 31, 32, 33]. In other papers one �ndsthe quantization of a super Minkowski `phase space', which, although it is not exactlythe problem that we examine here, it is also of interest [34, 35, 36]. Quantizations forother superspaces such as supergroups [37, 38, 39], their coadjoint orbits [40] or otherhomogeneous superspaces Pm|n are constructed in Ref. [41].

The principle that guides us in choosing a particular deformation is that we want topreserve the action of the corresponding symmetry groups and we also ask that thequantum Minkowski superspace appears as the big cell (appropriately de�ned in algebraicterms) inside the quantum conformal superspace. Obviously, all of these requests haveto be made precise in the framework of deformations.

Conformal invariance is not a symmetry of all the physical theories (it is a symmetryof electromagnetism, for example), so it should be an explicitly broken symmetry. Aspointed out in Ref. [2], one can write down any �eld theory in the twistor formalism

7

CONTENTS

and then the terms that break the invariance appear isolated. This could then clarifythe mechanisms for the explicit breaking of the symmetry. In mathematical terms, onepasses from the Minkowski space to conformal space by a compacti�cation and viceversaby taking the big cell of the conformal space. So one could think on a non conformallysymmetric �eld theory as a conformal theory broken down to the big cell by some extraterms.Moreover, conformal symmetry has a fundamental role in the gauge/gravity correspon-dence [4] (for a recent review see Ref. [5]) which relates gravity theories to conformallyinvariant gauge theories de�ned on a boundary of spacetime.

In the original papers [1, 2], Penrose believed that twistor theory could help to introducethe indetermination principle in spacetime. The points had to be `smeared out' and intwistor formalism a point of spacetime is not a fundamental quantity, but it is secondaryto twistors. The twistor space is C4 and the Grassmannian Gp2, 4q � C4.

Nevertheless, all the twistor construction is classical, and the formalism does not providewith a quantum formulation of gravity.

In order to introduce the quantum indetermination principle in spacetime, one has tointroduce noncommutativity in the algebra of functions over spacetime.

This can be done by deforming the original algebra, this deformation induces a starproduct on elements of classical space with the scalars extended to the ring of formalpower series Cq.

We give an explicit formula for the star product among two polynomials in the bigcell of the quantum Grassmannian, which is the quantum Minkowski space. Since thequantum algebras that we present here are deformations of the algebra of polynomialson Minkowski space, the star product that we obtain is algebraic. Nevertheless, weshow that this deformation can be extended to the set of smooth functions in terms ofa di�erential star product. The Poisson bracket (the antisymmetrized �rst order termin h with q � eh) of the deformation is a quadratic one, so the Poisson structure is notsymplectic (nor regular).

Examples of such transition from the category of algebraic varieties to the category ofdi�erential manifolds in the quantum theory are given in Refs. [12, 13, 40]. In thesereferences, the varieties under consideration are coadjoint orbits and the Poisson bracketis linear. It was shown in that paper that some algebraic star products do not havedi�erential counterpart (not even modulo and equivalence transformation).

It is interesting that one of the algebraic star products that do not have di�erentialextension is the star product on the coadjoint orbits of SUp2q associated to the standardquantization of angular momentum. For algebraic star products and their classi�cationsee also Ref. [15].

8

1.1. BASIC ALGEBRA

Preliminaries

The principal mathematical framework used in the Thesis is introduced in this chapter.It can be skipped to come back to it if needed when reading each chapter.The organization of this chapter is as follows:In section 1.1 we provide some de�nitions and examples of basic algebra.In section 1.2 we present some basic algebraic geometry de�nitions and examples.In section 1.3 we provide some de�nitions, properties and examples of category theory.In section 1.4 we give some de�nitions, properties and examples of basic sheaf theory .In section 1.5 we present basic scheme de�nition, some properties and examples of it.In section 1.6 we provide the de�nition, some properties and examples of the functor ofpoint .In section 1.8 we give the Hopf algebras de�nition, some properties and examples of it.

1.1 Basic algebra

In this section we follow the Ref.[55]

De�nition 1.1.1. A group G is a �nite or in�nite set of elements together with a binaryoperation, G�GÑ G given by pa, bq ÞÑ ab, that satis�es the next four properties:

1. Closure: If a and b are two elements in G, then the product ab is also in G.

2. Associativity: For all a, b, c P G, then pabqc � apbcq.

3. Identity: There is an identity element e such that ea � ae � a for all a P G.

4. Inverse: There must be an inverse of each element. Therefore, for each element aof G, the set contains an element b � a�1 such that aa�1 � a�1a � e.

A subset H of the group G is a subgroup of G if and only if it is nonempty and closedunder group operation and inverses. �

Example 1.1.2. The cyclic subgroup of order k for the group G, generated by b P Gis given by Gk � tb, b2, . . . , bk | bk�1 � b, for k P Zu. We can easily see, it is closed underthe exponent addition, associative, and each element has unique inverse. �

De�nition 1.1.3. A group G is called an abelian or commutative group if for all aand b in G then ab � ba. �

De�nition 1.1.4. A map between two groups which preserves the identity and the groupoperation is called a homomorphism . If a homomorphism has an inverse which is alsoa homomorphism, then it is called an isomorphism and the two groups are calledisomorphic. �

9

CONTENTS

De�nition 1.1.5. If G is a group and X is a set then a left group action of G on Xis a map

G�X Ñ X

pg, xq ÞÑ gx

that satis�es the following two axioms:

1. Associativity: pghqx � gphxq for all h, g P G and x P X.

2. Identity: There is an identity element e such that ex � x for all x P X.

�

De�nition 1.1.6. In complete analogy, one can de�ne a right group action of G onX by the map X � G Ñ X, px, gq ÞÑ xg such that xpghq � pxgqh for all g, h P G andx P X. There is also e, such that xe � x for all x P X. �

De�nition 1.1.7. The left group action of G on the set X is called transitive if X isnon-empty and if for any x, y P X there exist an g P G such that gx � y. �

De�nition 1.1.8. Consider a group G acting on a set X. The orbit of a point x P X isa set of elements of X to which x can be moved by the elements of G. The orbit of x isdenoted Gx and de�ned by

Gx � tgx | g P G and x P Xu.

�

De�nition 1.1.9. Some elements of a group G acting on a set X may �x a point x.These group of elements form a subgroup called the isotropy group, i.e., Ix � tg PG and x P X | gx � xu. �

De�nition 1.1.10. A ring is a set R equipped with two binary operations� : R�RÑ Rand � : R � R Ñ R, called addition and multiplication respectively. The set R andtwo operations pR,�, �q must satisfy the following requirements:

1. pR,�q is an abelian group, where the identity element is denoted by 0.

2. pR, �q is a monoid , that is, it satis�es the three following properties:

(a) Closure: for all a, b P R, then a � b is also in R.

(b) Associativity: for all a, b, c P R, the equation pa � bq � c � a � pb � cq holds.

(c) Identity: there exists an element 1 P R, such that 1 � a � a � 1 � a holds, forall a P R.

3. The distributive laws:

(a) For all a, b and c in R, the equation a � pb� cq � pa � bq � pa � cq holds.

10

1.1. BASIC ALGEBRA

(b) For all a, b and c in R, the equation pa� bq � c � pa � cq � pb � cq holds.

A subset of a ring that forms a ring with respect to the operations of ring is known as asubring . �

Example 1.1.11. Considering the ring Z. It is a subring of the ring of polynomials withinteger coe�cients Zrxs when we identify Z as the constant polynomials. The ring Z hasnot subrings (with multiplicative identity) others than itself. �

De�nition 1.1.12. A ring homomorphism is a map f : R Ñ R between the rings Rand R such that:

1. fpx� yq � fpxq � fpyq (so that f is a homomorphism of abelian groups),

2. fpxyq � fpxqfpyq,

3. fp1q � 1.

In other words, f respects addition, multiplication and the identity element. �

De�nition 1.1.13. A subring I of a ring R is a left (right) ideal if ax P I (xa P I),for all a P R and x P I. If R is an abelian group then all ideals are left and right. A leftand right ideal is a two-sided ideal . �

De�nition 1.1.14. Let I be a left ideal of the ring R. If for every x P I we can writex �

°iPJ aixi where ai P R and txiuiPJ � I, we say that txiuiPJ generates I. �

De�nition 1.1.15. A zero-divisor in a ring R is an element a � 0 which �divides 0�,i.e., for which there exists b � 0 in R such that a � b � 0. �

De�nition 1.1.16. A ring with no zero-divisors � 0 (and in which 1 � 0) is called anintegral domain . �

Example 1.1.17.

1. R,C,Z are integral domains.

2. Rings of polynomials are integral domains if the coe�cients come from an integraldomain. For instance, the rings, Rrxs,Crxs,Zrxs are integral domains. �

De�nition 1.1.18. An element a P R is nilpotent if an � 0 for some n ¡ 0, n P N. �

De�nition 1.1.19. A unit in R is an element a which �divides 1�, i.e., an element asuch that a � b � 1 for some b P R. The element b is then uniquely determined by a, andis written a�1. �

The units in R form a (multiplicative) abelian group.

De�nition 1.1.20. A principal ideal is an ideal m in a ring R that is generated by asingle element m of R. �

11

CONTENTS

De�nition 1.1.21. A �eld is a commutative ring which contains a multiplicative inverseof every non-zero element. Equivalently, a �eld is a ring whose non-zero elements forman abelian group under multiplication. �

Example 1.1.22.

1. R and C are �elds.

2. Z is not a �eld.

�

De�nition 1.1.23. An ideal P of a abelian ring R is prime if it has the following twoproperties:

1. If a, b P R, and a � b is an element of P, then a P P or b P P.

2. P is not equal to the whole ring R.

�

De�nition 1.1.24. Given a ring R and a proper ideal I of R (that is I � R, t0u), I isa maximal ideal of R if any of the following equivalent conditions holds:

1. There exist no other proper ideal K of R so that I � K.

2. For any ideal K with I � K, either K � I or K � R.

3. The quotient ring R{I is a simple ring (that is, it has no ideal besides the zeroideal and itself.)

�

Example 1.1.25. In the ring Z, the maximal ideals are the principal and prime idealsgenerated by a prime.

De�nition 1.1.26. A local ring is a ring R that contains a single maximal ideal m. Thequotient K � R{m is a �eld called the residue �eld of R. �

De�nition 1.1.27. A principal ideal domain is an integral domain in which everyideal is principal. �

De�nition 1.1.28. Let R be any ring. A multiplicatively closed subset of R isa subset S of R such that 1 P S and S is closed under multiplication. It de�nes anequivalent relation � on R � S as follows:

pr, sq � pr1, s1q if and only if prs1 � r1sqs2 � 0 for some s2 P S, s2 � 0.

Let r{s denote the equivalence class of pr, sq, the localization of R at S denoted byS�1R or RS is the set of equivalence classes. RS has ring structure by de�ning additionand multiplication of these �fractions� r{s in the same way as in elementary algebra

r

s�r1

s1�rs1 � r1s

ss1,

r

s�r1

s1�rr1

ss1.

�

12

1.1. BASIC ALGEBRA

Example 1.1.29. If p is a prime ideal of R, then S � R z p is a multiplicative system,and the corresponding localization is denoted by Rp. If f is an element of R, thenS � R { tfn |n ¥ 1u is a multiplicative system, and the corresponding localization isdenoted by Rf . �

Remark 1.1.30. The elements a{s with a P p form an ideal m in Rp. If b{t R m, thenb R p, hence b P S and therefore b{t is a unit in Rp. It follows that if q is an ideal in Rp

and q � m, then q contains a unit and it is therefore the whole ring. Hence m is the onlymaximal ideal in Rp; in other words Rp is a local ring. �

De�nition 1.1.31. Let R be a commutative ring. A left R-module is pair pM,µq,where M is a abelian group and a map µ : R �M Ñ M given by µpa, xq � ax (scalarmultiplication), the following axioms are satis�ed:

1. Distributive laws: For all a, b P R and x, y PM , the equations,

(a) apx� yq � ax� ay,

(b) pa� bqx � ax� ay, holds.

2. Associativity: For all a, b P R and x PM , the equation, pabqx � apbxq holds,

3. Identity: For 1 P R and every x PM the equation 1x � x holds.

A submodule M 1 of M is a subgroup of M which is closed under multiplication byelements of R. �

Example 1.1.32. For the ring Z and G any abelian group, G is a Z-module, with themap µ : Z�GÑ G given by pn, xq ÞÑ µpn, xq � nx � x� . . .� x (n-summands), where0x � 0 and p�nqx � �pnxq. �

De�nition 1.1.33. Let M,N be R-modules. A map f : M Ñ N is an R-module

homomorphism (or is R-linear) if for all a P R and all x, y PM :

1. f preserves the addition, i.e., fpx� yq � fpxq � fpyq,

2. f preserves the scalar multiplication, i.e., fpaxq � afpxq.

�

Example 1.1.34. HompM,Nq the set of all R-module homomorphisms from M to Nwith the map µ : R � HompM,Nq Ñ HompM,Nq such that,

pa, f � gqpxq � afpxq � agpxq,

for all a P R, f, g P HompM,Nq and x PM is a R-module. �

De�nition 1.1.35. If M,N are R-modules, their direct sum M ` N is the set of allpairs px, yq PM �N . This is an R-module if we de�ne addition and scalar multiplicationin the obvious way:

13

CONTENTS

1. px1, y1q � px2, y2q � px1 � x2, y1 � y1q for xi PM and yi P N,

2. apx, yq � pax, ayq for a P R, x PM and y P N .

More generally, if tMiuiPI is any family of R-modules, we can de�ne their direct sumÀiPIMi. Its elements are the families txiuiPI such that xi P Mi for all i P I and almost

all the xi are zero. �

De�nition 1.1.36. A free R-module is one which is isomorphic to an R-module of theform

ÀiPIMi, where each Mi � R (as an R-module). A �nitely generated free R-module

is therefore isomorphic to R ` � � � `R. �

De�nition 1.1.37. A R-module V is called a vector space if R � k, for k a �eld. �

De�nition 1.1.38. Let V be a vector space over k of dimension n ¥ 2 and 0   d   n bean integer. The Grassmannian Gpd, nq over k is de�ned as the set of all d-dimensionalsubspaces of V i.e.,

Gpd, nq � tW |W is a k- subspace of V of dimension du.

�

De�nition 1.1.39. Let R be a commutative ring. An R-algebra is an R-module Atogether with a A-multiplication p , q : A�AÑ A, wich satis�es pax� by, zq � apx, zq �bpy, zq, pz, ax� byq � apz, xq � bpz, yq for all a, b P R and x, y, z P A. �

De�nition 1.1.40. An R-algebra homomorphism f : A Ñ A1 is a ring homomor-phism which is also an R-module homomorphism. �

1.2 Classic Algebraic Geometry

In this section we follow the Ref.[53]

1.2.1 A�ne varieties

De�nition 1.2.1. A topological space is a set X with a distinguished collection ofsubsets, to be called the open sets. These open sets must satisfy the following:

1. Both X and the empty set are open.

2. If U and V are open sets, then so is their intersection U X V .

3. The union of any collection of open sets is open.

These open subsets de�ne the topology of X. �

14

1.2. CLASSIC ALGEBRAIC GEOMETRY

De�nition 1.2.2. If the X is a topological space, then a cover C � tUα |α P Iu of Xis an indexed family of sets such that X �

�αPI Uα. We say that C is an open cover if

each of its members is an open set. �

De�nition 1.2.3. The order of a cover is the largest integer n such that there are n�1members of the cover that have a nonempty intersection. A cover C 1 is a re�nement ofa cover C if every member of C 1 is contained in some member of C. �

De�nition 1.2.4. A topological space X is said to be �nite dimensional if there issome integer n such that for every open cover C of X, there is an open cover C 1 thatre�nes C and has order at most n � 1. The topological dimension is de�ned as thesmallest value of n which this statement holds. �

De�nition 1.2.5. An a�ne n-space over the �eld k (usually R, C), denoted Ank orsimply An, is the set of all n-tuples of elements of k. An element p P An will be called apoint and if p � pa1, ..., anq with ai P k, then the ai will be called the coordinates of p.

�

Let A � krx1, . . . , xns be the polynomial ring in n variables over k. We will interpret theelements of A as functions from the a�ne n-space to k, by de�ning fppq � fpa1, ..., anqwhere f P A and p P An. Thus if f P A is a polynomial, we can talk about the set ofzeros of f , namely Zpfq � tp P An|fppq � 0u. More generally, if T is any subset of A,we de�ne the zero set of T to be the common zeros of all the elements of T , namely

ZpT q � tp P An | fppq � 0 for all f P T u.

De�nition 1.2.6. A subset Y of An is an algebraic set if exists a subset T � A suchthat Y � ZpT q. �

De�nition 1.2.7. We de�ne the Zariski topology on An by taking the open subsetsto be the complements of the algebraic sets. This is a topology, because the intersectionof two open sets is open and the union of any family of open sets is open. Furthermore,the empty set and the whole space are both open and closed. �

De�nition 1.2.8. A nonempty subset Y of a topological space X is irreducible if itcannot be expressed as the union Y � Y1 Y Y2 of two proper subsets, each one of whichis closed in Y . The empty set is not considered to be irreducible. �

De�nition 1.2.9. An a�ne algebraic variety (or simply a�ne variety) is an irre-ducible closed algebraic subset of An. The topology in the a�ne variety is the inducedZariski topology. �

De�nition 1.2.10. So for any subset Y � An, let us de�ne the ideal of Y in A by

IpY q � tf P A | fppq � 0 for all p P Y u.

�

15

CONTENTS

Proposition 1.2.11. Let Y � An be an algebraic set. Then Y is irreducible if and onlyif IpY q is a prime ideal.Proof . See [54]. �

Lemma 1.2.12. Let p � pa1, . . . , anq P An. The ideal

mp :� px1 � a1, . . . , xn � anq P A

is maximal.Proof . See [54]. �

Theorem 1.2.13. Let T � A and MT the set of all maximal ideals m � A with T � m.Then the map

ZpT q ÝÑMT

pa1, . . . , anq ÞÑ px1 � a1, . . . , xn � anq

is a bijection.Proof . See [54]. �

De�nition 1.2.14. If Y � An is an a�ne algebraic set, we de�ne the a�ne coordinate

ring OpY q of Y , to be A{IpY q. �

Remark 1.2.15. By Hilbert's Nullstellensatz, the points of a�ne variety Y are in oneto one correspondence with the maximals of its coordinate ring OpY q, through the mapgiven in the above theorem. �

De�nition 1.2.16. We de�ne the dimension of an a�ne variety to be its dimension asa topological space. �

De�nition 1.2.17. Let Y be an a�ne variety. A function f : Y Ñ k is regular at a pointp P Y if there is an open set U with p P U � Y , and polynomials g, h P A � krx1, . . . , xns,such that h is nowhere zero on U , and f � g{h on U . We say that f is regular on Y if itis regular at every point of Y . �

Remark 1.2.18. The elements of the coordinate ring OpY q are so called the regularfunctions of the variety Y . �

1.2.2 Projective varieties

De�nition 1.2.19. Let k be our �xed �eld. We de�ne projective n-space over k,denoted Pnk , or simply Pn, to be the set of equivalence classes of pn�1q-tuples pa0, . . . , anqof elements of k, not all zero, under the equivalence relation given by pa0, . . . , anq �pλa0, . . . , λanq for all λ P k, λ � 0. �

Another way of saying this is that Pn, as a set, is the quotient of the set An�1ztp0, . . . , 0quunder the equivalence relation which identi�es points lying on the same line through theorigin.An element of Pn is called a point . If p is a point, then any pn� 1q-tuple pa0, . . . , anq inthe equivalence class p is called a set of homogeneous coordinates for p.

16

1.2. CLASSIC ALGEBRAIC GEOMETRY

De�nition 1.2.20. A graded ring is a ring S, together with a decomposition S �Àd¥0 Sd of S into a direct sum of abelian groups Sd, such that for any d, e ¥ 0, Sd �Se �

Sd�e. An element of Sd is called a homogeneous element of degree d. �

Thus any element of S can be written uniquely as a (�nite) sum of homogeneous elements.

De�nition 1.2.21. An ideal a � S is a homogeneous ideal if a �
d¥0paX Sdq. �

An ideal is homogeneous if and only if it can be generated by homogeneous elements. Thesum, product and intersection of homogeneous ideals are homogeneous. To test whethera homogeneous ideal is prime, it is su�cient to show for any two homogeneous elementsf, g that f � g P a implies f P a or g P a.

The polynomial ring A � krx0, . . . , xns is a graded ring with the standard polynomialgrade1.If f P A is an polynomial, we cannot use it to de�ne a function on Pn, because ofthe non uniqueness of the homogeneous coordinates. However, if f is a homogeneous

polynomial of degree d, then fpλa0, . . . , λanq � λdfpa0, . . . , anq, so that the property off being zero or not depends only on the equivalence class of pa0, . . . , anq.

Thus we can talk about the zeros of a homogeneous polynomial , namely Zpfq �tp P Pn|fppq � 0u.

De�nition 1.2.22. If T is any set of homogeneous elements of A we de�ne the zero set

of T to be

ZpT q � tp P Pn | fppq � 0 for all f P T u.

�

De�nition 1.2.23. If a is a homogeneous ideal of A, we de�ne Zpaq � ZpT q, where Tis the set of all homogeneous elements in a. �

De�nition 1.2.24. A subset Y of Pn is an algebraic set if there exists a set T ofhomogeneous elements of A such that Y � ZpT q. �

Proposition 1.2.25. The union of two algebraic sets of Pn is an algebraic set. Theintersection of any family of algebraic sets is an algebraic set. The empty set and thewhole space are algebraic sets.Proof . See [53] �

De�nition 1.2.26. We de�ne the Zariski topology on Pn by taking the open sets tobe the complements of algebraic sets. �

Once we have a topological space, the notions of irreducible subset and the dimension ofa subset, which were de�ned for the a�ne case, in the same way apply.

1The degree of a nonzero polynomial f is the largest total degree of a monomial occurring in f with

nonzero coe�cients.

17

CONTENTS

De�nition 1.2.27. A projective algebraic variety (or simply projective variety)is an irreducible algebraic set in Pn, with the induced Zariski topology. �

De�nition 1.2.28. If Y is any subset of Pn, we de�ne the homogeneous ideal of Yin A, denoted IpY q, to the ideal generated by tf P A | f is homogeneous and fppq �0 for all p P Y u. If Y is an algebraic set, we de�ne the homogeneous coordinate ring

of Y to be O � A{IpY q. �

Proposition 1.2.29. Every projective variety has an open covering by a�ne varieties.Proof . See [53] �

De�nition 1.2.30. Let Y � Pn be a projective variety. A function f : Y Ñ k is regularat a point p P Y if there is an open neighborhood U with p P U � Y , and homogeneouspolynomials g, h P A � krx0, . . . , xns, of the same degree, such that h is nowhere zero onU , and f � g{h on U . We say that f is regular on Y if it is regular at every point. �

Note that in this case, even though g and h are not functions on Pn, their quotient is awell-de�ned function whenever h � 0, since they are homogeneous of the same degree.

1.3 Category theory

De�nition 1.3.1. A category is a quadruple C � pOC,Hom, id, �q, where

1. OC is a class, the members of which are called objects .

2. Hom, assigns to each ordered pair pA,Bq of objects a set HompA,Bq, the membersof which are called themorphisms from A (domain) to B (codomain) and denotedin the form f : AÑ B.

3. id assigns to each object C P OC a morphism idC : C Ñ C, called the identity onC.

4. � is an operator, called composition , which assigns to each pair of morphismsf : A Ñ B and g : B Ñ C a morphism g � f : A Ñ C (it is often abbreviated asgf).

These data are subject to the following three axioms:

1. For each f : AÑ B, idB � f � f � f � idA.

2. The composition � is associative i.e., hpgfq � phgqf for all f : AÑ B, g : B Ñ Cand h : C Ñ D.

3. If pA,Bq � pC,Dq, then HompA,Bq and HompC,Dq are disjoint. �

Example 1.3.2. The category Set consisting of all sets as objects and maps betweensets as morphisms. Identities are identity maps, and composition is just composition ofmaps. �

18

1.3. CATEGORY THEORY

Example 1.3.3. Taking all groups as objects and homomorphisms between groups asmorphisms produces a category Grp. Similarly, topological spaces and continuous mapsform a category Top. �

De�nition 1.3.4. The opposite category C opis obtained by reversing the morphism of

the category C, while keeping the same objects. �

De�nition 1.3.5. Let C and D be categories. A functor F from C to D consists of a ob-

ject mapOCFÝÑ OD and a family of morphism maps F : HompA,Bq Ñ HompFpAq,FpBqq,

where A y B range over OC such that:

1. FpidAq � idFpAq, for each object A in C and

2. FpgqFpfq � Fpgfq, for each pair of morphisms f : AÑ B and g : B Ñ C in C. �

Of course, one has an identity functor IA from A to A on each category A (which actsas the identity on objects as well as on morphisms), and the composite of two functorsin the obvious sense is again a functor.

De�nition 1.3.6. A functor F : AÑ B is called covariant if it preserves the directionsof morphisms, i.e., every morphism f : AÑ B in the category A is mapped to a morphismFpfq : FpAq Ñ FpBq in the category B. �

De�nition 1.3.7. A functor F : A Ñ B is called contravariant if it reverses thedirections of morphisms, i.e., every morphism f : AÑ B in the category A is mapped toa morphism Fpfq : FpBq Ñ FpAq. �

Note that a contravariant functor F : AÑ B is the covariant functor F : A opÑ B.

Example 1.3.8. Given an object A in the category A one has a covariant functorHompA,_q : AÑ Set which maps an object B to HompA,Bq and a morphism f : B Ñ Cto the map HompA,Bq Ñ HompA,Cq whose value at g : A Ñ B is f � g : A Ñ C.Dually, there is a contravariant functor Homp_, Aq : A Ñ Set , which maps a mor-phism f : B Ñ C to the map HompC,Aq Ñ HompB,Aq whose value at g : C Ñ A isg�f : B Ñ A. These functors are uni�ed in a covarian functor Homp_,_q : A op

�AÑ Setwhich maps a pair pA,Bq of objects to HompA,Bq and a pair pf, gq of morphisms to themap h ÞÑ g � h � f . �

De�nition 1.3.9. A functor F : A Ñ B is called an embedding if on morphisms isinjective. �

De�nition 1.3.10. Let F ,G : A Ñ B be functors. A natural transformation Φ :F Ñ G is a family of B-morphisms ΦA : FpAq Ñ GpAq, where A ranges over OA, suchthat, for each morphism f : AÑ B in A, ΦBFpfq � GpfqΦA, i.e., the following diagramcommutes.

19

CONTENTS

FpAq GpAq

FpBq GpBq

-ΦA

?

Fpfq

?

Gpfq

-ΦB

More over, if every ΦA is an isomorphism the Φ is called a natural isomorphism . �

It is easily veri�ed that the composition of two natural transformations is again a naturaltransformation.

De�nition 1.3.11. A natural transformation Φ : G Ñ F of functors from a category C tothe category Set is injective if for every object A the induced map of sets GpAq Ñ FpAqis injective. In this case we will say G is a subfunctor of F . �

De�nition 1.3.12. F ,G,H are functors from a category C to the category Set and ifΦ : G Ñ F and Ψ : H Ñ F are natural transformations, the �ber product of G and Hover F , denoted G �F H is the functor from C to Set de�ned by setting, for any object Aof C

pG �F HqpAq � tpx, yq P GpAq �HpAq |Φpxq � Ψpyq in FpAqu

and de�ned on morphisms in the obvious way. �

De�nition 1.3.13. We de�ne C and D to be equivalente categories if there exist functorsF : CÑ D and G : DÑ C such that GF � I C and FG � ID where � denotes the naturalisomorphism of functors. �

1.4 Sheaf theory

In this section we follow the Ref.[53]

De�nition 1.4.1. Let X be a topological space. A presheaf F of abelian groups on Xconsists of the data:

1. For every open subset U � X, F pUq is an abelian group.

2. For every inclusion V � U of open subsets of X, there is a morphism of abeliangroups ρUV : F pUq Ñ F pV q, usually called the restriction map, subject to the nextconditions:

(a) F p∅q � t0u, where ∅ is the empty set.

(b) ρUU is the identity map F pUq Ñ F pUq.

(c) If W � V � U are three open subsets, then ρUW � ρVW � ρUV .

�

20

1.4. SHEAF THEORY

De�nition 1.4.2. (Alternative)

For any topological space X, we de�ne a category OppXq whose objects are the opensubsets of X, and where the only morphism is the inclusion map. Thus HompV, Uq isempty if V � U where U and V are objects of OppXq and HompV, Uq has just one elementif V � U . Now a presheaf is just a contravariant functor from the category OppXq tothe category AbpXq of abelian groups. �

De�nition 1.4.3. We de�ne a presheaf with values in any �xed category C, byreplacing the words �abelian group� in the above de�nitions by �ring�, �set�, or �object ofC � respectively. �

We will stick to the case of abelian groups in this section, and the reader can make thenecessary modi�cations for the case of rings, sets, etc.

As a matter of terminology, if F is a presheaf on X, we refer to F pUq as the sectionsof the presheaf F over the open set U , and we sometimes use the notation Γ pU, F q todenote the group F pUq, and s|V instead of ρUV psq, if s P F pUq.

De�nition 1.4.4. A presheaf F on a topological space X is a sheaf if it satis�es thefollowing two supplementary conditions:

1. If U is an open set, tViuiPI is an open covering of U , and s P F pUq is an elementsuch that s|Vi � 0 for all i P I, then s � 0.

2. Gluing property. If U is an open set, tViuiPI is an open covering of U , and wehave elements si P F pViq for each i P I, with the property that for each i, j P I,si|ViXVj � sj|ViXVj , then there is an element s P F pUq such that s|Vi � si for eachi P I. �

Example 1.4.5. Let X be an algebraic variety over the �eld k. For each open set U � X,let OXpUq be the ring of regular functions from U to k, it has a sheaf structure: for eachopen set V � U , the map ρUV : OXpUq Ñ OXpV q is a restriction map (in the usualsense). We call OX the sheaf of regular functions or structural sheaf on X.

De�nition 1.4.6. If F is a presheaf on X, and if p is a point of X, we de�ne the stalkFX,p of F at p in the following way:

A element of FX,p is represented by a pair pU, sq (called germ), where U is an openneighborhood of p, and s is an element of F pUq. Two such pairs pU, sq and pV, tq de�nethe same element of FX,p if and only if there is an open neighborhood W of p withW � U X V , such that s|W � t|W . Thus we may speak of elements of the stalk FX,p asgerms of sections of F at the point p. In the case of a variety X and its structural sheafOX , the stalk OX,p at a point p is just the local ring of p on X. �

In other words the stalk FX,p of F at p to be the direct limit of groups F pUq over all openneighborhoods U of p P X.

21

CONTENTS

De�nition 1.4.7. If F and G are presheaves on X, a morphism of presheaves ϕ : FÑ G

is given by morphisms of abelian groups ϕU : FpUq Ñ GpUq for each open set U , such thatwhenever V � U is an inclusion, the diagram commutes.

FpUq GpUq

FpV q GpV q

-ϕpUq

?

ρUV

?

ρ1UV

-ϕpV q

where ρ to ρ1 are the restriction maps in F and G. Also a morphism of presheaves is anatural transformation between the corresponding functors.If F and G are sheaves on X, we use the same de�nition for a morphism of sheaves. Anisomorphism is a morphism which has an inverse. �

Proposition 1.4.8. Let ϕ : F Ñ G be a morphism of sheaves on a topological space X.

Then ϕ is an isomorphism if and only if the induced map on the stalk ϕp : FX,p Ñ GX,p

is

an isomorphism for every p P X.Proof . See Ref.[53] �

De�nition 1.4.9. Let f : X Ñ Y be a continuous map of topological spaces. For anysheaf F on X, we de�ne the direct imagen sheaf f� F on Y by pf� F qpV q � F pf�1pV qqfor any open set V � Y . �

1.5 Schemes

1.5.1 A�ne schemes

Remark 1.5.1. (Prime spectrum.)Let R be a ring and X be the set of all prime ideals of R. For each subset H of R, letVpHq denote the set of all prime ideals of R which contain H, then:

1. If H is the ideal generated by H, then VpHq � VpHq.

2. Vpt0uq � X and VpXq � ∅.

3. If tHiuiPI is any family of subsets of R, then

Vp¤iPI

Hiq �£iPI

VpHiq.

4. VpFpHq XGpHqq � VpFpHqq YVpGpHqq for any ideals F,G of R and H � R.

22

1.5. SCHEMES

These results show that the setsVpHq satisfy the axioms of the closed sets of a topologicalspace. The resulting topology is called Zariski topology . For each f P R, let Dpfq � Df

denote the complement of Vpfq. The sets Dpfq are open. The topological space X iscalled the prime spectrum of R, and is written SpecR. �

Example 1.5.2.

1. SpecZ � tprimes p P Zu Y t0u,

2. The Gaussian integers ring Zris :� ta � ib | a, b P Zu has a prime spectrum:

SpecZris � units

$'&'%

p, where p � 4n� 3 is a prime in N.α � a� bi, where Npαq is either 2 or a prime of Ncongruent to 1mod 4.

Where the units of Zris are �1,�i and the map N is the norm de�ned by Npαq � |α|2 �αα � a2 � b2. �

De�nition 1.5.3. SpecR.

The spectrum of ring R is the pair pSpecR,OSpecRq � SpecR consisting of the topologicalspace SpecR, together with its structural sheaf of rings OSpecR, i.e.,

Df

OSpecR

ÝÝÝÝÑ Rf

where Df is any open set in SpecR and Rf is the localization of R at the prime ideal f(see 1.1.28), i.e., OSpecR is sheaf of regular functions on SpecR (see 1.4.5). �

Proposition 1.5.4. Let R be a ring and SpecR its spectrum.

1. For any f P SpecR, the stalk OSpecR,f of the sheaf OSpecR is isomorphic to the localring Rf.

2. Γ pSpecR,OSpecRq � R.

Proof . See Ref.[53] �

De�nition 1.5.5. A ringed space X is a pair pX,OXq consisting of a topological spaceX and the structure sheaf of rings OX on X. �

De�nition 1.5.6. Amorphism of ringed spaces from pX,OXq to pY,OY q is a pair pf, fqof a continuous map f : X Ñ Y and a map f : OY Ñ f�OX of sheaves of rings on Y . �

De�nition 1.5.7. The ringed space X is a locally ringed space and it is denoted Xl

if for each point p P X, the stalk OX,p is a local ring. �

23

CONTENTS

De�nition 1.5.8. A morphism of locally ringed spaces Xl and Yl is a morphism pf, fqof ringed spaces, such that for each point p P X, the induced map of local rings fp :OY,fppq Ñ OX,p is a local homomorphism of local rings.

We explain the last condition: Given a point p P X, the morphism of sheaves f : OY Ñf�OX induces a homomorphism of rings OY pV q Ñ OX

�f�1pV q

�, for every open set V in

Y . As V ranges over all open neighborhoods of fppq, f�1pV q ranges over a subset of theneighborhoods of p.

Taking direct limits, we obtain a map

OY,fppq � limÝÑV

OY pV q Ñ limÝÑV

OX�f�1pV q

�� OX,p.

Thus we have an induced homomorphism fp : OY,fppq Ñ OX,p, and we require that this

be a local homomorphism , i.e., f�1p pmpq � mfppq, where mfppq and mp respectively are

the maximal ideals in the stalks OY,fppq and OX,p. �

De�nition 1.5.9. An isomorphism of locally ringed spaces is a morphism with aninverse. Thus a morphism pf, fq is an isomorphism of locally ringed spaces if and onlyif f is a isomorphism of the underlying topological space, and f is an isomorphism ofsheaves. �

Proposition 1.5.10.

1. Let be R a ring then SpecR is a locally ringed space.

2. if ϕ : A Ñ B is a homomorphism of rings, then ϕ induces a natural morphism oflocally ringed spaces

pf, fq : pSpecB,OSpecBq ÝÑ pSpecA,OSpecAq.

3. If A and B are rings, then any morphism of locally ringed spaces from SpecB toSpecA is induced by a homomorphism of rings ϕ : AÑ B as above.

Proof . See Ref.[53] �

De�nition 1.5.11. An a�ne scheme is a locally ringed space pX,OXq which is iso-morphic to SpecR for some ring R. We say that X is a scheme if pX,OXq is a locallyringed space, which is locally isomorphic to an a�ne schemes. In other words, for eachp P X, there exists an open set Up � X such that pUp,OX |Upq is an a�ne scheme, werefer to pUp,OX |Upq as an open a�ne subscheme of X. A morphism of schemes isjust a morphism of locally ringed spaces. �

De�nition 1.5.12. A schemes morphism pi, iq : pY,OY q Ñ pX,OXq is a closed immer-

sion if, as a topological space, ipY q is a closed subspace of X, and i : OX Ñ i�OY is asurjective map of rings. �

24

1.5. SCHEMES

De�nition 1.5.13. A closed subscheme of pX,OXq is an equivalence class of closedimmersions into pX,OXq, i.e., the morphisms of schemes pi, iq : pY,OY q Ñ pX,OXq andpi1, i1q : pY 1,OY 1q Ñ pX,OXq are equivalent if there is an isomorphism pα, αq : pY,OY q ÑpY 1,OY 1q such that i1 � α � i and i1 � α � i. �

Remark 1.5.14.

1. There is an equivalence of categories (see 1.3.13) between the category of commuta-tive rings Ring and the category of a�ne schemes Asch. This equivalence is de�nedon the objects by

R ÞÑ SpecR.

A morphism of rings AÑ B corresponds contravariantly to a morphism SpecB ÑSpecA of the corresponding a�ne schemes.

2. Since any a�ne varietyX is completely described by the knowledge of its coordinatering OpXq (the ring of regular functions on the whole variety), we can associateuniquely to an a�ne variety X the a�ne scheme SpecOpXq.

�

1.5.2 Projective schemes

Next we will de�ne an important class of schemes, constructed from graded rings, whichare analogous to projective varieties.

In this section we follow to Ref.[27]

De�nition 1.5.15. ProjS.Let S be a graded ring (see (1.2.20)). We denote by S� the ideal

Àd¡0 Sd. We de�ne the

set ProjS as the set of all homogeneous prime ideals p which do not contain S� (theseare sometimes called relevant homogeneous prime ideals). For each homogeneous ideala of S, one de�nes the closed set

V paq � tp P ProjS|p � au.

This gives ProjS a topology, a base of open subset consisting of Dpfq, for each f homo-geneous element of S, Dpfq is the set of p P ProjS with f R p. We consider the sheaf ofrings OProjS on ProjS, with stalk at p P ProjS,

OProjS,p � th

g|h, g are homogeneous elements of the same grade in S and g R p.u

which is isomorphic to the local ring Sppq (the elements of zero degree in the localizationSp). And for any homogeneous f P S, pDpfq,OProjS|Dpfqq is isomorphic to SpecSpfq.Then pProjS,OProjSq is a scheme and it is denote by ProjS. �

25

CONTENTS

De�nition 1.5.16. If R is a ring, we de�ne the projective n-space over R to be thescheme ProjRrx0, . . . , xns. In particular, ifR is the �eld k � R,C, then Projkrx0, . . . , xns �Pn is a scheme, whose subspace of closed points of underlying topological space is natu-rally homeomorphic to the space of pn� 1q-tuples which de�ne the variety likewise calledprojective n-space Pn (see 1.2.19). �

De�nition 1.5.17. An open subscheme of a schemeX is a scheme U , whose topologicalspace is an open subset ofX, and whose structure sheaf OU is isomorphic to the restrictionOX |U of the structure sheaf of X. �

De�nition 1.5.18. The dimension of a scheme X, is its dimension as a topologicalspace. �

1.6 Functor of points

De�nition 1.6.1. Let Schopdenote the opposite category of schemes and X be a scheme.

The functor of points HX of X is the covariant functor such that, over objects,

SchopHX

ÝÝÝÑ SetY ÞÝÑ HXpY q � HompY ,Xq.

And for reverse morphisms f op : Z Ñ Y in the category Schopof the morphism of schemes

f : Y Ñ Z, the morphism of sets HXpZq Ñ HXpY q is given by HXpfopqpgq � g � f P

HXpY q. �

De�nition 1.6.2. Let FuncpSchop,Set q denote the category of functors from Schop

to Set.Then the functor of points of FuncpSchop

,Set q is a covariant functor,

H : SchÑ FuncpSchop,Set q

which assigns to every object X the functor HX and for every morphism of schemesf : X Ñ X 1, the natural transformation Φ : HX Ñ HX 1 which for each reverse morphismof schemes gop : Z Ñ Y satis�es ΦY �HXpg

opq � HX 1pgopq � ΦZ (see 1.3.10). �

De�nition 1.6.3. For a scheme X, the elements of HXpY q are called the Y -points ofX. If Y � SpecR for a commutative ring R, this is usually abbreviated as the R-pointsof X. �

De�nition 1.6.4. If F : SchopÑ Set is a functor, then F is representable if there is a

scheme X such that F � HX , where the isomorphism is in the category FuncpSchop,Set q

(that is, a natural transformation from F to HX with an inverse). �

Remark 1.6.5. The restriction of F : SchopÑ Set to a�ne schemes is not in general

representable. However, since, the category of a�ne schemes is equivalent to Ring, thereis a scheme X such that its functor of points He

X given by

HeX : Ringop

Ñ Set, HeXpRq � HompSpecR,Xq � R � points of X.

26

1.6. FUNCTOR OF POINTS

is representable.Notice that when X is a�ne, X � SpecOpXq and we have

HeXpRq � HompSpecR, SpecOpXqq � HompOpXq, Rq.

In this case, the functor HeX is representable. �

Proposition 1.6.6. Consider the a�ne schemes X � SpecOpXq and Y � SpecOpY q.There is a one-to-one correspondence between the scheme morphisms X Ñ Y and thering morphisms OpY q Ñ OpXq.Proof . See Ref.[59] �

Proposition 1.6.7. The functor of points HX of a scheme X is completely determinedby its restriction to the category of a�ne schemes.Proof . See Ref.[59] �

De�nition 1.6.8. Let Lrs denote the category of locally ringed spaces, and let Xl anobject in Lrs. We de�ne the functor of points of the locally ringed space Xl as therepresentable functor

HXl: Lrsop

ÝÑ Set,Yl ÞÝÑ HompYl, Xlq,

as before, HXlis de�ned on morphisms as follows: HXl

pf opqpgq � g � f . �

Theorem 1.6.9. (Yoneda's lemma.)Let C be a category and let X, Y be objects in C and let HX : C op

Ñ Set be therepresentable functor de�ned on the objects as HXpY q � HompY,Xq and, as usual, onthe morphisms as HXpf

opqpgq � g � f for f : Y Ñ Z, g P HompZ,Xq.

1. If F : C opÑ Set then we have a one-to-one correspondence between the sets

tΦ : HX Ñ F |Φ a natural transformationu ðñ FpXq.

2. HX � HY if and only if X � Y and the natural transformations HX Ñ HY are inone-to-one correspondence with the morphisms X Ñ Y .

Proof . See [59] �

Corollary 1.6.10. Two schemes are isomorphic if and only if their functors of points areisomorphic.Proof . See [59] �

Theorem 1.6.11. (Representability theorem.)A functor F : Ringop

Ñ Set is of the form FpRq � HompSpecR,Xq for a scheme X if andonly if

27

CONTENTS

1. F is local . This means that for each ring R and each open covering of X � SpecRby a�ne open sets Di, the functor F satis�es the sheaf axiom, over the localizationsRfi (see 1.5.3), that is, for every collection of elements αi P FpRfiq such that, αiand αj maps to the same element in FpRfifjq, there is a unique element α P FpRqmapping to each of the αi.

2. F is cover by the open a�ne subfunctors . This means that there exists a familytUiuiPI of subfunctors of F such that for every ring R and natural transformationsΦ : HSpecR Ñ F there is a collection of open subschemes Ui � SpecR (see 1.5.11)such that the �ber product Ui �F HSpecR � HUi

where tUiu cover SpecR.

Proof . See Ref.[59]. �

1.7 Coherent sheaves

In this section we will develop the basic properties of quasi-coherent and coherent sheaves.

De�nition 1.7.1. Let X be a ringed space. A sheaf of OX-modules is a sheaf F on X,such that for each open set U � X, the group F pUq is an OXpUq-module, and for eachinclusion of open sets V � U , the restriction homomorphism F pUq Ñ F pV q is compatiblewith the module structures via the ring homomorphism OXpUq Ñ OXpV q. �

De�nition 1.7.2. A morphism of OX-modules is a morphism F Ñ G of sheaves, such

that for each open set U the map FpUq ÞÑ GpUq is a homomorphism of OX-modules. �

Now that we have the general notion of a sheaf of modules on a ringed space, we specializeto the case of schemes. We start by de�ning the sheaf of modules OM on SpecR associatedto a module M over a ring R.

De�nition 1.7.3. Let R be a ring and let M be an R-module. We de�ne the sheafassociated to M on SpecR, denoted by OM as follows. For each prime ideal f � R, let Mf

be the localization of M at f. For any open set Df � SpecR we de�ne the group OMpDfqto be the set of germs of regular functions on SpecR near f. �

Proposition 1.7.4. LetM be a module for a commutative ring R. The sheaf OM de�nedabove has the following properties:

1. OMpDq is an OR-module.

2. OM,f �Mf, for all f P SpecR, i.e., the stalk at any prime f of the sheaf OM coincideswith the localization of M at f.

3. OMpSpecRq � M , i.e., the global sections of the sheaf coincide with the R-moduleM .

Proof . See Ref.[53] �

28

1.8. HOPF ALGEBRAS

De�nition 1.7.5. Let X � pX,OXq be a scheme, O a sheaf on X of OX-modules. Inother words, F pDq is an OXpDq-module for allD open in X and the restriction morphismsbehave nicely with respect to the OX-module structure. We say F is quasi-coherent ,if there exists an open a�ne cover tDi � SpecRiuiPI of X such that F |Di

� OMifor a

suitable Ri-module Mi. F is coherent if the a�ne cover can be chosen so that the Mi'sare �nitely generated Ri-modules. �

1.8 Hopf algebras

In this section we follow Ref.[60]

Quantum groups are non commutative nor cocommutative Hopf algebras. We are goingto start with a series of de�nitions that lead to the de�nition of Hopf algebra. Unlessotherwise stated, the �eld k under consideration is understood to be R or C.De�nition 1.8.1. An algebra pA,m, ηq is a vector space A over the �eld k such that:

1. The linear map m : Ab AÑ A (the product) is associative.

2. η is the linear map η : k Ñ A by ηp1q � 1A (the unit in the algebra).

In terms of commutative diagrams, these maps satisfy the properties:

Ab Ab A

Ab A Ab A

A

HHHHj

idbm�����

mbid

HHHHHjm

������ m

,

Ab A

Ab k A

@@@@@R

m6

idbη

-�

,

Ab A

k b A A

@@@@@R

m6

ηbid

-�

�

De�nition 1.8.2. A algebra map f between two algebras A and A1 is a map thatrespects the algebra structure, mA1 � pf b fq � f �mA and ηA1 � f � ηA. �

De�nition 1.8.3. A coalgebra pC,∆, εq is a vector space C over a �eld k such that:

1. The linear map ∆ : C Ñ C b C (the coproduct) is coassociative.

2. There is ε is a linear map C Ñ k (the counit).

In terms of commutative diagrams, the above properties are

C b C b C

C b C C b C

C

����*∆bid

HHHHY idb∆

HHHHHY

∆ �����*

∆

,

C b C

k b C C?

εbid

-�

@@

@@@I

∆,

C b C

C b k C?

idbε

-�

@@

@@@I

∆

�

29

CONTENTS

De�nition 1.8.4. A coalgebra map f : C Ñ C 1 is a map between two coalgebras thatrespects the coalgebra structure, pf b fq �∆C � ∆C1 � f and εC1 � f � εC . �

De�nition 1.8.5. A bialgebra pH,m, η,∆, εq is a vector space H over a �eld k suchthat:

1. pH,m, ηq is an algebra.

2. pH,∆, εq is a coalgebra.

3. ∆ and ε are algebra maps, where H bH has the tensor product algebra structure

phb gqph1 b g1q � hh1 b gg1,

for all h, h1, g, g1 P H.

These maps satisfy the compatibility conditions:

H bH H H bH

H bH bH bH H bH bH bH?

∆b∆

-m -∆

-idbτbid

6

mbm ,

H k

H bH

-ε

6m

�����

εbε,

k H

H bH

@@@@R

ηbη

-η

?

∆ ,

where τ is the map H bH Ñ H bH such that ab b ÞÑ bb a for all a, b P H. �

De�nition 1.8.6. A morphism of bialgebras f : H Ñ H 1 is a linear map which is bothan algebra and a coalgebra morphism. �

De�nition 1.8.7. A Hopf algebra pH,m, η,∆, ε, Sq is a vector space H over a �eld ksuch that:

1. pH,m, η,∆, εq is a bialgebra.

2. The map S : H Ñ H (the antipode) satis�es the compatibility condition.

H k H

H bH H bH?

∆

-ε -η

-idbS, Sbid

6m

�

The antipode of a Hopf algebra is unique and it satis�es

Sphgq � SpgqSphq, Sp1Hq � 1H i.e., is an antialgebra map.

If S2 � id the Hopf algebra is said to be involutive .

30

1.9. SUPERGEOMETRY

De�nition 1.8.8. A Hopf algebra morphism f : H Ñ K is a morphism of bialgebras,which preserve the antipode, i.e., f � SH � SK � f . �

De�nition 1.8.9. A Hopf algebra is commutative if it is commutative as an algebra,that is m � τ � m. It is cocommutative if it is cocommutative as a coalgebra, i.e., if

τ �∆ � ∆.

�

De�nition 1.8.10. A bialgebra or Hopf algebra H acts on an algebra A (one says thatA is a H-module algebra) if

1. H acts on A as a vector space (α : H b AÑ A).

2. The product map m commutes with the action of H.

3. The unit map η commutes with the action of H.

This is depicted in the following commutative diagrams:

H b Ab A H b A A Ab A

H bH b Ab A H b AbH b A?

∆bidbid

-idbm -idbα � m

-idbτbid

6αbα

H k A

H b A

@@@@R

ι

-ε -η

�����α

where ιphq � hb 1A the inclusion. �

1.9 Supergeometry

In this section we want to recall some basic de�nitions and facts in supergeometry. Formore details see Refs. [24, 26, 27, 40, 46].

1.9.1 Supervectorial spaces and superalgebras

De�nition 1.9.1. A super vector space is a Z2-graded vector space V � V0 ` V1,where elements of V0 are called �even� and elements of V1 are called �odd �. �

31

CONTENTS

De�nition 1.9.2. The parity of v P V, denoted by ppvq or |v|, is de�ned only onnon-zero homogeneous elements, that is, elements of V0 or V1:

ppvq � |v| �

#0 if v P V0

1 if v P V1.

�

Since any element may be expressed as the sum of homogeneous elements, it su�ces toconsider only homogeneous elements in the statement of next de�nitions, theorems andproofs.

De�nition 1.9.3. The super dimension of a super vector space V is the pair pp, qqwhere dimpV0q � p and dimpV1q � q as ordinary vector spaces. We simply write dimpVq �p|q. One can �nd a basis te1, . . . , epu of V0 and a basis tε1, . . . , εqu of V1 so that Visomorphic to the free k-module generated by the te1, . . . , ep, ε1, . . . , εqu. �

De�nition 1.9.4. A morphism between super vector spaces V and W is a linear mapfrom V to W preserving the Z2-grading. �

Example 1.9.5. Let sHompV,Wq be denoted the super vector space of morphismsV ÑW, with the s given by

rsHompV,Wqs0 � tα : V ÑW |α preserves parityu,

rsHompV,Wqs1 � tβ : V ÑW | β reverse parityu.

We will called α P rsHompV,Wqs0 has degree 0 as morphism and β P rsHompV,Wqs1has degree 1 as morphism. �

De�nition 1.9.6. A super algebra over the �eld k is a super vector space A over ktogether with a multiplication morphism m : AbAÑ A such that the image of AibAjlies in Ai�j, where the subscripts i, j are read module 2.We say that a super algebra A is super commutative if the product of homogeneouselements obeys the rule,

mpab bq � ab � p�1q|a||b| ba, for all homogeneous elements a, b P A.

Similarly we say that A is associative if the product m, satisfy

m � pmb idq � m � pidbmq

on A bA bA. In other words pabqc � apbcq. We also say that A has a unit if there isan even element 1 so that

mp1b aq � mpab 1q � a,

for all a P A, that is a1 � 1a � a. �

32

1.9. SUPERGEOMETRY

De�nition 1.9.7. (Alternative.)Let the �eld k � R,C. A super algebra A is a Z2-graded algebra, A � A0 ` A1.Where A0 is an algebra, while A1 is an A0-module. The parity on non-zero homogeneouselements is given by (1.9.2). �

The category of commutative superalgebras will be denoted by Salg.Remark 1.9.8. The tensor product A b B of two superalgebras A and B is again asuperalgebra, with multiplication de�ned as

pab bqpcb dq � p�1q|b||c|pacb bdq.

�

Example 1.9.9. Let V be a super vector space. The tensor superalgebra is the supervector space TpVq �

Àn¥0 V

bn with the Z2-grading,

rTpVqs0 �nà

m�0

n!

m!pn�mq!V0

bpn�mq ` V bm1 , for m even,

rTpVqs1 �nà

m�1

n!

m!pn�mq!V0

bpn�mq ` V bm1 , for m odd,

together with the product de�ned, as usual, via the ordinary bilinear map φr,s : Vbr �Vbs Ñ Vbpr�sq,

φr,spvi1 b � � � b vir , wj1 b � � � b wjsq � vi1 b � � � b vir b wj1 b � � � b wjs .

Then TpVq is an associative superalgebra with unit, which is noncommutative exceptwhen V is even or one-dimensional. �

Example 1.9.10. Polynomial superalgebra.We de�ne the polynomial superalgebra kn|m over the �el k as the super commutativealgebra, given by

kn|m � krx1, ..., xn, θ1, ..., θms � krx1, ..., xns b

©pθ1, ..., θmq,

where krx1, ..., xns is the ordinary polynomial algebra over the �eld k in the even inde-terminates x1, ..., xn and

�pθ1, ..., θmq is the exterior algebra or Grassmannian algebra,

generated by the odd indeterminates θ1, ..., θm (Grassmannian coordinates, θiθj � �θjθi).The Z2-grading is

rkn|ms0 �

$&%f0 �

¸|I| even

fI b θI | I � t1 ¤ i1   ...   ir ¤ mu

,.- ,

rkn|ms1 �

$&%

¸|J | odd

fJ b θJ | J � t1 ¤ j1   ...   jr ¤ mu

,.- ,

33

CONTENTS

where θI � θi1 � � � θir and f0, fI P krx1, ..., xns.Even indeterminates will always be denoted by latin letters while odd indeterminates willalways be denoted by greek letters. �

De�nition 1.9.11. We call Ar to be the reduced algebra associated with A, de�ne byAr � A{Iodd, with Iodd the (two-sided) ideal generated by the odd nilpotents. �

De�nition 1.9.12. An associative superalgebra A is graded if for all integers i ¥ 0, wehave a subspace Ai � A such that:

1. 1 P Ai,

2. AiAj � Ai�j,

3. A �
i¥0Ai.

The elements in Ai are said to be Z-homogeneous of degree i or just homogeneous

when there is no ambiguity. We also say the grading is compatible with the superalgebrastructure (A � A0 `A1) if π

ipA0q � A0 XAi where πi : AÑ Ai the natural projections.Then we have that A0, the even part of A, is a graded algebra,

A0 �ài¥0

Ai0, Ai0 � A0 X Ai,

and A1 is a graded A0-module. �

De�nition 1.9.13. A super Lie algebra L is an object in the category Svec, togetherwith a morphism r , s : L b L Ñ L, often called the super bracket , or simply, thebracket , which satis�es the following two conditions:

1. Anti-symmetry: rx, ys � p�1q|x||y|ry, xs � 0 for x, y P L homogeneous.

2. The Jacobi identity: rx, ry, zss�p�1q|x||y|�|x||z|ry, rz, xss�p�1q|y||z|�|x||z|rz, rx, yss � 0.

�

Example 1.9.14. We de�ne the associative super algebra sEndpVq as the supervector space sHompV,Vq:

sEndpVq � rsHompV,Vqs0 ` rsHompV,Vqs1.

Taking the composition map as product. sEndpVq is a Lie super algebra with bracket

rτ, ρs � τρ� p�1q|τ ||ρ|ρτ

where the bracket as usual is de�ned only on homogeneous elements and then extendedby linearity. �

34

1.9. SUPERGEOMETRY

1.9.2 Modules for superalgebras

For this section, let A be a superalgebra, not necessarily commutative.

De�nition 1.9.15. A left A-module is a super vector space M with a morphism AbM ÝÑ M obeying the usual identities, see (1.1.31), replacing the ring R for the superalgebra A. A right A-module is de�ned similarly. Note that if A is commutative, aleft A-module is also a right A-module if we de�ne (the sign rule)

ma � p�1q|m||a| am

for m PM and a P A. �

Let us now turn our attention to free A-modules. We already have the notion of thesuper vector space kp|q over k, and so we de�ne Ap|q :� Ab kp|q where

rAp|qs0 � A0 b rkp|qs0 ` A1 b rkp|qs1,

rAp|qs1 � A1 b rkp|qs0 ` A0 b rkp|qs1.

De�nition 1.9.16. We say that an A-module M is free if it is isomorphic to Ap|q forsome pp, qq (in the category of A-modules). �

This de�nition is equivalent to saying that there are p even elements te1, ..., epu and qodd elements tε1, ..., εqu such that:

M0 � spanA0te1, . . . , epu ` spanA1

tε1, . . . , εqu,

M1 � spanA1te1, . . . , epu ` spanA0

tε1, . . . , εqu.

We shall also say that M is the free module generated over A by the even e1, . . . , ep andthe odd elements ε1, . . . , εq.

De�nition 1.9.17. Let α : Ap|q ÝÑ Ar|s be a morphism of free A-modules and writetep�1, ..., ep�qu for the odd generators tε1, ..., εqu. Then, on the basis elements te1, ..., ep�quwe have

αpeiq �r�s

k�1

fktki .

Hence τ can be represented as a matrix of size pr � sq � pp� qq

α �

�α1 α2

α3 α4

(1.1)

where α1 is an r � p matrix consisting of even elements of A, α2 is an r � q matrix ofodd elements, α3 is an s � p matrix of odd elements, and α4 is an s � q matrix of evenelements.We said that α1 and α4 are even blocks and that α2 and α3 are odd blocks . �

35

CONTENTS

Note that the fact that α is a morphism of super A-modules means that it must preserveparity, and therefore the parity of the entries of the matrix is determined.

De�nition 1.9.18. Let M � Ap|q, the free A-module generated by p even and q oddvariables, let

GLpAp|qq � tα : Ap|q Ñ Ap|q | τ invertibleu,

denotes the super general linear group of automorphisms of M, we may also use thenotation GLpAp|qq � GLpp|qq. �

Next we de�ne the generalization of the determinant, called the Berezinian over elementsin GLpAp|qq.

De�nition 1.9.19. Let α P GLpp|qq then we have the standard block form (1.1). TheBerezinian of α is given by

Berpαq � detpα1 � α2α�14 α3q detpα4q

�1

where “ det ” is the usual determinant. �

De�nition 1.9.20. Let M � Ap|q, the free A-module generated by p even and q oddvariables, the super special linear group of automorphisms of M, are the elements αof GLpp|qq such that Berpαq � 1, and it is denote by SLpp|qq. �

1.9.3 Hopf superalgebras

De�nition 1.9.21. We say that the super algebra A (not necessarily commutative) is aHopf super algebra if A has the following properties:

1. A is a super algebra, with the morphisms of super vectorial spaces: multiplicationm : A b A Ñ A and unit η : k Ñ A, that satisfy the commutative diagrams of(1.8.1).

2. A is a super coalgebra, that is, the morphisms of super vector spaces: comultiplica-tion ∆ : AÑ AbA and counit ε : AÑ k that, satisfy the commutative diagramsof (1.8.3).

3. The multiplication m and the unit η are super coalgebra morphisms (morphisms ofsuper vector space which are coalgebra maps too) or equivalently the comultiplica-tion ∆ and the counit ε are super algebra morphisms (morphisms of super vectorspace which are algebra maps too).

4. A is equipped with an superalgebra and supercoalgebra morphism S : A Ñ Acalled the antipode such that satis�es the commutative diagram of (1.8.7).

�

A super algebra A satisfying the �rst four properties is called a super bialgebra .

36

1.9. SUPERGEOMETRY

De�nition 1.9.22. A Hopf super algebra morphism is a linear map φ : A Ñ Bwhich is a morphism of both the super algebra and super coalgebra structure of A andB, and in addition,

SB � φ � φ � SA,

where SA and SB denote respectively the antipodes in A and B. �

De�nition 1.9.23. We say that I is a Hopf ideal of a Hopf superalgebra A if I is atwo-sided ideal and

∆pIq � IbA�Ab I, εpIq � 0, SpIq � I.

�

1.9.4 Superspaces

From now on all superalgebras are assumed to be commutative unless otherwise speci�ed.

De�nition 1.9.24. A ring pR,�, �q is a super ring R if R has two subgroups R0 andR1 such that R � R0 ` R1 and Ri � Rj � Ri�j. We call a commutative super ring ifa � b � p�1q|a||b| ba for all homogeneous elements a, b P R. �

Note that a R-superalgebra A is a super ring with the morphism multiplication AbAÑA restrict to the product of the ring R for R � k a �eld.

De�nition 1.9.25. A super ringed space denoted by S is a pair pS,OSq where S is atopological space endowed with a structural sheaf OS of commutative super rings. �

Notice that S0 � pS, rOSs0q is an ordinary ringed space as in (1.5.5), where rOSs0 isthe structural sheaf of ordinary rings on S. Notice also that rOSs1 de�nes a sheaf ofrOSs0-modules on S.

De�nition 1.9.26. A superspace denoted by S is a super ringed space S with theproperty that the stalk OS,x is a local super ring for all homogeneous element x P S. �

Example 1.9.27. Let M be a di�erentiable manifold over the �eld k, M its underlyingtopological space over k and OM , the sheaf of OM -functions on M .

We de�ne the sheaf of super commutative k-algebras as (for V �M open)

V ÞÝÑ OMpV q :� OMpV qrθ1, . . . , θqs,

where OMpV qrθ1, . . . , θqs � OMpV q b

�pθ1, . . . , θqq , OMpV q is the ring of di�erentiable

functions on an open set V of M and�pθ1, . . . , θqq is the Grassmann algebra or exterior

algebra for the odd (anticommuting) indeterminates θ1, ..., θq. pM,OMq is a super ringedspace and a superspace too.

In the special case M � Rp, we obtain the superspace Rp|q � pRp,ORprθ1, . . . , θqsq. �

37

CONTENTS

From now on, with an abuse of notation, Rp|q will denote both, the super vector spaceRp `Rq and the superspace pRp,ORprθ1, . . . , θqsq. The meaning should be clear from thecontext.

Example 1.9.28. Let Mp|q � Rp2�q2|2pq. This is the superspace corresponding to the

super vector space of p|q� p|q matrices, the underlying topological space being Rp2�q2�

Mp �Mq. In other words, as super vector space

Mp|q �

"�A BC D

*, rMp|qs0 �

"�A 00 D

*, rMp|qs1 �

"�0 BC 0

*,

where A,B,C,D are respectively p�p, p�q, q�p, q�q matrices with entries in Rp2�q2|2pq.Hence as a superspace Mp|q has p

2� q2 even coordinates tij, 1 ¤ i, j ¤ p or p� 1 ¤ i, j ¤p� q and 2pq odd ones θkl, 1 ¤ k ¤ p, p� 1 ¤ l ¤ p� q or p� 1 ¤ k ¤ p� q, 1 ¤ l ¤ p.The structure sheaf on Mp|q is the assignment

V ÞÝÑ ORp2�q2 pV qrθkls, for all V open subset in Mp �Mq.

Now, let us consider in the topological space Rp2�q2, the open sets U consisting of the

points for which detptijq1¤i,j¤p � 0 and detptijqp�1¤i,j¤p�q � 0. We de�ne the superspaceGLp|q :� pU,

�ORp2�q2 b

�rθkls

�|Uq, the open subspace of Mp|q associate to the open

subset U . We will call it the general linear superspace . And the elements Mp|q inGLp|q such that BerpMp|qq � 1 (see de�nition 1.9.19). �

De�nition 1.9.29. A morphism of superspaces S and T is a pair pϕ, ϕq, where ϕ :S ÝÑ T is a map of underlying topological spaces and ϕ : OT ÝÑ ϕ�OS is a localhomomorphism (see 1.4.9 and 1.5.8). �

De�nition 1.9.30. A super manifold of dimension p|q is a superspace M � pM,OMqwhich is locally isomorphic to the super space Rp|q, i.e., if for all x PM there exists opensets Vx �M and U � Rp such that

OM |Vx � ORp |U .

�

De�nition 1.9.31. A super space S � pS,OSq is a super scheme if pS, rOSs0q is anordinary scheme and rOSs1 is a quasi-coherent sheaf of rOSs0-modules. �

De�nition 1.9.32. SpecA.Let A in Salg. Since A0 is an algebra, we can consider the topological space

SpecpA0q � tprime ideals p � A0.u

Also A is an A0-module and it has indeed the structural sheaf OSpecA0 on SpecA0, withstalk OSpecA0,p at any prime p � A0, to be the localization of A at p. SpecA :�pSpecA0,OSpecA0 q is a super scheme. For more details see Ref. [53] II �5. �

38

1.9. SUPERGEOMETRY

It is not hard to see that Spec is also a functor from Salg to the category of super schemesSsch. The proof is very similar to the ordinary setting (see [59] Ch. II) and can be foundin [27] Ch. 10.

De�nition 1.9.33. An a�ne superscheme denoted S is a superscheme that is iso-morphic to SpecA for some superalgebra A. An a�ne algebraic supervariety is asuperscheme isomorphic to SpecA for some a�ne superalgebra A, i.e., �nitely gen-erated superalgebra such that Ar � A{Iodd has no nilpotents, we will call this a�nesuperalgebra A � OpS q the coordinate superring of the supervariety S. �

We have the following:

Proposition 1.9.34. A superspace S is a superscheme if and only if it is locally isomor-phic to SpecA for some superalgebra A, i.e., if for every x P S, there exists Ux � S opensuch that pUx,OS|Uxq � SpecA for some superalgebra A (that clearly depends on Ux).Proof . See Ref. [27] �5. �

Example 1.9.35. For a superalgebra A, we de�ne An|m :� SpecAn|m, for the A-module

An|m � A b kn|m (see 1.9.10). An|m is an a�ne super scheme of dimension n|m. Itsunderlying scheme is the a�ne space An of dimension n with its structural sheaf. �

Next we want to introduce the concept of the representable functor of points of a super-scheme, following Sec. 1.6 but now in the super setting.

De�nition 1.9.36. The functor of points of a super scheme X in the category Ssch isthe representable functor

HX : SschopÑ Set,

Y ÞÑ HXpYq � HompY,Xq

and HXpfopqg � g � f for any reverse morphism f op : Z Ñ Y and for the sheaves

morphism HXpfopqg � f � g. The elements in HXpYq are called the Y-points of X. �

The facts detailed in the following observation are proven in Ref.[27]. They will beimportant in the sequel.

Observation 1.9.37.

1. The functor of points of a super scheme is determined by its restriction to the categoryof a�ne super schemes.

2. The category of a�ne superschemes is equivalent to the category of a�ne superalgebrasdenoted by ASalg, hence the functor of point of a superscheme X can be equivalentlyde�ned as a functor

HX :ASalgopÑ Set,

Y ÞÑ HompSpecY,Xq

39

CONTENTS

and Hpf opqpgq � g � f 1op, where f 1op : SpecY Ñ SpecZ is the induced schemes morphismof the reverse morphism f op : ZÑ Y of any morphism f : Y Ñ Z and g P HXpZq.When X is itself an a�ne superscheme, the functor of points can be written as follows:

HX :ASalgopÑ Set,

Y ÞÑ HompSpecY, SpecAq � HompA,Yq � HompOpX q,Yq

for some a�ne superalgebra A, which is the coordinate ring of X (see 1.9.33).

3. The functor of points F of a super scheme X seen as HX : ASalgopÑ Set is a

local functor i.e., it has the sheaf property. In other words, let A P ASalg and lettfi, i P Iu a system of generators, that is, pfiq � p1q � A. Let the induced schemesmorphism φ1i : SpecAfi ÝÑ SpecA of φi : A ÝÑ Afi the natural map of the a�ne superalgebra A to its localization Afi , and let the map ϕij : Afifj Ñ Afi . Then, for a familyαi P HXpAfiq, such that αi�ϕ

1ij � αj �ϕ

1ji

op, there exists α P HXpAq such that α�φ1i � αi.

4. Yoneda's Lemma.Given superschemes S and T, the natural transformations HS ÝÑ HT are in one-to-onecorrespondence with the superscheme morphisms S ÝÑ T. Consequently two super-schemes are isomorphic if and only if their functor of points are isomorphic.

�

1.9.5 Projective supergeometry

We will consider projective superschemes and supervarieties. We want to generalize theconstruction made in subsection (1.5.2) to superalgebras.

Notation. We shall use the lower indices to indicate the Z2-gradation, while the upperindices will indicate the Z-gradation. When we say `graded ' we shall always mean Z-graded, while for the Z2-graded objects we shall use the word `super '. �

De�nition 1.9.38. ProjA.Let A be a graded superalgebra (see 1.9.12). We consider the structural sheaf of gradedsuperalgebras OProjA0 on the topological space ProjA0 (see 1.5.15), with stalk at p PProjA0

OProjA0,p �

"f

g| f, g P A, g homogeneous the same grade P A0z p

*.

One can check that pProjA0,OProjA0q is a superscheme, and we will denote it with ProjA(see [53] Ch. II �5 for more details). �

Let us see two important examples: the projective superspace and the projective super-varieties.

Example 1.9.39. Projective superspace.

40

1.9. SUPERGEOMETRY

Consider the graded polinomial superalgebra over the �eld k, S � krx0, . . . , xm, ξ1, . . . , ξns(see 1.9.10). We want to describe ProjS explicitly as a superscheme.For each i, 0 ¤ i ¤ m, we consider the graded superalgebra

Sris � krx0, . . . , xm, ξ1, . . . , ξnsrx�1i s, degpx�1

i q � �1.

The subalgebra rSriss0 � Sris of degree 0 is

rSriss0 � kru0, . . . , ui, � � � , um, η1, . . . , ηns, us �xsxi, ηr �

ξrxi, (1.2)

(the label ` ˆ ' means that this generator is omitted). Let the set Ui � ProjSris0 �SpecSris00, ProjS0 is covered by a�ne open subsets Ui's.We denote by Ui the superscheme

Ui � pUi � rSpecSris0s0,OUi

q, OUi� OProjS0 |Ui

b©

rξ1, . . . , ξns � OPm |Uib©

rξ1, . . . , ξns.

where Pm is the classical projective space of dimension m and OPm its structural sheaf.where,

OUi|UiXUs � OUs |UiXUs .

We conclude that there exists a unique sheaf on ProjS0 that we denote as OPm|n �OPm b

�rξ1, . . . , ξns, whose restriction to Ui is OUi

and ProjS � pProjS0,OPm|nq.

We will call the superscheme ProjS � Pm|n the super projective space of dimensionm|n.

�

Example 1.9.40. Projective supervarieties.

Let I � S � krx1, . . . , xm, ξ1, . . . , ξns be a homogeneous ideal then S{I is also a gradedsuperalgebra and we can repeat the same construction as above. First of all, we noticethat the reduced algebra pS{Iqr corresponds to an ordinary projective variety.We look at an a�ne cover of ProjrS{Is0. We de�ne the open sets

Vi � Proj

�krx0, . . . xm, ξ1 . . . ξns

Irx�1i s

�0

� Spec

�krx0, . . . xm, ξ1 . . . ξns

Irx�1i s

�0

0

� Spec

�kru0, . . . , ui, . . . um, η1 . . . ηns

Iloc

�0

,

where Iloc are the elements of zero degree in Irxis�1. And de�ned the superscheme

Vi � Spec

�kru0, . . . , ui, . . . um, η1 . . . ηns

Iloc

One can check that the supersheaves OVi are such that OVi |ViXVj � OVj |ViXVj . Hence as be-fore there exists a supersheaf OProjrS{Is0 on ProjrS{Is0 whose restriction to Vi is OVi . The

supervariety corresponding to S{I is equipped with a projective embedding in Pm|n, en-coded by the morphism of the graded superalgebra SÑ S{I, hence pProjrS{Is0,OProjrS{Is0qis called a projective supervariety . �

41

CONTENTS

1.9.6 The functor of points of projective superspace

We now want to understand the functor of points of the projective superspace Pm|n andof its subvarieties. The situation is essentially the same as in the classical case. We willbrie�y sketch it in the super setting. (For more details of the classical case see Ref. [59]pg.111).

In Example 1.9.39 we have given the projective superscheme Pm|n and an open a�necovering. We want to give now explicitly its functor of points, namely

HPm|n : SalgopÑ Set

A ÞÑ HPm|npAq � HompSpecA,Pm|nq.

Using the locality property (see 3 of Facts 1.9.37), one can prove that a morphismψ : SpecA ÝÑ Pm|n is determined by a family of morphisms ψi : SpecAfi ÝÑ Pm|n,where pfi, i P Iq � p1q � A, provided they induce the same map on intersections (seeProposition III-39 in Ref. [59]). If we denote by φ1ij : SpecAfifj ÝÑ SpecAfi , the mapsof superschemes induced by φij : Afi ÝÑ Afifj , then we have that

ψi � φ1ij � ψj � φ

1ji.

We want now to give a description of the morphisms SpecA ÝÑ Pm|n when A is a localsuperalgebra (Proposition III-36 in Ref. [59]).

Proposition 1.9.41. LetA a local graded super algebra. Then the morphisms SpecAÑ

Pm|n are in one to one correspondence with the set of pm�1|nq-tuples pa0, . . . , am, α1, . . . , αnq PAm�1|n with at least one ai invertible, modulo multiplication by an invertible element inA.Proof .The proof is the same as the classical one, we brie�y sketch it. Consider the elementpa0, . . . , am, α1, . . . , αnq P Am�1|n with ai (i �xed) a unit. We want to write the corre-sponding morphism by its restriction to the cover open subfunctorsUi of Pm|n (see 1.9.39).We have the scheme morphism ϕ1i : SpecA Ñ SpecAris00, induced by the superalgebramorphism

ϕi : Aris00 � rAriss0 � kru1, . . . , ui, . . . , um, η1, . . . , ηns ÝÑ Auj ÞÑ aj{aiηk ÞÑ αk{ai

This is well de�ned with respect to the equivalence relation. Moreover it de�nes a sheafmorphism ϕ 1

i : OUiÝÑ ϕ 1

i�OSpecA0 , with Ui � SpecAris00. One can check that ϕ 1i|UiXUj

�

ϕ 1j|UiXUj

hence the pairs ϕ 1i's de�ne a sheaves morphism

ϕ : OPm|n ÝÑ ϕ 1�OSpecA0 , such that ϕ|Ui

� ϕi.Conversely, assume that we have a morphism ψ : SpecA ÝÑ Pm|n. The topological mapsends the maximal ideal m of A0 into some Ui, ψpmq P Ui. We claim that all SpecA0

42

1.9. SUPERGEOMETRY

is mapped inside Ui. In fact ψ�1pUiq is an open set containing m hence necessarily thewhole SpecA0. Hence ψpSpecA0q � Ui. We will denote by ψi the morphism ψ restrictedto the Ui. Then we have the superschemes morphism ψ1i : SpecA Ñ SpecAris00 inducedby the superalgebras morphism

krx0{xi, . . . , xm{xi, ξ1{xi, . . . , ξn{xis Ñ A

xk{xi ÞÑ bk,

ξµ{xi ÞÑ βl.

So this map determines: pb0, . . . , bi � 1, . . . bm, β1, . . . , βnq P Am�1|n. One can check thatthis is well de�ned. In fact if ψpmq is also in Uj we have that it corresponds to a di�erentpm� 1|nq-uple which is a multiple of the previous one. �

The next observation characterizes the functor of points of subvarieties of the projectivesuperspace, that is projective super varieties .

Observation 1.9.42. Let X � Pm|n be a projective supervariety de�ned by the homoge-neous ideal I � pf1, . . . , frq. As we saw in above proposition, for A a local superalgebra,the A-points of Pm|n are the pm � 1|nq-tuples pa0, . . . , am, α1, . . . , αnq P Am�1|n with atleast one ai invertible. We have that such an pm� 1|nq-tuple corresponds to an A-pointof X P HompProjA,Xq if and only if it is a zero of all the polynomials f1, . . . , fr. Formore details on the ordinary setting, see [59] Ch. III. �

1.9.7 The functor of points of the Grassmannian superscheme

In this section we want to construct the functor of points of the Grassmannian super-scheme (for a general and more detailed treatment of functor of points and Grassmanniansee [27] �3).

We want to stress the importance of the functorial treatment since it is more general andit gives geometric intuition to the problem. In fact it allows to recover the description ofGrassmannian superscheme as the set of submodules of rank r|s inside some free pm|nq-module. Let us see more in detail this construction.

Consider the functor HGr : SalgopÝÑ Set, with HGrpAq for the superalgebra A the set

of projective A-modules of rank r|s of Am|n, that is,

HGrpAq � HompSpecAm|n,Grq � tα : Am|n Ñ L, with L a projective A�module of rank r|su(1.3)

where α � α1 if and only if they have the same kernel.To complete the de�nition of functor of points we need to specify HGr on morphismsψ : A ÝÑ B. Let its induced morphism ψ1 : SpecBm|n Ñ SpecAm|n then HGrpψ

opq :

HompBm|n,Grq Ñ HompAm|n,Grq given by g � ψop 1 for g P HompBm|n,Grq.

43

CONTENTS

Let the reverse morphism ψop : B Ñ A, we can give to B the structure of A-module bysetting

a � b � ψpaqb, a P A, a P B,

then ψop is an A-module morphism. Also, given an B-module L, we can construct theA-module LbB.

We want to brie�y motivate why HGr is the functor of points of a superscheme, sendingthe reader to [27] and [58] for more details and the complete proof. We use the followingresult [27] �3:

The next theorem is the super version of (1.6.11).

Theorem 1.9.43. Representability theorem.

A functor F : SalgopÝÑ Set is the functor of points of a superscheme X if and only if it

has the sheaf property and it admits a cover by open a�ne subfunctors.

Proof . See Ref. [27] �

We will start by showing that HGr admits a cover by open a�ne subfunctors. Considerthe multiindex I � pi1, . . . , ir|µ1, . . . , µsq and the map φI : Ar|s ÝÑ Am|n de�ned by

φIpx1, . . . , xr|ξ1, . . . , ξsq � pm|nq � tuple with

x1, . . . , xr occupying the position i1, . . . , ir,

ξ1, . . . , ξs occupying the position µ1, . . . , µs

and the other positions are occupied by zero.

(For example, let m � n � 2 and r � s � 1. Then φ1|2px, ξq � px, 0|0, ξq).

It is possible to de�ne subfunctors VI of HGr that on local superalgebras look like

VIpAq � tα : Am|n ÝÑ L |α � φI is invertibleu.

It turns out that the VI 's are representable and they correspond to the a�ne superspaceAr|s�m|n. Moreover they cover HGr.

As for the sheaf property of HGr, notice that classically we have a functorial equivalencebetween the categories of projective �nitely generated A-modules and coherent sheaveson SpecA0 which are locally of constant rank (see [53] pg. 111). One can check that thisclassical equivalence translates to the super category, hence our functor can be identi�edwith

HGrpAq � t F � OAm|n | F is a subsheaf, of locally constant rank r|su,

where OAm|n � km|n b OSpecA0 .

By its very de�nition this functor is local. Hence we have proven that HGr is the functorof points of a superscheme Grassmannian Gr.

44

1.9. SUPERGEOMETRY

1.9.8 Supergroup functors

De�nition 1.9.44. A supergroup functor is a group valued functor

G : SalgopÑ Set.

�

Remark 1.9.45. Saying that G is a group valued is equivalent to have the followingnatural transformations.

1. Multiplication µ : G � G Ñ G, such that

G � G � G G � G

G � G G

-µ�id

?

id�µ

?

µ

-µ

2. Unit ε : Ek Ñ G, where

Ek : SalgopÑ Set,

A ÞÑ EkpAq :� 1A,

such thatG � Ek G � G Ek � G

G

-id�ε

@@@@@@R ?

µ

���

���

�ε�id

3. Inverse ι : G Ñ G, satisfying

G G � G

Ek G

-id�ι

? ?

µ

-ε

�

If G is the functor of points of a superscheme X, i.e., G � HX we say that X is asupergroup scheme . An a�ne supergroup scheme , that we will denoted by Gis a supergroup scheme which is an a�ne superscheme, i.e., G � Am|n for some localsuperalgebra A. To make the terminology easier we will drop the word �scheme� whenspeaking of supergroups schemes.

45

CONTENTS

Remark 1.9.46. The functor of points of an a�ne supergroup G is a representablefunctor. It is represented by its coordinate ring OpG q (see 1.9.33 and 2 of 1.9.37 ).This superalgebra has a Hopf superalgebra structure, so we identify the category of a�nesupergroups with the category of commutative Hopf superalgebras. �

Example 1.9.47. Supermatrices.Consider the funtor of points of supermatrices Mm|n, discussed in (1.9.28)

HMm|n:ASalgop

Ñ Set

A ÞÑ

�A BC D

,

where A and D are m�m, n� n even block matrices with entries in A0, while B and Care m�n, n�m odd block matrices with entries in A1. HMm|n

is a representable functor,represented by the superalgebra of polynomials krxij, ξkls for suitable indices i, j, k, l. Thefunctor HMm|n

is a group-valued, in fact any HMm|npAq has an additive group structure,

where the addition of matrices. �

The last remark (1.9.46) for the supergroup SLpm|nq implies it is represented by thesuperalgebra

OpSLpm|nqq � krxij, ξkl, T sLtTBer� 1u,

where xij's and ξkl's are respectively even and odd variables with 1 ¤ i, j ¤ m or m�1 ¤i, j ¤ m � n, 1 ¤ k ¤ m m � 1 ¤ l ¤ m � n or m � 1 ¤ k ¤ m � n, 1 ¤ l ¤ m wheredetpxijq1¤i,j¤m � 0 and detpxijqm�1¤i,j¤m�n � 0 then Berezinian Ber de�ned in (1.9.19)is nonzero.

1.9.9 Actions of Supergroups

In this section we brie�y summarize the de�nition and main properties of the stabilizerfunctor for the action of a supergroup on a superscheme. Since a supervariety is inparticular a superscheme, all our statements hold if we replace the word `superscheme'with `supervariety'.

De�nition 1.9.48. We say that a supergroup G acts on a superscheme X, if we have anatural transformation between their functor of points

HGpAq �HXpAq ÝÑ HXpAq

pg, xq ÞÑ g � x

satisfying the usual axioms for an action

1. pg � ph � xqq � gh � x, for all g, h P HGpAq, x P HXpAq;

2. 1G � x � x, for all x P HXpAq.

46

1.9. SUPERGEOMETRY

Once a supergroup action is de�ned, we can talk about the stabilizer of a subfunctorof HX. We are not assuming the subfunctor to be representable or in general to be asub-superscheme of X. In the following we are inspired by [56] Ch. 1, �2.

De�nition 1.9.49. Let Y be a subfunctor of HX. We call StabY the stabilizer of Ythe following supergroup subfunctor of HG:

StabYpAq :� g P HGpAq

�� g � YpA1q � YpA1q, for all A� superalgebrasA1(

Notice that by this de�nition there is no guarantee that StabY is the functor of pointsof a superscheme and this even in the case in which Y is a subscheme of X. In fact, asit happens already in the ordinary setting, we have examples for Y an open subschemeof X, for which StabY is not representable. Consider for example the natural actionof the multiplicative group G � A1|0 z t0u (for the A-module A1|0) on X � A1|0 bymultiplication and let the subfunctor Y � HA1|0 z t0u. This is the functor of points of anopen subscheme of X. Geometrically it is clear that the only points stabilizing the opensubset A1|0 z t0u are the units in GpAq, that is, the elements that have inverse. Howeverif one computes the stabilizer of YpAq one �nds that StabYpAq consists of the elementsf � n, with f a unit and n a nilpotent element of A and one can prove this functor isnot representable.

Despite this complication, we however have some positive answer to the question whetherStabYpAq is the functor of points of a superscheme (see [27] Ch. 6 for more details andthe proof of this statement).

Proposition 1.9.50. Let the notation be as above and assume Y is the functor of pointsof a closed subscheme of X. Then StabY is representable and it is a closed subgroup ofthe supergroup G.

47

CONTENTS

48

Chiral Superspaces

In this chapter we present an overview on the physical question that originates our dis-cussion (Section 2.1). We are after a rigorous mathematical description of the chiralsuper�elds and their quantization, making sure to preserve the natural supergroup ac-tions of the superconformal and super Poincaré groups.In section 2.2 we review brie�y the classical theory of the Grassmannian manifold asembedded in the projective space. This is the standard Plücker embedding . We thengeneralize these structures to the super setting.In Section 2.3 we use the technology developed in the series of papers [7, 8, 9], to quantizethe Grassmannian and �ag supervarieties by replacing the symmetry supergroup by aquantum supergroup. We then discuss quantum supergroups and their homogeneousspaces [37, 38, 39, 23], by looking at the corresponding non commutative superalgebras.One result of this approach is that although the superalgebras become non commutative,the group law, represented by the comultiplication in the quantum supergroup, is notdeformed. This can be interpreted by saying that the physical symmetry principle remainsintact in the process of quantization.In Section 2.4 we give a de�nition of the quantum big cell inside the quantum Grassman-nian, presenting a coaction of the quantum super Poincaré group.

2.1 Real and chiral super�elds in Minkowski superspace

We want to devote this to introduce real and chiral super�elds as they are used in physicsas well as to motivate the importance of having them quantized. For this purpose we willconsider super�elds in super Minkoswski space.

2.1.1 Scalar super�elds

We consider the complexi�ed Minkowski space C4. The scalar super�elds are elementsof the commutative superalgebra

OpC4|4q � C8pC4q b

©rθ1, θ2, θ

91, θ92s,

where C8pC4q is the ring of di�erentiable functions on C4 and

�rθ1, θ2, θ

91, θ92s is the

exterior algebra o Grassmann algebra generated by the odd variables θ1, θ2, θ91, θ

92.

49

CHAPTER 2. CHIRAL SUPERSPACES

Giving this superalgebra is equivalent to giving the superspace C4|4 (see 2 of 1.9.37).We will denote the generators of the superspace as

xµ, µ � 0, 1, 2, 3 p even coordinatesq,

θα, θ 9α, α, 9α � 1, 2 podd coordinatesq,

and a super�eld, in terms of its �eld components, as

Ψpx, θ, θq �ψ0pxq � ψαpxqθα � ψ1

9αpxqθ9α � ψαβpxqθ

αθβ � ψµpxqθασµ

α 9βθ

9β (2.1)

� ψ9α 9βpxqθ

9αθ9β � ψ1αβ 9γpxqθ

αθβ θ 9γ � ψα 9β 9γpxqθαθ

9β θ 9γ � ψαβ 9γ 9δpxqθαθβ θ 9γ θ

9δ.

De�nition 2.1.1. A conjugation on a superalgebra A (not necessarily commutative)is a map µ : AÑ A such that:

µpαf � βgq � µpαqµpfq � µpβqµpgq � α�µpfq � β�µpgq, (antilinear)

µ � µ � Id, (involutive)

µpf � gq � p�1qpfpgµpgq � µpfq, (2.2)

where α, β P C and f, g P A, pf is the parity of the element f . Here we take the conventionof Ref.[26]. �

On C8pC4q, there exists the standard complex conjugation, denoted as

µpfq � f�, f P C8pC4q.

Hence, to give a conjugation on OpC4|4q is enough to give it on the odd generators. Thisis done formally in the following way

pθαq� � θ 9α, pθ 9αq� � θα. (2.3)

This is then extended by (2.2) to the whole superalgebra.

A real super�elds then belong to OpR4|4q can be de�ned apply the conjugation mapde�ned above to the �eld components (2.1). Then a super�eld Ψpx, θ, θq is real if andonly if

ψ�0 � ψ0, ψ�α � ψ19α, ψ�αβ � ψ

9α 9β, ψ�µ � ψµ,

ψ1�αβ 9γ � ψγ 9α 9β, ψ�

αβ 9γ 9δ� ψγδ 9α 9β.

2.1.2 Action of the Lorentz group SO(1,3)

The spin group SpinpC1,3q � Spinp1, 3q for the complex quadratic vector space C1,3 isde�nded as the universal cover of SOpC1,3q � SOp1, 3q. Since the fundamental group ofSOp1, 3q is Z2, it follows that in this case Spinp1, 3q is the double cover of SOp1, 3q. We

50

2.1. REAL AND CHIRAL SUPERFIELDS IN MINKOWSKI SUPERSPACE

will call C � CpC1,3q for the Cli�ord algebra of C1,3 and C� � CpC1,3q its even part.There is an imbedding of Spinp1, 3q inside C� as a complex group which lies as doublecover of SOp1, 3q. The key property of the imbedding is that the restriction map gives abijection between simple C�-modules and two irreducibles C�|Spinp1,3q-modules which aredenoted by S� and they are called spin modules (for all details see [24]).

There is an action of Spinp1, 3q � SLp2,Cq � SLp2,Cq over C4|4 such that:

xµ ÞÑ Λµνx

ν ,

θα ÞÑ pS�qαβ θβ,

θ 9α ÞÑ pS�q 9α9βθ

9β,

where xµ are even coordinates and Λµν are the Lorentz transformations , while θ and θ

Weyl spinors. In fact, for the real form Spinp1, 3q, S� and S� are complex, and theyare related by complex conjugation, so this is consistent with the rule (2.3). In physicsnotation S� � p1{2, 0q and S� � p0, 1{2q.The scalar super�elds are invariant under the action of the Lorentz group,

Ψpx, θ, θq � pRΨqpΛ�1x, pS�q�1θ, pS�q�1θ q,

where RΨ is the super�eld obtained by transforming the the �eld components

Rψ0pxq � ψ0pxq, Rψαpxq � ψβpxqpS�αβq�1, . . . .

The Hermitian matrices

σ0 �

�1 00 1

, σ1 �

�0 11 0

, σ2 �

�0 �ii 0

, σ3 �

�1 00 �1

, (2.4)

de�nes a Spinp1, 3q-morphism

S� b S� ÝÑ C1,3

sα b t 9α ÞÝÑ sασµα 9αt9α.

2.1.3 Derivations

De�nition 2.1.2. We called left derivation of degree m � 0, 1 of a super algebra Ato the linear map DL : AÑ A such that

DLpΨ � Φq � DLpΨq � Φ� p�1qmpΨΨ �DLpΦq.

�

Graded left derivations span supervector spaces.We can see from de�nition (1.9.5), for the case of derivations of a commutative superal-gebra, an even derivation has degree 0 as a linear map and an odd derivation has degree1 as a linear map.

51

CHAPTER 2. CHIRAL SUPERSPACES

De�nition 2.1.3. In the same way, we called right derivation of degree m � 0, 1 of asuper algebra A to the linear map DR : AÑ A such that

DRpΨ � Φq � p�1qmpΦDRpΨq � Φ�Ψ �DRpΦq.

�

Notice that derivations of degree zero are both, right and left derivations. Moreover,given a left derivation DL of degree m one can de�ne a right derivation DR also of degreem in the following way

DRΨ � p�1qmppΨ�1qDLΨ. (2.5)

Let us now focus on the commutative superalgebra OpC4|4q.

De�nition 2.1.4. The standard left derivations over the super�eld Ψpx, θ, θq are de�neby

BLαΨ � ψα � 2ψαβθβ � ψµσ

µ

α 9βθ

9β � 2ψ1αβ 9γθβ θ 9γ � ψα 9β 9γ θ

9β θ 9γ � 2ψαβ 9γ 9δθβ θ 9γ θ

9δ,

BL9αΨ � ψ1

9α � ψµθασµα 9α � 2ψ

9α 9β θ9β � ψ1γβ 9αθ

γθβ � 2ψβ 9α 9γθβ θ 9γ � 2ψγβ 9α 9δθ

γθβ θ9δ.

�

With our convention (2.3), one has that

pBLαΨq� � BL9αΨ�.

Also, using (2.5) one can de�ne BRα , BR9α . They have the same property than the left

derivatives under complex conjugation.We consider now the odd left derivations

QLα � �iBLα � σµα 9αθ

9αBµ, QL9α � iBL

9α � θασµα 9αBµ.

They satisfy the anticommutation rules

tQLα, Q

L9αu � �2iσµα 9αBµ, tQL

α, QLβu � tQL

9α, QL9βu � 0.

QL and QL are called the supersymmetry charges or supercharges . Together with

P µ � �iB

Bxµ,

they form a Lie superalgebra, the super translation algebra , which then acts on thesuperspace C4|4.Let us de�ne another set of (left) derivations,

DLα � BLα � iσµα 9αθ

9αBµ, DL9α � �BL

9α � iθασµα 9αBµ,

52

2.1. REAL AND CHIRAL SUPERFIELDS IN MINKOWSKI SUPERSPACE

with anticommutation rules

tDLα , D

L9αu � 2iσµα 9αBµ, tDL

α , DLβ u � tDL

9α , DL9βu � 0.

They also form a Lie superalgebra, isomorphic to the supertranslation algebra. This canbe seen by taking

QL Ñ �iDL, QL ÝÑ �iDL.

It is easy to see that the supercharges anticommute with the derivations DL and DL. Forthis reason, DL and DL are called supersymmetric covariant derivatives or simplycovariant derivatives , although they are not related to any connection form.

�

De�nition 2.1.5. A chiral super�eld is a super�eld Φ such that

DL9αΦ � 0. (2.6)

�

Because of the anticommuting properties of D1s and Q1s, we have that

DL9αΦ � 0 ñ DL

9αpQLβΦq � 0, DL

9αpQL9βΦq � 0.

This means that the supertranslation algebra acts on the space of chiral super�elds.Due to the derivation property,

DL9αpΦ �Ψq � DL

9αpΦq �Ψ� p�1qpΦΦ � DL9αpΨq,

we have that the product of two chiral super�elds is again a chiral super�eld.

2.1.4 Shifted coordinates

One can solve the constraint (2.6) in the following way. Notice that the quantities

yµ � xµ � iθασµα 9αθ9α, θα, (2.7)

satisfy

DL9αyµ � 0, DL

9αθα � 0,

and using the derivation property, any super�eld of the form

Φpyµ, θq, satis�es DL9αΦ � 0

and so it is a chiral super�eld. This is the general solution of (2.6).We can make the change of coordinates

xµ, θα, θ 9α ÝÑ yµ � xµ � iθασµα 9αθ9α, θα, θ 9α.

53

CHAPTER 2. CHIRAL SUPERSPACES

A super�eld may be expressed in both coordinate systems

Φpx, θ, θq � Φ1py, θ, θq.

The covariant derivatives and supersymmetry charges take the form 1

DLαΦ1 �

BLΦ1

Bθα� 2iσµα 9αθ

9αBLΦ1

ByµDL

9αΦ1 � �BLΦ1

Bθ 9α,

QLαΦ1 � �i

BLΦ1

Bθα� 2θ 9ασµα 9α

BLΦ1

ByµQL

9αΦ1 � iBLΦ1

Bθ 9α.

In the new coordinate system the chirality condition is simply

BL9αΦ1 � 0,

so it is similar to a holomorphicity condition on the θ's.This shows that chiral scalar super�elds are elements of the commutative superalgebra

OpC4|2q � C8pC4q b

©rθ1, θ2s.

We shall realize this superspace as the big cell inside the chiral conformal superspace,which is the Grassmannian of 2|0-subspaces of C4|1.

The complete (non chiral) conformal superspace is in fact the �ag supervariety of 2|0-subspaces inside 2|1-subspaces of C4|1. On this supervariety one can put a reality condi-tion, and the real Minkowski superspace is the big cell inside the super�ag. It is instructiveto compare Eq. (2.7) with the incidence relation for the big cell of the �ag manifold inEq. (12) of Ref. [23]. We can then be convinced that the Grassmannian that we use todescribe chiral super�elds is inside the (complex) super�ag.

2.1.5 Supersymmetric theories

Wess-Zumino models are supersymmetric models for one or several chiral super�elds.These were the �rst type of supersymmetric theories that were written down [42]. Chiralsuper�elds also appear in super Yang-Mills theories [43, 44], where the parameter of thegauge transformation is itself a chiral super�eld.The study of the chiral super Minkowski space is then justi�ed from the physical pointof view. Of course, most of the theories make use of the real super Minkowski space, andone needs also to consider real �elds to formulate supersymmetric theories.Also, it is important to consider the embedding of super Minkowski space inside conformalsuperspace, since some theories (for example, some Wess-Zumino models and N=4 superYang-Mills theory) have this symmetry. In fact, for the classic (non quantum) case, thishas been done in the modern language of supergeometry in [23].

1Derivation changes B

Bxµ � B

Byµ ,B

Bθα � iσµα 9αθ9α B

Byµ � B

Bθα ,B

Bθ 9α � iθασµα 9αB

Byµ � B

Bθ 9α .

54

2.2. THE SUPER GRASSMANNIAN VARIETY

In this work though, we want to consider a quantization of these superspaces that pre-serves the action of the corresponding supergroups. It has been particularly di�cultto �nd deformations of the space of chiral super�elds involving also the odd variables[31, 33]. Up to now, this has prevented to formulate Wess-Zumino or Yang-Mills modelsin a non commutative superspace with a non trivial deformation of the odd part and pre-serving the supersymmetry. Essentially, what happened in previous formulations is thatthe covariant derivatives were not anymore derivations of the noncommutative product,and then the ring of chiral super�elds did not extend to a quantum chiral ring. Someproposals to keep a chiral ring (but not an antichiral one) include the partial (explicit)breaking of supersymmetry [32, 31].

In our formulation, we start with the classical chiral ring and �nd a quantum chiralring in a natural way. We substitute the supergroup by a quantum supergroup andpreserving the relations among all the elements of the construction. As it is well known,the comultiplication is not deformed when going from the classic to the quantum group,which means that the supersymmetry algebra is preserved without deformation althoughnow it is realized on a non commutative superspace. Mathematically, this is already a nontrivial problem, and physically it is a problem that must be solved in order to formulatethe theories that use chiral super�elds in non commutative spaces.

Our approach will be complete once we extend it to the real super Minkowski space. Inorder to do this, one has to deal with the �ag supermanifold. The Grassmannian thensits inside the complexi�ed �ag supermanifold.

One can certainly extend the same philosophy of quantization to the �ag supervariety.

Nevertheless, the problem is non trivial, presents its own complications and will be thesubject of a forthcoming paper.

Finally, since the superconformal symmetry is implicit in our approach, we expect toobtain in the future a basis to formulate conformal theories in a non commutative space.This will include for example N � 4 super Yang-Mills.

2.2 The super Grassmannian variety

We will start by reviewing the Plücker embedding of the Grassmannian variety. Wethen turn to a description of the same object as a quotient space for the action of thespecial linear supergroup and we give an explicit description of the big cell inside theGrassmannian supervariety. The big cell is especially important since it is identi�ed withthe complex Minkowski superspace, while the subgroup of SLp4|1q stabilizing the big cellcontains the Poincaré times the dilations supergroup .

Our point of view is purely algebraic, since it is the most suitable for the quantization.The reader however must be aware that these objects have a natural di�erential structure,as always is the case for supergroups and the supervarieties on which they are acting [47].

55

CHAPTER 2. CHIRAL SUPERSPACES

2.2.1 Plücker embedding of the Grassmannian variety and the big cell

The Grassmannian variety Gp2, 4q, the set of 2-planes inside a four dimensional space C4

and its embeddeding in the projective space Pp�2 C4q � P5. A plane π can be given by

two linearly independent vectors a and b,

π � pa, bq � spanta, bu, a, b P C4. (2.8)

If spanta, bu � spanta1, b1u they de�ne the same point of the Grassmannian. This meansthat we can take linear combinations of the vectors a and b

pa1, b1q � pa, bqh, h P GLp2,Cq, (2.9)

in matrix notation ����a11 b11a12 b12a13 b13a14 b14

��� �

����a1 b1

a2 b2

a3 b3

a4 b4

��� �h11 h12

h21 h22

,

to represent the same plane π.

What relates the Grassmannian to the conformal group is that there is a transitive action(see Def. 1.1.7) of GLp4,Cq on Gp2, 4q,

g P GLp4,Cq, gπ � pga, gbq.

One can take SLp4,Cq instead and the action is still transitive. Then, the GrassmannianGp2, 4q is a homogeneous space of SLp4,Cq.Let te1, e2, e3, e4u be the canonical basis of C4 and π0 � spante1, e2u the 2-plane generatedby e1 and e2. Then

Gp2, 4q � SLp4,Cq{P0,

where the isotropy group (see Def. 1.1.9) P0 is the subgroup that leaves invariant π0, thisis

P0 �

"�L M0 R

P SLp4,Cq

*,

with L,M,R being 2� 2 matrices, and L and R invertible.

The conformal group in dimension four and Minkowskian signature is the orthogonalgroup SOp2, 4q, its spin group is SUp2, 2q. Considering the complexi�cation, SOp6,Cq,SLp4,Cq respectively. We have then that the spin group of the complexi�ed conformalgroup acts transitively on the Grassmannian Gp2, 4q.

To obtain the Plücker map, one starts by considering the vector space E ��2pC4q � C6

with basis tei ^ ejui j, i, j � 1, . . . , 4. Then, for any plane π � spanta, bu we have

a^ b �¸i j

yijei ^ ej.

56

2.2. THE SUPER GRASSMANNIAN VARIETY

A change of basis pa1, b1q � pa, bqu, where u P GLp2,Cq, produces a change

a1 ^ b1 � detpuqa^ b,

so the image π ÞÑ ra^bs is well de�ned into the projective space PpEq ≈ P5. The Plückermap is an embedding.

We can observe that the image of the Plücker map over Gp2, 4q is the set of all totallydecomposable vector p in

�2pC4q according to the following de�nition.

De�nition 2.2.1. Let V a vector space and p P�d V is totally decomposable if there

exist linearly independent vectors v1, ..., vd P V so that p � v1 ^ � � � ^ vd. �

Lemma 2.2.2. For p P�2 V , where V is a vector space over C is totally decomposable

if and only if p^ p � 0 . �

In the case when V � C4 we get exactly one quadratic Plücker relation.

Lemma 2.2.3. Let V be a 4-dimensional vector space over C. Then a vector p P�2 V

is totally decomposable if and only if the corresponding Plücker coordinates satisfy therelation

y12y34 � y13y24 � y14y23 � 0, p with yij � �yjiq. (2.10)

Proof. See [54] �

The last equation is called the quadratic Plücker relation or Klein quadric.In algebraic terms, the fact that the Grassmannian is embedded in the projective space isre�ected in the fact that the ideal IP generated by the relation (2.10) in the polynomialalgebra PolpC6q is homogeneous.

2.2.2 The Poincaré group plus dilatations and the big cell

We give an open cover for the projective variety Gp2, 4q. The plane

π � spanta, bu �

����a1 b1

a2 b2

a3 b3

a4 b4

���

has rank 2, since the two vectors are independent, so at least one of the 6 minors yij �aibj � biaj, i   j is di�erent from zero. The sets

Uij � pa, bq P C4 � C4 | yij � 0

(, (2.11)

are open a�ne sets of Gp2, 4q and cover it. U12 is called the big cell .

By a change of basis a plane π in the big cell can always be represented by the matrix

π �

�11A

, A �

�a11 a12

a21 a22

, (2.12)

57

CHAPTER 2. CHIRAL SUPERSPACES

so U12 �M2pCq � C4, and U12 is again the a�ne space C4.

Given an element of SLp4,Cq, since the columns are linearly independent, we can choosethe �rst two columns to be the vectors a and b representing a 2-plane π. If the plane isin the big cell, then there is a transformation in the isotropy group P0 such that bringsthe matrix of SLp4,Cq to the form �

112 0A 112

. (2.13)

The big cell is left invariant by the subgroup of SLp4,Cq consisting of the matrices of theform

Pl �

"�x 0Tx y

| x, y invertible, detx � det y � 1

*. (2.14)

The bottom left entry is arbitrary but we have written it like that for convenience. Theaction on U12 is then

A ÞÑ T � yAx�1, (2.15)

so Pl has the structure of semidirect product Pl � H M22, where M2 � tT u is the set

of 2� 2 matrices acting as translations, and

H �

"�x 00 y

| x, y P GLp2,Cq, detx � dety � 1

*.

The subgroup H is the direct product SLp2,Cq�SLp2,Cq�C�. But SLp2,Cq�SLp2,Cqis the spin group of SOp4,Cq, the complexi�ed Lorentz group and C� acts as a dilation.Then Pl is the Poincaré group times dilations.

In the basis of the Pauli matrices (2.4) an arbitrary matrix A can be written like

A �

�a11 a12

a21 a22

� x0σ0 � x1σ1 � x2σ2 � x3σ3 �

�x0 � x3 x1 � ix2

x1 � ix2 x0 � x3

,

and px0, x1, x2, x3q are the ordinary coordinates of Minkowski space. Moreover,

detA � px0q2 � px1q2 � px2q2 � px3q2.

For all this we can interpret the big cell as the complexi�cation of the Minkowski space.

2.2.3 The Plücker embedding for the super Grassmannian

We are interested in the super Grassmannian of p2|0q-planes inside the superspace C4|1.Since we are only concerned with this particular Grassmannian, we will just denote it asGr � Gp2|0; 4|1q. In the chapter one (1.9.7) we give a description of its functor of points,HGr. Here, we will use the fact that we can work on local algebras.

2H is a normal subgroup of Pl, i.e., for each h P H and each g P Pl, the element ghg�1 is still in H.

58

2.2. THE SUPER GRASSMANNIAN VARIETY

Then the projective modules are free modules and the description is greatly simpli�ed.On a local algebra A,

HGrpAq � tL to be a free module of A4|1 of rank 2|0u.

One such module can be speci�ed by a couple of independent even vectors, which in thecanonical basis te1, e2, e3, e4, E5u are

π � spanta, bu � span

$''''&''''%

������a1 b1

a2 b2

a3 b3

a4 b4

α5 β5

�����

,////.////-, (2.16)

with ai, bi P A0 for 1 ¤ i ¤ 4 and α5, β5 P A1.

We want now to work out the expression for the Plücker embedding.

We want to give a natural transformation among the functors

Φ : HGr Ñ HPpEq,

where E is the super vector space E ��2C4|1 � C7|4 and HPpEqpAq are the A-modules

in P6|4 of rank 1|0.We recall that for an arbitrary super vector space V ,©2

V ��V b V

�{xub v � p�1qpupvv b uy, u, v P V.

Given, the canonical basis for C4|1 we construct a basis for E

e1 ^ e2, e1 ^ e3, e1 ^ e4, e2 ^ e3, e2 ^ e4, e3 ^ e4, E5 ^ E5, (even)

e1 ^ E5, e2 ^ E5, e3 ^ E5, e4 ^ E5, (odd). (2.17)

If L is a A-module of rank 2|0, then�2L �

�2A4|1 and�2L is a projective A-module of

rank 1|0. In other words, it is an element of HPpEqpAq. Hence we have de�ned a naturaltransformation

Φ : HGrpAq ÝÑ HPpEqpAq

L ÞÑ©2

L,

with A P Salg.

Let a, b be two even independent vectors in A4|1, for any superalgebra A, they generatea free submodule of A4|1 of rank 2|0. The natural transformation described above is asfollows

ΦA : HGrpAq ÝÑ HPpEqpAq

spanAta, bu ÞÑ spanAta^ bu,

59

CHAPTER 2. CHIRAL SUPERSPACES

The map ΦA is clearly injective.

The image ΦApHGrpAqq is the subset of even elements in HPpEqpAq decomposable in

terms of two even vectors of A4|1. We are going to �nd the necessary and su�cientconditions for an even element Q P HPpEqpAq to be decomposable in terms of evenvectors.In terms of the canonical basis (2.17) we have

Q � q � λ^ E5 � a55E5 ^ E5, with

q � q12e1 ^ e2 � � � � � q34e3 ^ e4, qij P A0,

λ � λ1e1 � � � � � λ4e4, λi P A1. (2.18)

Q is decomposable if and only if

Q � pr � ξE5q ^ ps� θE5q with

r � r1e1 � � � � � r4e4, s � s1e1 � � � � � s4e4, ri, si P A0, ξ, θ P A1,

which meansQ � r ^ s� pθr � ξsq ^ E5 � ξθE5 ^ E5.

This is equivalent toq � r ^ s, λ � θr � ξs, a55 � ξθ,

and these are in turn equivalent to the following

q ^ q � 0, q ^ λ � 0, λ^ λ � 2a55q, λa55 � 0.

Plugging (2.18) we obtain

q12q34 � q13q24 � q14q23 � 0, (classical Plücker relation)

qijλk � qikλj � qjkλi � 0, 1 ¤ i   j   k ¤ 4

λiλj � 2a55qij 1 ¤ i   j ¤ 4

λia55 � 0. (2.19)

These are the super Plücker relations .As we shall see in the next section the superalgebra

OpGr q � krqij, λk, a55s{IP , (2.20)

is associated to the supervariety Gr in the Plücker embedding described above, whereIP denotes the ideal of the super Plücker relations (2.19).

Remark 2.2.4. We want to stress that although OpGr q does not represent the functorof points HGr, one can recover all the information about the supervariety from OpGr qby the procedure described in Chapter 1 (see 1.9.40 in the notation used there one hasthat S � OpGrq). �

60

2.2. THE SUPER GRASSMANNIAN VARIETY

2.2.4 The superstraightening algorithm

We want to take a small digression to explain why the relations (2.19) generate the idealIP of all the relations among the coordinates qij, λk, a55. This could be done with adirect calculation as in [51], however we prefer to justify it by resorting to the theory ofsemistandard tableaux and the superstraightening algorithm. The algorithm has intereston his own, but we will also need these notions later, in the quantum setting.

De�nition 2.2.5. A Young diagram is a �nite collection of boxes, or cells, arrangedin left-justi�ed rows, with the row lengths weakly decreasing (each row has the sameor shorter length than its predecessor). Listing the number of boxes in each row givesa partition of a non-negative integer n, the total number of boxes of the diagram. AYoung tableau is obtained by �lling in the boxes of the Young diagram with symbolstaken from some alphabet, which is usually required to be a totally ordered set. Usuallythe alphabet consists of the �rst natural numbers.Let us assume now that the set of indices is separated into two disjoint subsets: theeven and the odd indices . A tableau is called semistandard or superstandard thefollowing conditions are satis�ed

• The entries are non decreasing along each row.

• The rows have no repeated even entries.

• The entries are non decreasing down each column.

• The columns have no repeated odd indices.

Some authors take the opposite convention, interchanging rows and columns. �

To describe the super Grassmannian we take indices 1, 2, 3, 4, 5 with 1, 2, 3, 4 beingeven and 5 being odd. Each of the generators of the super Grassmannian is associated toa pair of multi-indices, that we shall write using the letterplace notation [48], [49]. The�rst pair of indices indicates the rows and the second pair indicates the columns thatdetermine the submatrix whose minor is associated to the generator. Then we have, alsoin terms of the semistandard tableaux,

qij ÝÑ pi, j|1, 2q ÝÑ i j , 1 2

λk ÝÑ pk, 5|1, 2q ÝÑ k 5 , 1 2

a55 ÝÑ p5, 5|1, 2q ÝÑ 5 5 , 1 2 .

We can suppress the second tableau, which indicates the columns, since it is going to bethe same in what follows.

A monomial in the generators can be encoded in a tableau where each line correspondsto the indices of the coordinates, in the order in which they are written.

61

CHAPTER 2. CHIRAL SUPERSPACES

For example the monomials q12q34λ3a55, q12q23λ55 correspond to the tableaux

1 23 43 55 5

1 22 35 5

The second table is semistandard, while the �rst one is not, because the odd elements 3and 5 in the columns are repeat.

The semistandard tableaux are very important since the monomials associated to themare a basis for the superring OpGr q.

The next theorem provides us with a presentation of such ring and with a basis. Itsproof is based on the straightening algorithm in the super setting, which we are unableto describe here, since it would take us too far from our purpose. We refer the reader tothe beautiful works [48], [49], where the full details are discussed.

Theorem 2.2.6.

1. The Grassmannian super ring is given in terms of generators and relations as

OpGr q � Crqij, λj, a55s {IP , 1 ¤ i   j ¤ 5 and i � j � 5

where IP is the two-sided ideal generated by the super Plücker relations (2.19).

2. The Grassmannian super ring is the free super ring over k generated by the mono-mials in the variables qij, λj, a55 whose indices form a semistandard tableau.

�

Remark 2.2.7. In our special case a semistandard tableaux means that the indices ofthe variables forming the monomial and appearing in the tableau in two consecutive linesare such that pik, jkq   pik�1, jk�1q lexicographically, but the even pairs of type p1, 4q andp2, 3q (and more generally pi, jq, pk, lq, with i   k   l   j) are not allowed to appear:such pairs shall be disposed using the (super) Plücker relations, through the straighteningalgorithm [48]. �

The ring OpGr q is our starting point for the quantization: we will quantize the Grass-mannian together with its embedding into PpEq by obtaining a quantization of the ringOpGr q.

We will obtain the quantized superalgebra as a sub superalgebra of the quantum super-group SLqp4|1q. In this way, the quantized Grassmannian Grq will carry a natural actionof the quantum supergroup SLqp4|1q, just as in the classical case.

62

2.2. THE SUPER GRASSMANNIAN VARIETY

2.2.5 The conformal and Poincaré supergroups and the big cell

We start by describing the natural action of the special linear supergroup SLp4|1q (con-formal supergroup) on the Grassmannian supervariety Gr.The functor of points of the supergroup, in terms of local algebras, is

HSLp4|1qpAq �

$''''&''''%g �

������g11 g12 g13 g14 γ15

g21 g22 g23 g24 γ25

g31 g32 g33 g34 γ35

g41 g42 g43 g44 γ45

γ51 γ52 γ53 γ54 g55

����� , Berpgq � 1

,////.////-, (2.21)

where gij, g55 P A0 and γi5, γ5i P A1. Ber stands for the Berezinian or superdeterminantof the matrix g (1.9.19).We can describe the natural action of the special linear supergroup SLp4|1q on the Grass-mannian supervariety as a natural transformation of the functors,

HSLp4|1qpAq �HGrpAq ÝÑ HGrpAq, A local,

which in this language is simply given by the multiplication of matrices (2.21) and (2.16).

In fact this de�nition, given for local superalgebras, can be extended to all superalgebras(see [27] Ch. 9).

Also, we refer the reader to Preliminaries (1.9.9) for de�nitions concerning actions andstabilizers.As in the classical case, let te1, e2, e3, e4, E5u the canonical base for the superspace C4|1,we may take p2|0q-plane, π0 � spante1, e2u. The stabilizer of this point is the upperparabolic sub-supergroup Pu, whose functor of points is

HPupAq �

$''''&''''%

������c11 c12 c13 c14 ρ15

c21 c22 c23 c24 ρ25

0 0 c33 c34 ρ35

0 0 c43 c44 ρ45

0 0 δ53 δ54 d55

�����

,////.////-. (2.22)

Then, the Grassmannian can be identi�ed with the quotient

HGrpAq � HSLp4|1qpAqLHPupAq.

The description of homogeneous spaces for super Lie groups is done in detail in [23].

These algebras are commutative Hopf superalgebras (see Sec. 1.8). Where the comulti-plication is given as usual by matrix multiplication, (see for example Refs. [38, 39], alsofor the counit and antipode), by organizing the generators in matrix form

∆

�gij γi5γ5j g55

�

�gik γi5γ5k g55

b

�gkj γk5

γ5j g55

.

63

CHAPTER 2. CHIRAL SUPERSPACES

The superalgebra OpGr q is a sub superalgebra (not a Hopf sub superalgebra ) of Op SLp4|1q q.It is in fact the superalgebra generated by the corresponding minors, and the Plücker re-lations are all the relations satis�ed by these minors in Op SLp4|1q q.

As in the non super case, the super Grassmannian admits an open cover in terms ofa�ne superspaces. In terms of the functor of points we say that HGr admits a cover byopen a�ne subfunctors. This is explained in detail in the Chapter 1 (see 1.9.7), and itgeneralizes the open cover of the non super case given in Section 2.2.1.

As we have detailed in the previous section, we shall concentrate our attention just onlocal algebras. We will describe HU12

, the functor of points of the big cell U12.

First of all, we write an element of HSLp4|1qpAq in blocks as��C1 C2 ρ1

C3 C4 ρ2

δ1 δ2 d55

� .

Assuming that detC1 is invertible, we can bring this matrix, with a transformation ofHPupAq, to the form �

�112 0 0A 112 0α 0 1

� . (2.23)

The assumption that detC1 invertible means that the matrix is in the open set U12 �GrXV12, where V12 is the a�ne open set de�ned by taking in PpEq the coordinate q12 � 0.So if the column vectors of �

�C1

C3

δ1

�

with detC1 invertible represent a p2|0q-module in the big cell, the same module can berepresented by a matrix of the form�

�112

Aα

� , A �

�a11 a12

a21 a22

, α � pα1, α2q,

with the entries of A in A0 and the entries of α in A1. The big cell U12 of Gr is then thea�ne superspace (see Remark 1.9.46) associated with the superalgebra

OpU12q � Craij, αjs � OpC4|2q. (2.24)

We are now interested in the super subgroup of HSLp4|1qpAq that preserves the big cellU12.This is the stabilizer functor ST U12

. According to the De�nition 1.9.49 we have

ST U12pAq � tg P HSLp4|1qpAq | g �HU12

pA1q � HU12pA1q for all A-algebras A1 u.

64

2.2. THE SUPER GRASSMANNIAN VARIETY

As we remarked after De�nition 1.9.49, ST U12is not in general representable, and it

does not correspond to the functor of points of a supervariety. Nevertheless, there existsa subsupergroup StU12

of SLp4|1q, whose functor of points is the largest subgroup functorof ST U12

.

In our case, StU12is the lower parabolic sub supergroup Pl, whose functor of points is

given in suitable coordinates by matrices of the type

HPlpAq �

$&%�� x 0 0tx y yηdτ dξ d

� ,.- , (2.25)

where x and y are even, invertible 2� 2 matrices, t is an even, arbitrary 2� 2 matrix, ηa 2� 1 odd matrix and τ, ξ a 1� 2 odd matrix. d is an invertible even element given bythe superdeterminant equal 1 condition.

Let us see this. From the above de�nitions we have

HPlpAq � HStU12

pAq.

Since this embedding is functorial and by the Yoneda's Lemma (1.9.37), we have anembedding Pl � StU12

of the algebraic supergroups. Since any supergroup over C is alsoa supermanifold (see [47]) we have that this is also a supermanifold embedding.

Let us look at the superdimension of the stabilizer StU12and Pl. The superdimension

is well de�ned since these are supermanifolds. To compute them one can look at thetangent spaces. We then have

dimStU12 ¤ dimSLp4|1q � dimU12 � 42 � 1|2 � 4� 4|2 � 13|6,

but

dimPl � 22 � 22 � 22 � 1|2� 2� 2 � 13|6,

hence dimStU12 � dimPl. Now, the equality StU12 � Pl follows from the following theorem.

Theorem 2.2.8. Let M and N be two supermanifolds with the same dimension andsuch that M � N . If we have an embedding M � N then M � N.

Proof . See in [27] ch. 5. �

The action of the stabilizer supergroup Pl on the big cell U12 is as follows,

HPlpAq �HU12

pAq ÝÑ HU12pAq

���� x 0 0tx y yηdτ dξ d

� ,

��112

Aα

� � ÝÑ

��112

A1

α1

� ,

65

CHAPTER 2. CHIRAL SUPERSPACES

where, using a transformation of HPupAq to revert the resulting matrix to the standardform (2.23), we have �

�112

A1

α1

� �

�� 112

ypA� ηαqx�1 � tdpα � τ � ξAqx�1

� . (2.26)

The subgroup with ξ � 0 is the super Poincaré group times dilations (compare withEq. (14) in Ref [23]). In that case

d � detx det y.

2.3 Quantum super Grassmannian

Before giving the de�nition of quantum super Grassmannian, we need some preliminarieson quantum supergroups in general.

2.3.1 Quantum supergroups

In this section we follow Manin [37].

De�nition 2.3.1. The quantum matrix superalgebra Mqpm|nq is de�ned as

Mqpm|nq :� Cqxaijy{IM

where Cqxaijy denotes the free superalgebra over the polinomial ring in the q, q�1 inde-terminates Cq � Crq, q�1s generated by

aij �

$'''&'''%xij 1 ¤ i, j ¤ m, or m� 1 ¤ i, j ¤ m� n.

ξij 1 ¤ i ¤ m, m� 1 ¤ j ¤ m� n, or

m� 1 ¤ i ¤ m� n, 1 ¤ j ¤ m

where x's are even variables and ξ's are odd variables, such that ξ2kl � 0.

IM is an ideal generated by the relations [37],

66

2.3. QUANTUM SUPER GRASSMANNIAN

aijail � p�1qπpaijqπpailqqp�1qppiq�1

ailaij, for j   l (2.27)

aijakj � p�1qπpaijqπpakjqqp�1qppjq�1

akjaij, for i   k (2.28)

aijakl � p�1qπpaijqπpaklqaklaij, for i   k, j ¡ l (2.29)

or i ¡ k, j   l

aijakl � p�1qπpaijqπpaklqaklaij � p�1qπpaijqπpakjqpq�1 � qqakjail, (2.30)

for i   k, j   l

where πpaijq � ppiq � ppjq mod 2 denotes the parity of aij (with ppiq � 0 if 1 ¤ i ¤ mand ppiq � 1 otherwise). �

Remark 2.3.2. In our de�nition we take q to be an indeterminate. However this def-inition makes sense also for any value of q P C� (usually one asks away from roots ofone).

�

Mqpm|nq is a superbialgebra with the usual comultiplication and counit

∆paijq �¸

aik b akj, Epaijq � δij. (2.31)

Notice that the comultiplication, which encodes the matrix product law (see Appendix.A) is not deformed.In fact Mqpm|nq is a bialgebra with the commutative product (q � 1) and the samecomultiplication and counit.

De�nition 2.3.3. We de�ne the quantum minor of an element in Mqpm|nq obtainedby taking rows i1 � � � ir and columns j1 � � � jr as an element Dj1���jr

i1���irP Cqxaijy given by

Dj1���jri1���ir

:�¸

σ:pi1���irqÑpj1���jrq

p�qq�lpσqai1σpi1q � � � airσpirq

with 1 ¤ i1   � � �   ir ¤ m � n and 1 ¤ j1   � � �   jr ¤ m � n. Where σ runs over allthe bijections and lpσq is the length of the permutation σ, r is called the rank of Dj1���jr

i1���ir.

�

We are ready to de�ne the general linear supergroup.

LetD1 � D1���m

1���m, D2 � Dm�1���m�nm�1���m�n,

be the quantum determinants of the diagonal blocks.

67

CHAPTER 2. CHIRAL SUPERSPACES

De�nition 2.3.4. The quantum general linear supergroup GLqpm|nq is de�ned as

GLqpm|nq :� Mqpm|nqLtD1D1

�1 � 1, D�11 D1 � 1, D2D

�12 � 1, D�1

2 D2 � 1u.

�

De�nition 2.3.5. The quantum Berezinian Berq over elements of Mqpm|nq is de�nedby

Berq

�X11 Ξ12

Ξ21 X22

:� detpSqpX22qq detpX11 � Ξ12S

qpX22qΞ21q

where X11 � paijq1¤i,j¤m, X22 � paijqm�1¤i,j¤m�n, Ξ12 � paklq1¤k¤m,m�1¤l¤m�n, Ξ21 �paklqm�1¤k¤m�n,1¤l¤m are matrices of indeterminates, which satis�es the relations p2.27q,

p2.28q, p2.29q, p2.30q. And the quantum antipode is Sqpaijq � p�qqj�iD1���i���m�n

1���j���m�n.

�

For the properties of Berq refer to [39].

De�nition 2.3.6. The quantum special linear supergroup SLqpm|nq is de�ned as

SLqpm|nq :� Mqpm|nqLtBerq � 1u,

�

GLqpm|nq and SLqpm|nq are Hopf superalgebras. The comultiplication and the counitare the same as in Mqpm|nq. One must give also the comultiplication on D�1

1 and D�12 .

This has been done in detail in Refs. [38, 39].

The roles of GLqpm|nq and SLqpm|nq can be interchanged in what follows, as it happensfor the super non quantum setting. We then make the choice to use SLqpm|nq.

2.3.2 Presentation of the quantum super Grassmannian Grq

Let the notation be as above. We are going to de�ne the quantum super GrassmannianGrq in terms of generators and relations. Then we will prove that it is a deformation of thealgebra OpGrq given in the equation (2.20). We are now ready for the central de�nitionthe �rst part of this thesis, namely the quantum deformation of the super Grassmannian.

De�nition 2.3.7. The quantum super Grassmannian of p2|0q-planes in a p4|1q di-mensional superspace is the non commutative superalgebra Grq generated by the followingquantum super minors in SLqp4|1q

D12ij � Dij � ai1aj2 � q�1ai2aj1, 1 ¤ i   j ¤ 4,

D12i5 � Di5 � ai1a52 � q�1ai2a51, 1 ¤ i ¤ 4

D1255 � D55 � a51a52.

�

68

2.3. QUANTUM SUPER GRASSMANNIAN

For clarity we write explicitly all the generators

D12, D13, D14, D23, D24, D34, D55, (even)

D15, D25, D35, D45 (odd). (2.32)

Notice that D55 is an even nilpotent element.The parity is easily given by the rule

|Dij| � p�1q|i|�|j|, |i| � 0 for i � 1, . . . 4, and |i| � 1 for i � 5.

We will see that this subalgebra of SLqp4|1q is generated, as a vector space, by monomialsin the above determinants. This fact is not obvious at all, since the commutation relationsof the minors could introduce minors that are not included in (2.32).For q � 1 we recover the classical Grassmannian algebra OpGrq described in detail insection (2.2.1).

More precisely, we will �nd a presentation of Grq in terms of generators Dij and relations.In order to do so, we �rst work out the commutation relations of the minors. Then,as in the classical setting, there will be additional relations among the generators: thequantum super Plücker relations.

Let us start with the commutation relations. In Ref. [7] such commutation relations aregiven for the even minors Dij, with 1 ¤ i   j ¤ 4. As one can readily check, they holdalso when just one of the indices is 5. The reason is that the commutation of one evenand one odd variable in the matrix bialgebra generated by the aij's is the same as thecommutation of two even variables, and the expression of Dkl in terms of aij is formallythe same.The commutation relations are as follows

1. If i, j, k, l are not all distinct

DijDkl � q�1DklDij, pi, jq   pk, lq,

where ` ' refers to the lexicographic ordering 3.

2. If i, j, k, l are all distinct we have

DijDkl � q�2DklDij, 1 ¤ i   j   k   l ¤ 5, (2.33)

DijDkl � q�2DklDij � pq�1 � qqDikDjl, 1 ¤ i   k   j   l ¤ 5, (2.34)

DijDkl � DklDij, 1 ¤ i   k   l   j ¤ 5. (2.35)

3Given two partially ordered set A and B, the lexicographic order on the Cartesian product A�B is

de�ned as

pa, bq   pa1, b1q if and only if a   a1 or pa � a1 and b   b1q.

69

CHAPTER 2. CHIRAL SUPERSPACES

3. For 1 ¤ i   j ¤ 4

DijD55 � q�2D55Dij,

Di5Dj5 � �q�1Dj5Di5 � pq�1 � qqDijD55 � �qDj5Di5

Di5D55 � D55Di5 � 0.

We are ready to tackle the calculation of the quantum super Plücker relations.Similarly to the classic and the super settings, in the quantum case we have the Plückermap in the projective quantum spaces

Grq ãÑ Pp© 2

qC4|1q

we will take C4|1 over the ring Mqp4|1q with canonical base te1, e2, e3, e4, E5u. We recall

© 2

qC4|1 :� C4|1 b C4|1

Ltei b ej � qej b eiu with 1 ¤ i   j ¤ 5.

p P� 2

q C4|1 can be express,

p �D12e1 ^ e2 �D13e1 ^ e3 �D14e1 ^ e4 �D15e1 ^ E5

�D23e2 ^ e3 �D24e2 ^ e4 �D25e2 ^ E5

�D34e3 ^ e4 �D35e3 ^ E5

�D45e4 ^ E5 �D55E5 ^ E5.

The quantum version of Lemma (2.2.2), the solution of p ^ p � 0 give us �ve extrarelations

D12D34 � q�1D13D24 � q�2D14D23 � 0 (2.36)

DijDk5 � q�1DikDj5 � q�2Di5Djk � 0, 1 ¤ i   j   k ¤ 4, (2.37)

Also we can computed directly the relation

Di5Dj5 � qDijD55, 1 ¤ i   j ¤ 4. (2.38)

The relations (2.36),(2.37) and (2.38) are the quantum super Plücker relations .If one speci�es q � 1, the superalgebra becomes commutative and the quantum superPlücker relations become the standard ones (see equation 2.19).

Now we want to show that the commutation relations together with the quantum superPlücker relations are all the relations among the determinants. We do this in the followingproposition.

Proposition 2.3.8.

70

2.3. QUANTUM SUPER GRASSMANNIAN

1. The quantum Grassmannian superring is given in terms of generators and relationsas

Grq � CqxDijyLIGr, 1 ¤ i   j ¤ 5 and i � j � 5

where IGr is the two-sided ideal generated by commutation relations (see Subsect.2.3.2) and the quantum super Plücker relations (2.36), (2.37), (2.38). MoreoverGrq{pq � 1q � OpGrq (see Section 2.2.3).

2. The quantum Grassmannian ring is the free ring over Cq generated by the mono-mials in the quantum determinants

Di1j1 � � �Dirjr

where pi1, j1q, . . . , pir, jrq form a semistandard tableau (see 2.2.5).

Proof . Though the proof is based on the classical result and it is the same as [52] webrie�y sketch it, since its importance in our construction.The generic monomials in the quantum determinants Dij generate the ring Grq as Cq-module. Using the commutation relations we can certainly write any monomial as alexicographically ordered monomial, then using the quantum Plücker relations and thestraightening algorithm [61], we can rewrite any lexicographically ordered monomial as alinear combination of standard monomials.Notice that there are two obstacles to apply the straightening algorithm to the quantumsetting, but they are both easily overcome. The �rst is the presence of the coe�cients qin the Plücker relations. This is not a problem, since q is invertible. The second is thenon commutativity: when we are commuting two quantum determinants one may arguethat the pairs of the forbidden kind, that is pi, jq and pk, lq with i   k   l   j may arise.However a closer look to the commutation relations shows that this is never the case.

So both (1) and (2) will be done if we can show that the standard monomials in the Dij'sare linearly independent (i.e. there are no other relations among them apart the IGr ones).Assume there is a extra relation R among such monomials. Clearly R � 0 mod pq � 1qsince there are no relations among the standard classical monomials, hence R � pq�1qR1.Such relation evidently holds also in the bigger superalgebra SLqpm|nq, which is knownto be torsion free. Hence the quantum determinants satisfy R1 and repeating this sameargument enough times we obtain a non trivial relation among the classical monomials,hence the relation R we start with cannot exist. �

2.3.3 Grq as a homogeneous quantum space

We want to prove that the quantum super Grassmannian that we have constructed admitsa coaction of a quantum group on it, namely the quantum group SLqp4|1q, as it happensin the classical setting.

Proposition 2.3.9. Grq is a quantum homogeneous superspace for the quantum super-group SLqp4|1q, i.e., there is a coaction on Grq given via the restriction of the comultipli-cation on SLqp4|1q (2.31)

∆|Grq : Grq ÝÑ SLqp4|1q bGrq

71

CHAPTER 2. CHIRAL SUPERSPACES

Proof . We just have to check that the restriction is well de�ned. Then, the coactionproperty

Grq SLqp4|1q bGrq

SLqp4|1q bGrq SLqp4|1q b SLqp4|1q bGrq

-∆|Grq

?

∆|Grq

?

∆|Grqb Id

-Idb∆|Grq

is guaranteed by the coassociativity axiom satis�ed by the comultiplication,

Mqpm|nq Mqpm|nq bMqpm|nq

Mqpm|nq bMqpm|nq Mqpm|nq bMqpm|nq bMqpm|nq

-∆

?

∆

?

∆b Id

-Idb∆

So we need to verify that

∆pDijq, ∆pDi5q, ∆pD55q P SLqp4|1q bGrq.

Let us denote the generic 2� 2 quantum minors as

Dklij � aikajl � q�1ailajk, (2.39)

so in the previous notation Dij � D12ij .

In the purely even setting, we can proof it use the formula (see Ref. [7])

∆pD12ij q �

¸1¤k l¤4

Dklij bD12

kl . (2.40)

Dklij P SLqp4|1q and D

12kl P Grq.

In the super case, we can extend the sum to l � 5.

For the minors Di5, let

∆pD12i5 q � ∆pai1a52 � q�1ai2a51q � ∆pai1q∆pa52q � q�1∆pai2q∆pa51q

�¸k l

aika5l b pak1al2 � q�1ak2al1q � aila5k b pal1ak2 � q�1al2ak1q

�aika5k b ak1ak2 � q�1aika5k b ak2ak1

�¸k l

aika5l bD12kl � ailak5 bD12

lk

�¸k l

Dkli5 bD12

kl

72

2.4. QUANTUM DEFORMATION OF THE BIG CELL INSIDE THE SUPERGRASSMANNIAN

has been used ak1ak2 � q�1ak2ak1 (2.27) andD12ji � �q�1D12

ij for i   j. ButDkli5 P SLqp4|1q

and D12kl P Grq.

The proof for the ∆pD55q element is analogue to Di5. �

2.4 Quantum deformation of the big cell inside the super Grass-

mannian

2.4.1 Super setting

We want now to de�ne the analogue, in the quantum setting, of the superalgebra repre-senting the big cell of the Grassmannian supermanifold. At the classical level we obtainedit in subsection (2.2.2).

The superalgebra OpU12q � OpC4|2q in the subsection (2.2.5) is the superalgebra corre-sponding to the chiral Minkowski superspace (see Sec. 2.1).

In Section 2.2.5 we wrote the action of the lower parabolic supergroup Pl (see equation2.26), which includes the super Poincaré group times dilations, using the functor of points.We want now to translate it into the coaction language to make the generalization to thequantum setting.

The �rst step is to understand the Hopf superalgebra of the lower parabolic supergroupPl as the quotient, by a suitable ideal, of the algebra representing SLp4|1q

OpSLp4|1qq � Crgij, g55, γi5, γ5jsLtBer� 1u

see remark (1.9.46).

The generators can be written in matrix form�gij γi5γ5j g55

,

so to read the comultiplication as the matrix product.

Proposition 2.4.1. Let OpPl q be the superalgebra

OpPl q :� OpSLp4|1qqLI

where I is the (two-sided) ideal generated by

g1j, g2j, for j � 3, 4 and γ15, γ25.

This is the Hopf superalgebra of the lower parabolic subgroup, with comultiplicationnaturally inherited by OpSLp4|1qq.

73

CHAPTER 2. CHIRAL SUPERSPACES

Proof . It is enough to prove that the comultiplication and the counit are well-de�nedmaps on the quotient, which is the same as saying that the ideal I is a Hopf ideal (1.9.23).Let the canonical projection π : OpSLp4|1qq Ñ OpSLp4|1qq

LI. Then the comultiplication

in the quotient ∆pOpSLp4|1qqLIq is induced by the composition pπ b πq �∆

OpSLp4|1qq OpSLp4|1qq b OpSLp4|1qq

OpSLp4|1qqLI OpSLp4|1qq

LIb OpSLp4|1qq

LI

?

π

-∆

?

πbπ

And each element of I is an element in Ib OpSLp4|1qq ` OpSLp4|1qq b I.The counit satis�es

Epgij ` g1kq � δij ` δ1k � δij for 1 ¤ i, j ¤ 5, k � 3, 4.

so EOpSLp4|1qq

LIpgij ` Iq � Epgijq. �

In matrix form, the functor of points of OpPl q on local superalgebra A is given by,

HPlpAq �

$''''&''''%

������g11 g12 0 0 0g21 g22 0 0 0g31 g32 g33 g34 γ35

g41 g42 g43 g44 γ45

γ51 γ52 γ53 γ54 g55

�����

,////.////-� HSLpm|nqpAq. (2.41)

Remark 2.4.2. The superalgebra representing the big cell is in fact a sub superalgebra(not a Hopf subalgebra) of OpPl q.

It is more convenient to make a change of variables for the generators, so������g11 g12 0 0 0g21 g22 0 0 0g31 g32 g33 g34 γ35

g41 g42 g43 g44 γ45

γ51 γ52 γ53 γ54 g55

����� �

�� x 0 0tx y yητx dξ d

� .

The notation used here is slightly di�erent to the notation used in (2.25). We can de�ne

dτ :� τx, (2.42)

but we will see that having τ is essential to describe the bigcell. With this change ofvariable we have

HPlpAq �

�� x 0 0tx y yηdτ dξ d

� .

74

2.4. QUANTUM DEFORMATION OF THE BIG CELL INSIDE THE SUPERGRASSMANNIAN

Proposition 2.4.3. The Hopf superalgebra OpPl q is generated by the two alternativeset of variables

1. x, y, t, τ , ξ, η and d.

2. x, y, t, τ , ξ, η and d.

The sub superalgebra of OpPl q generated by pt, τq coincides with the big cell superringOpU12q as de�ned in (2.24).Moreover, there is a well de�ned coaction of OpPl q on OpU12q induced by the comulti-plication (see equation 2.45),

∆ : OpU12q ÝÝÝÑ OpPl q b OpU12q

which explicitly takes the form

∆tij � tij b 1� yiaSpxqbj b tab � yiaηaSpxqbj b τb, (2.43)

∆τj �pdb 1qpτa b 1� ξb b tba � 1b τaqpSpxqaj b 1q. (2.44)

(The reader should notice right away that this is the dual to the equation (2.26)).Proof . First of all it is clear that the two given sets generate the superring OpPl q and

the sub superalgebra generated by pt, τq is isomorphic to C4|2 because t is the even matrixin M2pCq and the τ is odd matrix 2� 1.

The coaction can be read by the comultiplication in matrix form, that is

∆

�� x 0 0tx y yητx dξ d

� �

�� x 0 0tx y yητx dξ d

� b

�� x 0 0tx y yητx dξ d

� . (2.45)

Then

∆x � xb x,

∆ptxq � txb x� y b tx� yη b τx,

∆pτxq � τxb x� dξ b tx� db τx.

In components

∆xij � xik b xkj,

∆ptijxjlq � p∆tijqp∆xjlq � tiaxab b xbl � yia b tabxbl � yiaηa b τbxbl �

ptij b 1� yiaSpxqbj b tab � yiaηaSpxqbj b τbqpxjp b xplq,

∆pτjxjlq � p∆τjqp∆xjlq � τjxjk b xkl � dξj b tjkxkl � db τjxjl �

pτj b 1� dξbSpxqaj b tba � dSpxqaj b τaqpxjk b xklq,

75

CHAPTER 2. CHIRAL SUPERSPACES

where S is the antipode,

Spxq � x�1 �1

detpxq

�x22 �x12

�x21 x11

.

From these equations we can read the coaction of the group on the big cell,

∆tij � tij b 1� yiaSpxqbj b tab � yiaηaSpxqbj b τb,

∆τj � τj b 1� dξbSpxqaj b tba � dSpxqaj b τa.

Now we can using the equations (2.43) and (2.44) with (2.26) to realize that the actionand the coaction are dual to each other. It is enough to use (2.42) but only in the �rstfactor

∆τj �dτaSpxqaj b 1� dξbSpxqaj b tba � dSpxqaj b τa �

pdb 1qpτa b 1� ξb b tba � 1b τaqpSpxqaj b 1q,

and we obtain the coaction in the desired form.

�

2.4.2 Quantum setting

We now turn to the quantum setting. We shall repeat all the classical arguments, exertinghowever extreme care, since in all of our calculations, the Manin commutation relations(see De�nition 2.3.1) now play a key role. In order to keep our notation minimal we usethe hat over the same letters as in the super case to denote the generators of the quantumbig cell and the quantum supergroups.

Proposition 2.4.4. Let OqpPlq be the superalgebra

OqpPlq :� OqpSLp4|1qqLIq

where Iq is the (two-sided) ideal in OqpSLp4|1qq generated by

g1j, g2j, for j � 3, 4 and γ15, γ25. (2.46)

This is the Hopf superalgebra of the quantum lower parabolic subgroup, with comultipli-cation the one naturally inherited from OqpSLp4|1qq.

Proof . Notice that the comultiplication is the same than in the classical case, so Iqis a Hopf ideal (see Proposition 2.4.1) and the Hopf superalgebra structure goes to thequotient. �

76

2.4. QUANTUM DEFORMATION OF THE BIG CELL INSIDE THE SUPERGRASSMANNIAN

Remark 2.4.5. The quantum lower parabolic supergroup is generated by the images inthe quotient of the generators gij and γij that are not listed in (2.46). In the quantumcase, this is a non trivial fact, because in the commutation relations among the generatorsof the ideal may appear generators other than the ones in (2.46), giving then a `bigger'ideal than in the classical case. One can check that this does not happen here, the Maninrelations (2.27), (2.28) and (2.29) have not add terms however the (2.30) added one, butthis still belongs to the ideal. (for details see Ref. [50]). �

As in the classical case, it is convenient to change coordinates������g11 g12 0 0 0g21 g22 0 0 0g31 g32 g33 g34 γ35

g41 g42 g43 g44 γ45

γ51 γ52 γ53 γ54 g55

����� �

�� x 0 0tx y yηˆτ x dξ d

� . (2.47)

Notice that in OqpPlq � OqpSLp4|1qq the elements D12 and D3434 are invertible (these are

the quantum determinants (2.39)).One can compute explicitly the inverse change of variables,

x �

�g11 g12

g21 g22

, t �

��q�1D23D

�112 D13D

�112

�q�1D24D�112 D14D

�112

y �

�g33 g34

g43 g44

, d � g55,

ˆτ ��g�1

55 γ51 g�155 γ51

�, ξ �

�g�1

55 γ53 g�155 γ54

�

η � y�1

�γ35

γ45

� pD34

34q�1

��qD45

34

D3534

where has been used

Spxq � x�1 � D�112

�g22 �qg12

�q�1g21 g11

and Spyq � y�1 � pD34

34q�1

�g44 �qg34

�q�1g43 g33

It is not hard to see that OpPl,qq is also generated by x, y, t, d, η, ξ and ˆτ .

The quantum Poincaré supergroup times dilations is the quotient of OqpPlq by the

ideal Iξ � 0. It is then generated by the images in the quotient of x, y, d, η and ˆτ . Here,the Remark 2.4.5 applies as well. In matrix form�

� x 0 0tx y yηˆτ x 0 d

� . (2.48)

77

CHAPTER 2. CHIRAL SUPERSPACES

De�nition 2.4.6. We de�ne the quantum big cell OqpU12q as the subring of OqpPlqgenerated by t and ˆτ . �

Proposition 2.4.7. The quantum big cell superring OqpU12q has the following presenta-tion

OqpU12q :� Cqxtij, ˆτklyLIU

where IU is the ideal generated by the relation

ti1ti2 � q�1ti2ti1,

t3j t4j � q�1 t4j t3j,

t31t42 � t42t31 � pq�1 � qqt41t32, (2.49)

t32t41 � t41t32,

ˆτ51ˆτ52 � �q�1 ˆτ52τ51,

tij ˆτ5j � q�1 ˆτ5jtij, (2.50)

ti1 ˆτ52 � ˆτ52ti1 � pq�1 � qqti2 ˆτ51,

ti2 ˆτ51 � ˆτ51ti2,

for 3 ¤ i ¤ 4 and 1 ¤ j ¤ 2.Proof . Direct check. �

As in the classical setting we have the following proposition.

Proposition 2.4.8. The quantum big cell OqpU12q admits a coaction of OqpPlq obtainedby restricting suitably the comultiplication in OqpPlq. Explicitly see (2.43),(2.44),

∆tij � tij b 1� yiaSpxqbj b tab � yiaηaSpxqbj b ˆτb,

∆ˆτj � pdb 1qpτa b 1� ξb b tba � 1b ˆτaqpSpxqaj b 1q.

by choosing as before the set of generators x, y, t, d, τ , ρ and ξ for OqpPlq and t, ˆτ for

OqpU12q with dτ � ˆτ x.Proof . This is so because the comultiplication is the same in the classical and thequantum group, given essentially by matrix multiplication. One has to be careful, though,when expressing the comultiplication in terms of the new generators, since the orderingappearing in the De�nition (2.47) has to be kept consistently. �

78

Quantum Twistors

In this chapter, since the underlying structure like Poisson algebras of the quantum groupsde�ned in Chapter 2, there are star products induced by the quantization of superalgebras.

In Section 3.1, we give the de�nition of star product associated to deformation made inSection 2.3 of Chapter 2.

In Section 3.2 we give the explicit formula for the star product among two monomials onMinkowski space.

In Section 3.3 we prove that the star product can be extended to smooth functions andcompute it explicitly up to order two in h.

In Section 3.4 we show that the coaction of the Poincaré group on the quantum Minkowskispace is representable by a di�erential operator (at least up to order one in h). To showthis, we need a technical result concerning the quantum Poincaré group, that we provein the appendix B.1.

In Section 3.5 we study the real forms of the quantum algebras that correspond to thereal forms of ordinary Minkowski and Euclidean space.

In Section 3.6 we write the quadratic invariant (the metric of the Minkowski space) inthe star product algebra.

3.1 Poisson algebra and �-product

Let us start from the commutative ring of formal power series kq: it is a ring extensionof the �eld k with elements of the form

a �8

n�0

anqn,

where an P k and q is undetermined.

De�nition 3.1.1. A Poisson algebra is a vectorspace A over the �eld k equipped witha commutative associative algebra structure (a multiplication map �) and a Lie algebrastructure (a Lie bracket t , u) which satisfy the compatibility condition

tfg, hu � f � tg, hu � tf, hu � g.

79

CHAPTER 3. QUANTUM TWISTORS

�

De�nition 3.1.2. Let pA, �, t , uq be a Poisson algebra over a �eld k. We say that theassociative algebra Arqs over k is a formal deformation of A if

1. There exists an isomorphism of modules ψ : Aq � t°8n�0 anq

n | an P Au Ñ Arqs,

2. ψpfgq � ψpfqψpgq mod(q),

3. ψpfqψpgq � ψpgqψpfq � q ψptf, guq mod(q2),

for all f, g P Aq �

Remark 3.1.3. The quantum groups de�ned in Section 2.3 of Chapter 2 are formaldeformation of polynomial superalgebra Cn|m, with the isomorphism ψ : Cn|m

q ÑMqpn|mqand the Manin relations 2.27, 2.28, 2.29, 2.30. �

De�nition 3.1.4. An associative product in Aq is de�ned by

f � g � ψ�1pψpfq � ψpgqq, f, g P Aq (3.1)

is called the star product on Aq induced by ψ. �

De�nition 3.1.5. (Alternative.)A star product on Aq can be also de�ned as an associative kq-linear product given by theformula

f � g � fg �B1pf, gqq �B2pf, gqq2 � � � � P Aq, f, g P Aq (3.2)

where the Bi's are bilinear operators. �

The associativity of � implies that tf, gu � B1pf, gq � B1pg, fq is a Poisson bracket onAq.

De�nition 3.1.6. Two star products on Aq, � and �1 are said to be equivalent or gauge

equivalent if there exists a linear map T : Aq Ñ Aq on the form

T � 11�¸n¡0

qnTn

with Tn linear operators on Aq, such that

f � g � T�1pT pfq �1 T pgqq

. �

Remark 3.1.7. Two star products that are equivalent are isomorphic and have the same�rst order term, so they are formal deformation of the same Poisson structure. �

De�nition 3.1.8. If A � C8pMq for M be a di�erentiable manifold over the �eld kq,

and the operators Bi's in (3.1.5) are bidi�erential operators we say that the star productis di�erential . If in addition A � C

8pMq and M is a real Poisson manifold, we will say

that � is a di�erential star product on Mq.

The set of (gauge) equivalence classes of di�erential star products on a manifold M hasbeen classi�ed by Kontsevich in terms of equivalence classes of formal Poisson structures(modulo formal di�eomorphisms).

80

3.2. ALGEBRAIC STAR PRODUCT ON MINKOWSKI SPACE

3.2 Algebraic star product on Minkowski space

Remark 3.2.1. Change notation

Below for convenience we will change the notation of some de�nitions of Chapter 2:

1. In Section 2.2, the generators of the 2-plane π in the big cell of the GrassmannianGp2, 4q, A, will be change by t in matrix form from (2.12) to

π �

����

1 00 1t31 t32

t41 t42

���

2. In Subsection 2.2.2 we gave a interpretation of the big cell U12 of the GrassmannianGp2, 4q as the compactifed and complex Minkowski space. Then we can replace thenotation U12 to M , which we will do in all cases (classic, super and quantum).

�

Theorem 3.2.2. We consider the algebra of the classical Minkowski space with thescalars extended to the ring Cq and the algebra of the complexi�ed quantum Minkowskispace OqpMq de�ned by the even generators in Proposition (2.4.7),

OpMqrq, q�1s � Cqrt41, t42, t31, t32s.

There is an module isomorphism

QM : Cqrt41, t42, t31, t32s Ñ OqpMq

ta41tb42t

c31t

d32 ÞÑ ta41t

b42t

c31t

d32,

QM is called an ordering rule or quantization map.Proof .See Ref. [16]. �

So OqpMq is a free module over Cq, with basis the set of standard monomials.

We can pull back the product on OqpMq to OpMqrq, q�1s. There, following (3.1.4) we cande�ne the star product as

f � g � Q�1M

�QMpfqQMpgq

�, f, g P OpMqrq, q�1s. (3.3)

By construction, the star product is associative. The algebra pOpMqrq, q�1s, � q is thenisomorphic to OqpMq.

Working on OpMqrq, q�1s has the advantage of working with `classic' objects (the polyno-mials), were one has substituted the standard point wise product by the noncommutativestar product. This is important for the physical applications.

81

CHAPTER 3. QUANTUM TWISTORS

Moreover, we can study if this star product has an extension to all the C8- functions,and if the extension is di�erential. If so, Kontsevich's theory [17] would then be relevant.

We want to obtain a formula for the star product. We begin by computing the auxiliaryrelations

tm42tn41 � q�mntn41t

m42,

tm31tn41 � q�mntn41t

m31,

tm31tn42 � tn42t

m31,

tm32tn42 � q�mntn42t

m32,

tm32tn31 � q�mntn31t

m32,

and after a (lengthy) computation we obtain

tm32tn41 � tn41t

m32 �

µ

k�1

Fkpq,m, nqtn�k41 tk42t

k31t

m�k32 ,

where µ � minpm,nq

Fkpq,m, nq � βkpq,mqk�1¹l�0

F pq, n� lq with F pq, nq �

�1

q2n�1� q

(3.4)

and βkpq,mq de�ned by the recursive relation

β0pq,mq � βmpq,mq � 1, and βkpq,m� 1q � βk�1pq,mq � βkpq,mqq�2k.

Moreover, βkpq,mq � 0 if k   0 or if k ¡ m. Using the above relations, we obtain thestar product of two arbitrary polynomials:

pta41tb42t

c31t

d32q � pt

m41t

n42t

p31t

r32q � q�mc�mb�nd�dpta�m41 tb�n42 tc�p31 td�r32

�

µ�minpd,mq¸k�1

q�pm�kqc�pm�kqb�npd�kq�ppd�kqFkpq, d,mq ta�m�k41 tb�k�n42 tc�k�p31 td�k�r32 (3.5)

3.3 Di�erential star product on the big cell

To compare the algebraic star product obtained above with the di�erential star productapproach, we consider a change in the parameter, q � exp h. The classic limit is obtainedas h Ñ 0. We will expand the right side of equation (3.5) in powers of h and we will showthat each term can be written as a bidi�erential operator.

3.3.1 Explicit computation up to order 2

We �rst take up the explicit computation of the bidi�erential operators up to order 2.Then we will argue that a di�erential operator can be found at each order.

82

3.3. DIFFERENTIAL STAR PRODUCT ON THE BIG CELL

We rewrite the �-product as

f � g � fg �8

j�1

hjCjpf, gq,

withf � ta41t

b42t

c31t

d32, g � tm41t

n42t

p31t

r32.

At order 0 in h we recover the commutative product. At order n in h we have contributionsfrom each of the terms with di�erent k in (3.5).

Cnpf, gq �

µ�minpd,mq¸k�0

Cpkqn pf, gq,

(the terms with k � 0 come from the �rst term in Equation 3.5).

Let us compute each of the contributions Cpkq1 :

• k � 0. We have

Cp0q1 � p�mc�mb� nd� dpq ta�m41 tb�n42 tc�p31 td�r32 .

It is easy to see that this is reproduced by the bidi�erential operator

Cp0q1 pf, gq � �pt41t31B31fB41g � t42t41B42fB41g � t32t42B32fB42g � t32t31B32fB31gq.

We will denote the bidi�erential operators by means of the tensor product (as it iscustomary). For example

Cp0q1 � �pt41t31B31 b B41 � t42t41B42 b B41 � t32t42B32 b B42 � t32t31B32 b B31q,

soCp0q1 pf, gq � C

p0q1 pf b gq.

• k � 1. Let us �rst compute the factor F1pq, d,mq � β1pq, dqF pq,mq. First, noticethat

β1pq, dq �1� q�2 � q�4 � � � � � q�2pd�1q �e�2dh � 1

e�2h � 1�

d� dpd� 1qh�1

3dp1� 3d� 2d2qh2 �Oph3q,

and thatF pq,mq � �2mh� 2mpm� 1qh2 �Oph3q,

so up to order h2 we have

β1pq, dqF pq,mq � �2md h� 2mdpd�m� 2qh2 �Oph3q.

83

CHAPTER 3. QUANTUM TWISTORS

Finally, the contribution of the k � 1 term to C1 is

Cp1q1 pf, gq � �2mdta�m�1

41 tb�n�142 tc�p�1

31 td�r�132 .

This is reproduced by the bidi�erential operator

Cp1q1 � �2t42t31B32 b B41.

• k ¥ 2 We have the factor

βkpq, dqF pq,mqF pq,m� 1q � � �F pq,m� k � 1q � Ophkq,

so the terms with k ¥ 2 do not contribute C1 because each F pq,m�pk� 1qq at lesscontribute with h term.

Summarizing,

C1 �Cp0q1 � C

p1q1 � �pt41t31B31 b B41 � t42t41B42 b B41�

t32t42B32 b B42 � t32t31B32 b B31 � 2t42t31B32 b B41q, (3.6)

so C1 is extended to the C8- functions. If we antisymmetrize C1 we obtain a Poissonbracket

tf, gu �t41t31pB41fB31g � B41gB31fq � t42t41pB41fB42g � B41gB42fq�

t32t42pB42fB32g � B42gB32fq � t32t31pB31fB32g � B31gB32fq�

2t42t31pB41fB32g � B41gB32fq (3.7)

We can express the Poisson bracket in terms of the usual variables in Minkowski space.Using the De�nition (2.4), the change of coordinates is�

t31 t32

t41 t42

� xµσµ �

�x0 � x3 x1 � ix2

x1 � ix2 x0 � x3

,

and the inverse change is

x0 �1

2pt31 � t42q, x1 �

1

2pt32 � t41q, x2 �

i

2pt32 � t41q, x3 �

1

2pt31 � t42q.

In these variables the Poisson bracket is

tf, gu �2i��px0q2 � px3q2

�pB1fB2g � B1gB2fq � x0x1pB0fB2g � B0gB2fq�

x0x2pB0fB1g � B0gB1fq � x1x3pB2fB3g � B2gB3fq�

x2x3pB1fB3g � B1gB3fq

(3.8)

We now compute the term C2. We sum the contributions to the order h2 of each term in(3.5)

84

3.3. DIFFERENTIAL STAR PRODUCT ON THE BIG CELL

• k � 0. The contribution to the order h2 is

Cp0q2 �

1

2pmc�mb� nd� dpq2 ta�m41 tb�n42 tc�p31 td�r32 .

This is reproduced by

Cp0q2 �

1

2t31t41 B31pt31B31q b B41pt41B41q � t42t31t41 B42B31 b B41pt41B41q�

t31t32t41t42 B31B32 b B41B42 � t231t32t41 B31B32 b B41B31�

1

2t42t41 B42pt42B42q b B41pt41B41q � t41t

242t32B42B32 b B41B42�

t41t42t31t32B42B32 b B41B31 �1

2t32t42t31B32pt32B32q b B42B31�

1

2t32t31 B32pt32B32q b B31pt31B31q � t32t42t31 B32pt32B32q b B42pt42B42q.

• k � 1. We have thatF1pq, d,mq � β1pq, dqF pq,mq.

Expanding both factors we have

β1pq, dq � d� dpd� 1qh�Oph2q,

F pq,mq � �2mh� 2mp1�mqh2 �Oph3q,

so we get

β1pq, dqF pq,mq � �2md h� 2md�pm� 1q � pd� 1q

�h2,

and

q�pm�1qc�pm�1qb�npd�1q�ppd�1q � 1��� pm� 1qc� pm� 1qb� npd� 1q � ppd� 1q

�h

the total contribution to order h2 is

h2�

2md�pm� 1q � pd� 1q � pm� 1qc� pm� 1qb� npd� 1q � ppd� 1q

�� ta�m�1

41 tb�n�142 tc�p�1

31 td�r�132 .

We reproduce that result with

Cp1q2 �2t32t42t31B

232 b B41 � 2t31t42t41B32 b B2

41 � 2t31t242t41B42B32 b B2

41�

2t42t231t41B31B32 b B2

41 � 2t31t242t32B

232 b B41B42 � 2t42t

231t32B

232 b B41B31.

• k � 2. We need calculate β2pq, dqF pq,mqF pq,m� 1q, but

F pq,mq � �2mh

F pq,m� 1q � �2pm� 1qh

85

CHAPTER 3. QUANTUM TWISTORS

then for have the h2 term, only need the zero orden in β2pq, dq. One can show that

β2pq, dq �dpd� 1q

2�Ophq,

soβ2pq, dqF pq,mqF pq,m� 1q � 2dpd� 1qmpm� 1qh2,

and the contribution of this term to the order h2 is

h2 2dpd� 1qmpm� 1q ta�m�241 tb�n�2

42 tc�p�231 td�r�2

32 .

This is given byCp2q2 � 2t242t

231B

232 b B2

41.

Summarizing we get

C2 �1

2t31t41 B31pt31B31q b B41pt41B41q � t42t31t41 B42B31 b B41pt41B41q�

t31t32t41t42 B31B32 b B41B42 � t231t32t41 B31B32 b B41B31�

1

2t42t41 B42pt42B42q b B41pt41B41q � t41t

242t32B42B32 b B41B42�

t41t42t31t32B42B32 b B41B31 �1

2t32t42t31B32pt32B32q b B42B31�

1

2t32t31 B32pt32B32q b B31pt31B31q � t32t42t31 B32pt32B32q b B42pt42B42q�

2t32t42t31B232 b B41 � 2t31t42t41B32 b B2

41 � 2t31t242t41B42B32 b B2

41�

2t42t231t41B31B32 b B2

41 � 2t31t242t32B

232 b B41B42 � 2t42t

231t32B

232 b B41B31�

2t242t231B

232 b B2

41.

3.3.2 Di�erentiability at arbitrary order

We are going to prove now the di�erentiability of the star product. We keep in mindthe expression (3.5), which has to be expanded in h. Our goal will be to show that, ateach order, it can be reproduced by a bidi�erential operator with no dependence on theexponents a, b, c, d,m, n, p, r.Let us �rst argue on a polynomial function of one variable, say x. For example, we have

m xm�1 � Bx�xm

�.

More generally, we have

mb xm � pxBxqb�xm

�and

mbpm� 1qc � � � pm� k � 1qd xm�k � BxpxBxqd�1 . . . BxpxBxq

c�1BxpxBxqb�1

�xm

�. (3.9)

86

3.3. DIFFERENTIAL STAR PRODUCT ON THE BIG CELL

Notice that in the last formula, we have b, c, . . . , d ¥ 1, otherwise the formula makes nosense. In fact, an arbitrary polynomial

ppxq �¸kPZ

fkpm,xqxm�k,

is not generically obtainable from xm by the application of a di�erential operator withcoe�cients that are independent of the exponents and polynomial in the variable x. Onecan try for example with ppxq � xm�1. We then have that

xm�1 �1

mBxpx

mq, or xm�1 �1

xxm.

So the right combinations should appear in the coe�cients in order to be reproduced bya di�erential operator with polynomial coe�cients.Let us see the contribution of the terms with di�erent k in (3.5). We start with the termk � 0. From

q�mc�mb�nd�dp ta�m41 tb�n42 tc�p31 td�r32

we only get terms of the form

bibcicdidmimninpip ta�m41 tb�n42 tc�p31 td�r32 .

Applying the rules (3.9), these terms can be easily reproduced by the bidi�erential oper-ators of the form

pt42B42qibpt31B31q

icpt32B32qid b pt41B41q

impt42B42qinpt31B31q

ip ,

applied tota41t

b42t

c31t

d32 b tm41t

n42t

p31t

r32.

We turn now to the more complicated case of k � 0. We have to consider the two factorsin (3.5)

q�pm�kqc�pm�kqb�npd�kq�ppd�kq, and Fkpq, d,mq.

Expanding both factors in powers of h it is easy to see that the coe�cients at each orderare polynomials in m,n, p, b, c, d, k. What we have to check is that this polynomials havea form that can be reproduced with a bidi�erential operator using (3.9). Let us startwith

Fkpq, d,mq � βkpq,mqk�1¹l�0

F pq,m� lq.

From the de�nition (3.4), we have that F pq, jq|j�0 � 0, so

F pq, jq � jGpq, jq,

with Gpq, jq a series in h with coe�cients that are polynomial in j. More generally, theproduct

Lkpq,mq �k�1¹l�0

F pq,m� lq � mpm� 1qpm� 2q � � � pm� k � 1qL1pq,mq.

87

CHAPTER 3. QUANTUM TWISTORS

The polynomials in L1pq,mq are easily obtained with combinations of di�erential operatorsof the form �

t41B41

�iptm41q.

The remaining factor mpm�1qpm�2q � � � pm�k�1qtm�k41 is adjusted with the di�erentialoperator

Bk41ptm41q � mpm� 1qpm� 2q � � � pm� k � 1q tm�k41 .

Let us work now with βkpq,mq. We have that

βkpq, dq � 0 for d   k,

soβkpq, dq � dpd� 1qpd� 2q � � � pd� k � 1qβ1kpq, dq,

with β1kpq, dq a series in h with coe�cients that are polynomial in d. The di�erentialoperator that we need is of the form

Bk32ptd32q � dpd� 1qpd� 2q � � � pd� k � 1q td�k32 .

Finally, the factor q�pm�kqc�pm�kqb�npd�kq�ppd�kq introduces factors of the form

bibcicpd� kqidpm� kqimninpip ta�m�k41 tb�k�n42 tc�k�p31 td�k�r32 ,

which are reproduced by

tk42tk31

�t42B42

�ib�t31B31

�ic�t32B32

�id b �t41B41

�im�t42B42

�in�t31B31

�ip,

acting onta41t

b42t

c31t

d�k32 b tm�k41 tn42t

p31t

r32.

This completes the proof of di�erentiability of the star product at arbitrary order.

3.4 Poincaré coaction

We would like to see how the coaction of the quantum Poincaré group plus dilations (2.48)over the Minkowski space looks in terms of the star product, and if it is also di�erential.

First of all, we notice that the subalgebra generated by txiju and tyabu are two copies ofthe algebra Mqp2q which commute among them (see Appendix B.6). The maps to thestandard quantum matrices are this time�

a bc d

Õ

�x11 x12

x21 x22

;

�a bc d

Õ

�y33 y34

y43 y44

,

as can be deduced from commutation relations (B.1) and (B.2). One can chose the Maninorder in each subset of variables,

y44   y43   y34   y33, x22   x21   x12   x11.

88

3.4. POINCARÉ COACTION

With this one can construct a quantization map (given by the standard monomials ba-sis) for the quantum Lorentz plus dilations group. We have now to include thetranslations to have the complete quantization map for the Poincaré group.It is clear that one can choose the Manin order also for the variables T , but, since thesevariables do not commute with the x and y we have to be careful in choosing a fullordering rule. This is a non trivial problem, but it can be solved. In Appendix B.1 weshow that the ordering

y44   y43   y34   y33   x22   x21   x12   x11   T41   T42   T31   T32

gives standard monomials that form a basis for the quantum Poincaré group OqpPlq.As for the Minkowski space star product (3.3), we extend the scalars of the commutativealgebra to Cq and de�ne a quantization map QG

OpPlqrq, q�1s

QGÝÝÝÑ OqpPlq

ya44yb43y

c34y

d33x

e22x

f21x

g12x

l11T

m41T

n42T

p31T

r32 ÝÝÝÑ ya44y

b43y

c34y

d33x

e22x

f21x

g12x

l11T

m41 T

n42T

p31T

r32.

(3.10)

If f, g P OpPlqrq, q�1s, then the star product is de�ned as for the Minkowski space,

f �G g � Q�1G pQGpfqQGpgqq.

Let us now consider the coaction ∆, formally as in Proposition 2.4.8.

De�nition 3.4.1. Using both quantization maps (QM and QG) we can de�ne a star

coaction ∆�,

OpMqrq, q�1s∆�ÝÝÝÑ Ñ OpGqrq, q�1s b OpMqrq, q�1s

f ÝÝÝÑ Q�1G bQ�1

M p∆pQMpfqq,

where ∆ is the coaction restricted to the even variables de�ned in Prop. 2.4.8. �

Proposition 3.4.2. The last star coaction ∆� and the star products �M , �G satis�ed thecompatibility property:

∆�pf �M gq � ∆�pfqp�G b �Mq∆�pgq, f, g P OpMqrq, q�1s. (3.11)

Proof .On one side

∆�pf �M gq � ∆�

�Q�1M

�QMpfqQMpgq

�� Q�1

G bQ�1M

�∆�QMpfqQMpgq

��Q�1

G

�QMpfqQMpgqp1q

�bQ�1

M

�QMpfqQMpgqp2q

�.

89

CHAPTER 3. QUANTUM TWISTORS

in the other side

∆�pfqp�G b �Mq∆�pgq ��Q�1G pQMpfqp1qq bQ�1

M pQMpfqp2qq�p�G b �Mq

�Q�1G pQMpgqp1qq bQ�1

M pQMpgqp2qq���

Q�1G pQMpfqp1qq �G Q

�1G pQMpgqp1qq

�b�Q�1M pQMpfqp2qq �M Q�1

G pQMpgqp2qq��

Q�1G

�QMpfqQMpgqp1q

�bQ�1

M

�QMpfqp2qQMpgqp2q

�.

where had been used a Sweedler notation. �

3.4.1 The coaction as a di�erential operator

We will restrict to the Lorentz group times dilations, that is, we will consider only thegenerators x and y.On the generators of Minkowski space the star coaction (see De�nition 3.4.1) is simply

∆�ptaiq � yabSpxjiq b tbj,

and, using the notationt�ami � tmi �M tmi �M � � � �M tmilooooooooooooomooooooooooooon

a times

,

for an arbitrary standard monomial the coaction is expressed as

∆�

�ta41t

b42t

c31t

d32

�� ∆�pt

�a41 �M t�b42 �M t�c31 �M t�d32q �

p∆�t41q�ap�G b �Mqp∆�t42q

�bp�G b �Mqp∆�t31q

�cp�G b �Mqp∆�t32q

�d.

We have used the exponent `�' to indicate`�M', `�G' or `�G�M' to simplify the notation.The meaning should be clear from the context.Contracting with µG�M (see the remark in Appendix A) we de�ne

τij � µG�M � ∆�ptijq � yabtbjSpxjiq;

Applying µG�M to the coaction, we get

µG�M � ∆�pta41t

b42t

c31t

d32q � τ �a41 �G�M τ �b42 �G�M τ �c31 �G�M τ �d32 . (3.12)

Notice that in each τ there is a sum of terms with factors ytSpxq that generically do notcommute. So we need to work out the star products in the right hand side of (3.12).

As we are going to see, the calculation is involved. We are going to make a change in theparameter q � exp h and expand the star product in power series of h. At the end, wewill compute only the �rst order term in h of the star coaction.

The star product �G�M is written, as usual (3.1.5),

f1 �G�M f2 �8

m�0

hmDmpf1, f2q, f1, f2 P OpG�Mqrq, q�1s.

90

3.4. POINCARÉ COACTION

For our purposes it will be enough to consider only functions f1 and f2 to be polynomialsin τ .

The generators x, y and t commute among themselves, so the star product in OpG �Mqrq, q�1s can be computed reordering the generators in each group x, y, and t in theManin ordering. The result will be terms similar to the star product (3.5), and in partic-ular, D1 will contain 3 terms of the type C1 (3.6), on for the variables x, another for thevariables y and another for the variables t. But C1 is a bidi�erential operator of order 1in each of the arguments, so it satis�es the Leibnitz rule

D1pf1, f2uq � D1pf1, f2qu�D1pf1, uqf2, u P OpPlqrq, q�1s

so we have for example

D1pτij, τaklq � aD1pτij, τklqτ

a�1kl , (3.13)

which will be used in the following.

In general, we have

τ �a41 � τ�b42 � τ

�c31 � τ

�d32 �

¸IPI

hMDi1pτ41, Di2pτ41, . . . Dia�1pτ41, Dj1pτ42,

Dj2pτ42, . . . Djb�1pτ42, Dl1pτ31, Dl2pτ31, . . . Dlc�1pτ31, Dm1pτ32,

Dm2pτ32, . . . Dmd�1pτ32, τ32q . . . q

Here M � i1 � . . . � ia � j1 � . . . � jb � l1 � . . . � lc �m1 � . . .md and we sum over allthe multiindices

I � pi1, . . . , ia�1, j1, . . . , jb�1, l1, . . . , lc�1,m1, . . . ,md�1q .

We are interested in the �rst order in h, so M � 1. This means that for any term in thesum we have only one D1 operator (the others are D0, which is just the standard productof both arguments). So we have the sum

¸k

�τ k41D1pτ41, τ

a�k�141 τ b42τ

c31τ

d32q � τa41τ

k42D1pτ42, τ

b�k�142 τ c31τ

d32q�

τa41τb42τ

k31D1pτ31, τ

c�k�131 τ d32q � τa41τ

b42τ

c31τ

k32D1pτ32, τ

d�k�132 q

.

91

CHAPTER 3. QUANTUM TWISTORS

using (3.13) we get

a�1

k�1

kτa�241 τ b42τ

c31τ

d32D1pτ41, τ41q �

a

k�1

bτa�141 τ b�1

42 τ c31τd32D1pτ41, τ42q�

a

k�1

cτa�141 τ b42τ

c�131 τ d32D1pτ41, τ31q �

a

k�1

dτa�141 τ b42τ

c31τ

d�132 D1pτ41, τ32q�

b�1

k�1

kτa41τb�242 τ c31τ

d32D1pτ42, τ42q �

b

k�1

cτa41τb�142 τ c�1

31 τ d32D1pτ42, τ31q�

b

k�1

dτa41τb�142 τ c31τ

d�132 D1pτ42, τ32q �

c�1

k�1

kτa41τb42τ

c�231 τ d42D1pτ31, τ31q�

c

k�1

dτa41τb42τ

c�131 τ d�1

32 D1pτ31, τ32q �d�1

k�1

kτa41τb42τ

c31τ

d�232 D1pτ32, τ32q.

These sums can be easily done. We then get the order h contribution to the action of thedeformed Lorentz plus dilations group:

apa� 1q

2D1pτ41, τ41qτ

a�241 τ b42τ

c31τ

d32 � abD1pτ41, τ42qτ

a�141 τ b�1

42 τ c31τd32�

bpb� 1q

2D1pτ42, τ42qτ

a41τ

b�242 τ c31τ

d32 � bcD1pτ42, τ31qτ

a41τ

b�142 τ c�1

31 τ d32�

cpc� 1q

2D1pτ31, τ31qτ

a41τ

b42τ

c�231 τ d32 � cdD1pτ31, τ32qτ

a41τ

b42τ

c�131 τ d�1

32 �

dpd� 1q

2D1pτ32, τ32qτ

a41τ

b42τ

c31τ

d�232 � acD1pτ41, τ31qτ

a�141 τ b42τ

c�131 τ d32�

adD1pτ41, τ32qτa�141 τ b42τ

c31τ

d�132 � bdD1pτ42, τ32qτ

a41τ

b�142 τ c31τ

d�132 .

This is reproduced by the di�erential operator

1

2D1pτ41, τ41qB

2τ41�D1pτ41, τ42qBτ41Bτ42 �

1

2D1pτ42, τ42qB

2τ42�

D1pτ42, τ31qBτ42Bτ31 �1

2D1pτ31, τ31qB

2τ31�D1pτ31, τ32qBτ31Bτ32�

1

2D1pτ32, τ32qB

2τ32�D1pτ41, τ31qBτ41Bτ31 �D1pτ41, τ32qBτ41Bτ32�

D1pτ42, τ32qBτ42Bτ32 .

Notice that the coe�cients have to match in order to get a di�erential operator, so theresult is again non trivial. For completeness, we write the values of D1pτij, τklq in termsof the original variables x, y, t:

92

3.5. THE REAL FORMS: THE EUCLIDEAN AND MINKOWSKIAN SIGNATURES

D1pτ41, τ41q � � 2py44y43s211t41t31 � y2

43s11s21t31t32 � y244s11s21t41t42�

y44y43s221t42t32 � 2y44y43s11s21t42t31 � y44y43s11s21t41t32q,

D1pτ41, τ42q � � py243s21s12t31t32 � y2

44s21s12t41t42 � 2y44y43s221t42t32�

2y44y43s11s12t41t31 � 2y44y43s21s12t42t31 � y44y43s21s12t41t32q�

y44y43s11s21t42t31q,

D1pτ42, τ42q � � py244s21s12t41t42 � 2y44y43s

212t41t31 � 2y2

43s12s22t31t32�

y44y43s21s12t42t31 � 2y44y43s12s22t42t31 � 2y44y43s12s22t41t32q�

3y44y43s21s22t42t32q,

D1pτ42, τ31q � ��y43y33s11s12t

231 � y44y33s11s12t41t31 � 2y43y34s11s12t41t31�

y44y34s11s12t241 � y43y34s21s12t41t32 � y43y34s21s12t41t32�

y44y33s11s21t42t31 � 2y44y34s11s21t41t42 � y43y33s21s12t31t32�

y43y33s11s22t31t32 � 2y43y34s21s12t42t31 � y43y34s11s22t42t31�

2y43y34s21s12t42t31 � y43y34s11s22t42t31 � y43y33s21s22t232�

2y43y34s21s22t42t32

�,

D1pτ31, τ31q � � 2py34y33s211t41t31 � y2

33s11s21t31t32 � y234s11s21t41t42�

y34y33s221t42t32 � 2y34y33s11s21t42t31 � y34y33s11s21t41t32q,

D1pτ32, τ32q � � 2py34y33s212t41t31 � y2

33s12s22t31t32 � y234s12s22t41t42�

� y34y33s222t42t32 � 2y34y33s12s22t42t31 � y34y33s12s22t41t32q,

D1pτ41, τ31q � ��y43y34s

211t41t31 � y43y34s

221t42t32 � 2y43y33s11s21t31t32�

y44y33s11s21t42t31 � 2y43y34s11s21t42t31 � y43y34s11s21t41t32�

2y44y34s11s21t41t42

�D1pτ41, τ32q � �

�y43y34s11s12t41t31 � y43y33s21s12t31t32 � y43y34s21s12t42t31�

y43y34s21s12t42t31 � y44y34s21s12t41t42 � y43y34s21s22t42t32

�,

D1pτ42, τ32q � ��y43y34s

212t41t31 � y43y34s

222t42t32 � y44y34s21s12t41t42�

2y43y33s12s22t31t32 � 2y43y34s12s22t42t31 � y43y34s12s22t41t32

�,

D1pτ31, τ32q � ��y2

33s21s12t31t32 � y234s21s12t41t42 � 2y34y33s11s12t41t31�

2y34y33s21s12t42t31 � y34y33s21s12t41t32�

y34y33s11s22t42t31 � 2y34y33s21s22t42t32

�.

3.5 The real forms: the Euclidean and Minkowskian signatures

93

CHAPTER 3. QUANTUM TWISTORS

3.5.1 The real forms in the classical case

Let A be a commutative algebra over C. An involution ι of A is an antilinear mapsatisfying, for f, g P A and α, β P C

ιpαf � βgq � α�ιf � β�ιg, (antilinearity) (3.14)

ιpfgq � ιpfqιpgq, (automorphism) (3.15)

ι � ι � Id. (3.16)

Let us consider the set of �xed points of ι,

Aι � tf P A | ιpfq � fu.

It is easy to see that this is a real algebra whose complexi�cation is A. Aι is a real formof A.

Example 3.5.1. The real Minkowski space.We consider the algebra of the complex Minkowski space OpMq � Crt31, t32, t41, t42s andthe following involution, �

ιMpt31q ιMpt32qιMpt41q ιMpt42q

�

�t31 t41

t32 t42

,

which can be also written simply as

ιMptq � tT .

Using the Pauli matrices (2.4)

t �

�t31 t32

t41 t42

� xµσµ �

�x0 � x3 x1 � ix2

x1 � ix2 x0 � x3

,

so

x0 �1

2pt31 � t42q, x1 �

1

2pt32 � t41q,

x2 �1

2ipt41 � t32q, x3 �

1

2pt31 � t42q,

are �xed points of the involution. In fact, it is easy to see that

OpMqιM � Rrx0, x1, x2, x3s.

�

Example 3.5.2. The Euclidean space.We consider now the following involution on OpMq�

ιEpt31q ιEpt32qιEpt41q ιEpt42q

�

�t42 �t41

�t32 t31

.

94

3.5. THE REAL FORMS: THE EUCLIDEAN AND MINKOWSKIAN SIGNATURES

Another way of expressing it is in terms of the matrix of cofactors,

ιEptq � cofptq.

The combinations

z0 �1

2pt31 � t42q, z1 �

i

2pt32 � t41q,

z2 �1

2pt41 � t32q, z3 �

i

2pt31 � t42q,

are �xed points of ιE, and as before,

OpMqιE � Rrz0, z1, z2, z3s.

�

We are interested now in the real forms of the complex Poincaré plus dilations

that have a coaction on the real algebras. So we start with (2.14)

OpPlq � Crxij, yab, TaisLtdetx � det y � 1u.

We then look for the appropriate involution in OpPlq, denoted as ιPl,M or ιPl,E `preserv-ing' the corresponding real form of the complex Minkowski space. This means that theinvolution has to satisfy

∆ � ιM � ιPl,M b ιM �∆,

∆ � ιE � ιPl,E b ιE �∆.

It is a matter of calculation to check that

ιPl,Mpxq � SpyqT , ιPl,Mpyq � SpxqT , ιPl,MpT q � T T ; (3.17)

ιPl,Epxq � SpxqT , ιPl,Epyq � SpyqT , ιPl,EpT q � cofpT q, (3.18)

are the correct expressions. It is not di�cult to realize that in the Minkowskian case thereal form of the Lorentz group (corresponding to the generators x and y) is SLp2,CqRand in the Euclidean case is SUp2q � SUp2q. One can further check the compatibility ofthese involutions with the coproduct and the antipode

∆ � ιPl,M � ιPl,M b ιPl,M �∆, S � ιPl,M � ιPl,M � S; (3.19)

∆ � ιPl,E � ιPl,E b ιPl,E �∆, S � ιPl,E � ιPl,E � S. (3.20)

95

CHAPTER 3. QUANTUM TWISTORS

3.5.2 The real forms in the quantum case

We have to reconsider the meaning of `real form' in the case of quantum algebras. Wecan try to extend the involutions (3.17, 3.18) to the quantum algebras. We will denotethis extension with the same name since they cannot be confused in the present context.

The �rst thing that we notice is that property (3.15) has to be modi�ed. In fact, theproperty that the involutions ιM, ιE satisfy with respect to the commutation relations(2.4.7) of the complex algebra OpMq is that they are antiautomorphisms, that is

ιMpfgq � ιMpgqιMpfq,

ιEpfgq � ιEpgqιEpfq.

This discards the interpretation of the real form of the non commutative algebra as theset of �xed points of the involution. The other two properties are still satis�ed.When considering the involutions ιPl,M and ιPl,E in the quantum group OqpPlq, we alsoobtain an antiautomorphism of algebras, but now the involution has to be compatiblealso with the Hopf algebra structure. The coproduct is formally the same and properties(3.19, 3.20) are still satis�ed (so the involutions are automorphisms of coalgebras). Onthe other hand, di�erently from the classical case, the involutions do not commute withthe antipode. This is essentially due to the fact that S2 � 1. One can explicitly checkthat

S2 � ιPl,M � S � S � ιPl,M,

S2 � ιPl,E � S � S � ιPl,E. (3.21)

Property (3.16) is still satis�ed, ιPl,M2 � 1 and ιPl,E

2 � 1. Using this fact, (3.21) can bewritten as

pιPlM � Sq2 � 11,

pιPl,E � Sq2 � 11.

All these properties de�ne what is known as a Hopf �-algebra structure (see for example[19]).

De�nition 3.5.3. Let A be a Hopf algebra. We say that it is a Hopf �-algebra if thereexists an antilinear involution ι on A which is an antiautomorphism of algebras and anautomorphism of coalgebras such that

pι � Sq2 � 11,

being S the antipode. �

For example, each real form of a complex Lie algebra corresponds to a �-algebra structurein the corresponding enveloping algebra, seen as a Hopf algebra.

96

3.6. THE DEFORMED QUADRATIC INVARIANT

Remark 3.5.4. Real forms on the star product algebra.The involutions can be pulled back to the star product algebra using the quantizationmaps QM : OpMqrq, q�1s Ñ OqpMq (see (3.2.2)), and QG (see (3.10)) and then extendedto the algebra of smooth functions. The Poisson bracket in terms of the Minkowski spacevariables (xµ) or the Euclidean ones (zµ) is purely imaginary (see (3.8)), as a consequenceof the antiautomorphism property of the involutions.In the case of the quantum groups, the whole Hopf �-algebra structure is pulled back tothe polynomial algebra and then extended to the smooth functions. �

3.6 The deformed quadratic invariant

Let us consider the quantum determinant in OqpMq

Cq � detq

�t32 t31

t42 t41

� t32t41 � q�1t31t42.

Under the coaction of OqpPlq with the translations put to zero (that is for the quantumLorentz times dilation group), the quantum determinant satis�es

∆pCqq � detqy Spdetqxq b Cq,

so if we suppress the dilations, then detqy � 1, detqx � 1 and the determinant is aquantum invariant ,

∆pCqq � 1b Cq.

The invariant Cq can be pulled back to the star product algebra with the quantizationmap QM

Cq � Q�1M pCqq � t41t32 � qt42t31. (3.22)

We can now change to the Minkowski space variables, and the quadratic invariant in thestar product algebra is

Cq � �qpx0q2 � qpx3q2 � px1q2 � px2q2. (3.23)

Cq is the quantum star invariant. Notice that the expressions (3.22, 3.23) depend uponthe quantization map or ordering rule chosen.

97

CHAPTER 3. QUANTUM TWISTORS

98

Conclusions

In this Thesis we have obtained a quantum chiral conformal superspace as a non com-mutative superalgebra that admits the action of the superconformal group SLqp4|1q. Thequantum chiral Minkowski superspace is realized as the big cell inside the quantum con-formal superspace, and the quantum super Poincaré group is properly de�ned as a subsupergroup of the conformal supergroup that preserves the big cell.

In particular, we have used the non obvious property of the chiral conformal superspace,the super Grassmannian of p2|0q-planes inside C4|1, of having an embedding in a super-projective space. The quantization has been performed using explicitly this embedding,thus giving implicitly a deformation of the projective superspace perhaps to be comparedwith the one in Ref. [41].

To obtain a supermanifold which admits the correct real form of the Minkowski andconformal superspaces, one has to go to a larger supergeometric object, namely the �agsupermanifold F p2|0, 2|1, 4|1q [46, 45, 23]. Luckily enough, this super�ag is also projective(which is not true for an arbitrary super�ag, contrary to the non super case), so the samemethod employed here to quantize the Grassmannian can be used for the super�ag.

We obtained a explicitly formula for a star product on polynomials on the complexi�edMinkowski space. This star product has several properties:

• It can be extended to a star product on the conformal space Gp2, 4q. This is doneby gluing the star products computed in each open set (2.11).

• It can be extended to act on smooth functions as a di�erential star product.

• The Poisson bracket is quadratic in the coordinates.

• There is a coaction of the quantum Poincaré group (or the conformal group in thecase of the conformal spacetime) on the star product algebra.

• It has at least two real forms corresponding to the Euclidean and Minkowski signa-tures.

• It can be extended to the superspace (to chiral and real super�elds).

99

CHAPTER 4. CONCLUSIONS

From the physical point of view, since �elds are smooth functions, the di�erentiabilityof the star product gives a hope that one can develop a quantum deformed �eld theory,that is, a �eld theory on the quantum deformed Minkowski space. The departure pointwill be to �nd a generalization of the Lapacian and the Dirac operator associated to thequantum invariant Cq.

One advantage of using the quantum group SLqp4,Cq is that the coalgebra structure isisomorphic to the coalgebra of the classical group SLp4,Cq (see for example Theorem 6.1.8in Ref. [18]). This means that the group law is unchanged, so the Poincaré symmetryprinciple of the �eld theory would be preserved in the quantum deformed case.

100

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106

Appendix 1

{OpGLp4,CqqHopf algebra In this appendix we will sketch the structure of a commutative,non cocommutative Hopf algebra of the polynomials of GLp4,Cq and intuitively why thecoproduct corresponds to the matrix multiplication on the groups itself.We �st consider the group GLp4,Cq, an algebraic group. An element of it is, generically,

g �

����g11 g12 g13 g14

g21 g22 g23 g24

g31 g32 g33 g34

g41 g42 g43 g44

��� , det g � 0.

The algebra of polynomials of GLp4,Cq is the algebra of polynomials in the entries of thematrix, and an extra variable d, which then is set to be the inverse of the determinant,thus forcing the determinant to be di�erent from zero:

OpGLp4,Cqq � Crgij, dsLtd � det g � 1u, i, j � 1, . . . , 4.

If we want to consider the algebra of SLp4,Cq we will have simply

OpSLp4,Cqq � CrgijsLtdet g � 1u, i, j � 1, . . . , 4. (A.1)

In both cases the group law is expressed algebraically as a coproduct, given on thegenerators as

OpGLp4,Cqq ∆ÝÝÝÑ OpGLp4,Cqq b OpGLp4,Cqq

gij ÝÝÝÑ°gik b gkj,

d ÝÝÝÑ db d

i, j, k � 1, . . . , 4, (A.2)

and extended by multiplication to the whole OpGLp4,Cqq. The coproduct is non cocom-mutative, since σ � ∆ � ∆, where σ : OpGLp4,Cqq b OpGLp4,Cqq Ñ OpGLp4,Cqq bOpGLp4,Cqq such that σpf b gq � g b f .

We also have the antipode S, (which corresponds to the inverse on GLp4,Cq),

OpGLp4,Cqq SÝÝÝÑ OpGLp4,Cqq

gij ÝÝÝÑ gij�1 � d p�1qj�iMji

d ÝÝÝÑ det g,

(A.3)

107

APPENDIX A. APPENDIX 1

where Mji is the minor of the matrix g with the row j and the column i deleted.

There is compatibility of these maps. For example, one has

∆pf1f2q � ∆f1∆f2, (A.4)

as well as the properties of associativity and coassociativity of the product and the co-product. There is also a unit and a counit (see Ref. [18], for example), and all this givesto OpGLp4,Cqq the structure of a commutative, non cocommutative Hopf algebra.

Remark A.0.1. Let us see intuitively why the coproduct corresponds to the matrixmultiplication on the group itself. We try now to see an element of OpGLp4,Cqq as afunction over the variety of the group itself. Let us denote the natural injection

O�GLp4,Cq

�b O

�GLp4,Cq

� µGÝÝÝÑ O�GLp4,Cq �GLp4,Cq

�f1 b f2 ÝÝÝÑ f1 � f2

such that pf1 � f2qpg1, g2q � pf1g1qpf2g2q. Then we have that

pµG �∆fqpg1, g2q � fpg1g2q, f P OpGLp4,Cqq.

�

108

Appendix 2

B.1 A basis for the Poincaré quantum group

In this appendix we prove that, given a certain speci�c ordering on the generators of thePoincaré quantum group, the ordered monomials form a basis for its quantum algebra.This is a non trivial result based on the classical work by G. Bergman [57].

B.1.1 Generators and relations for the Poincaré quantum group

Let us consider SLqp4|0q the quantum complex general linear group with even indeter-minates gij subject to the Manin relations (2.27, 2.28, 2.29, 2.30) and the de�nition ofquantum determinant (2.3.3)1 (see [11] sec. 7)2. Inside SLqp4|0q we consider the followingelements, which we write, as usual, in a matrix form:

x �

�g11 g12

g21 g22

, T �

��q�1D23D

�112 D13D

�112

�q�1D24D�112 D14D

�112

y �

�g33 g34

g43 g44

.

As in (2.47), the quantum Poincaré group, OqpPlq as the subring of SLqp4|0q generatedby the elements in the matrices x, y, T de�ned above.

In order to give a presentation for OqpPlq give the commutation relations among thegenerators xij, yab, Tai:

1In this appendix we write the noncommutative generators without the hat `ˆ ' to simplify the notation.2All of the arguments in this appendix hold replacing SLqp4|0q with the general linear quantum group

and the complex �eld with any �eld of characteristic zero.

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APPENDIX B. APPENDIX 2

x11x12 � q�1x12x11, x11x21 � q�1x21x11,

x11x22 � x22x11 � pq�1 � qqx21x12, x12x21 � x21x12,

x12x22 � q�1x22x12, x21x22 � q�1x22x21 (B.1)

y33y34 � q�1y34y33, y33y43 � q�1y43y33,

y33y44 � y44y33 � pq�1 � qqy43y34, y34y43 � y43y34,

y34y44 � q�1y44y34, y43y44 � q�1y44y43, (B.2)

T42T41 � q�1T41T42, T31T41 � q�1T41T31,

T32T41 � T41T32 � pq�1 � qqT42T31, T31T42 � T42T31,

T32T42 � q�1T42T32, T32T31 � q�1T31T32. (B.3)

and for i � 1, 2, a � 3, 4

x1iT32 � T32x1i, x1iT42 � T42x1i, x1iT31 � q�1T31x1i,

x1iT41 � q�1T41x1i, x2iT31 � T31x2i, x2iT41 � T41x2i,

x21Ta2 � q�1Ta2x21 � qpq�1 � qqx11Ta1

x22Ta2 � q�1Ta2x22 � qpq�1 � qqx12Ta1 (B.4)

y33T3a � qT3ay33, y34T3a � qT3ay34, y43T4a � qT4ay43, y44T4a � qT4ay44,

y33T4a � T4ay33, y34T4a � T4ay34, y43T3a � T3ay43, y44T3a � T3ay44. (B.5)

This can be checked by direct computation [10, 11].The entries in x (resp. y) satisfy the Manin commutation relations in dimension 2, thatis,

x �

�g11 g12

g21 g22

�

�a bc d

, y �

�g33 g34

g43 g44

�

�a bc d

ba � qab, ca � qac, db � qbd, dc � qcd,

cb � bc da � ad� pq�1 � qqbc. (B.6)

Moreover, they commute with each other:

xijykl � yklxij.

Similarly one can show that the entries in Tij satisfy the Manin relations, with the order

T �

�T32 T31

T42 T41

�

�a bc d

,

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B.1. A BASIS FOR THE POINCARÉ QUANTUM GROUP

but they do not commute with x and y (B.4, B.5).This provides a presentation of OqpPlq in terms of generators and relations.

OqpPlq � Cqxxij, ykl, TrsyLpIPl

, detqx � detqy � 1q,

where IPlis the ideal generated by the commutation relations (B.1, B.2, B.3, B.4, B.5).

B.1.2 The Diamond Lemma

Let us recall some de�nitions and theorems from the fundamental work by Bergman [57](see also [21] pg 103) 3.

De�nition B.1.1. Let Cqxxiy be the free associative algebra over Cq with generatorsx1, . . . , xn and let

X :� tXI � xi1 � � � xis | I � pi1, . . . , isq, ij P t1, . . . , nuu

be the set of all (unordered) monomials. X is clearly a basis for Cqxxiy. We de�ne on Xan order,  , such that given two monomials x and y, then x   y if the length of x is lessthan the length of y and for equal lengths we apply the lexicographical ordering. �

Let Π � tpXIk , fkq | k � 1, . . . , su be a certain set of pairs XIk P X and fk P Cqxxiy. Wedenote by JΠ the ideal

JΠ � pXIk � fk, k � 1, . . . , sq � OqpPlq.

In our application Π will yield the ideal of the commutation relations for the quantumPoincaré group.

De�nition B.1.2. We say that Π is compatible with the ordering   if fk consists of alinear combination of ordered monomials. �

For example if Mqp2q � Cqxa, b, c, dyLIM , where IM is the ideal of the Manin relations,

we have that

ΠM � tpba, qabq, pca, qacq, pcb, bcq, pdc, qcdq, pdb, qbdq, pda, ad� pq�1 � qqbcq u

is compatible with the ordering a   b   c   d.

We want to �nd a basis consisting of ordered monomials for a Cq-module Cqxxiy{JΠ.Clearly this is not possible for any chosen total order. However, when Π is compatiblewith the order, that is, when the relations XIk � fk behave nicely with respect to thegiven order, then we can device an algorithm to reduce any monomial to a standard

form (namely to writing it as a combination of ordered monomials). This is essentiallythe content of the Diamond Lemma for ring theory that we shall describe below.

We have two problems to solve: �rst, one has to make sure that any procedure to reducea monomial to the standard form terminates, and then one has to make sure that thechosen procedure gives a unique result.

3All of our arguments hold more in general replacing Cq with a commutative ring with 1.

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APPENDIX B. APPENDIX 2

De�nition B.1.3. Assume that we �x a generic set Π as above. Let x, y P X and let rxkybe the linear map of Cqxxiy sending the elements of the form xxiky to xfky and leavingthe rest unchanged. rxky is called a reduction and an element x P X (or more generallyin Cqxxiy) is reduced if rpxq � x for all reductions r. �

In general more than one reduction can be applied to an element. For example if we takethe quantum matrices Mqp2q and ΠM as above, we see that dcba is not reduced, and wehave several ways to proceed to reduce it. We want to make sure that that there are noambiguities, or, in other words, we want to make sure there is a unique reduced elementassociated with it.

De�nition B.1.4. Let x, y, z P X and xik , xil be the �rst elements of two pairs in Π.We say that px, y, z, xik , xilq form an overlapping ambiguity if xik � xy, xil � yz. Theambiguity is resolvable if there are two reductions r and r1 such that rpxikzq � r1pxxilq.In other words, if we can reduce xyz in two di�erent ways, we must obtain the sameresult.Similarly px, y, xik , xilq form an inclusion ambiguity if xik � xxily. The inclusionambiguity is solvable if there are two reductions r and r1 such that rpxikq � r1pxxilyq. �

Theorem B.1.5. (Diamond Lemma). Let R be the ring de�ned by generators andrelations as:

R :� CqxxiyLpXIk � fk, k � 1 . . . sq

If Π � tXIk , fkuk�1,...,s is compatible with the ordering   and all ambiguities are resolv-able, then the set of ordered monomials is a basis for R. Hence R is a free module overCq.Proof . See [57]. �

B.1.3 A basis for the Poincaré quantum group

In this section, we want to apply the Diamond Lemma, to obtain an explicit basis for thequantum algebra of the Poincaré quantum group. Let us �x a total order on the variablesx, y, t as follows:

t32 ¡ t31 ¡ t42 ¡ t41 ¡ x11 ¡ x12 ¡ x21 ¡ x22 ¡ y33 ¡ y34 ¡ y43 ¡ y44.

One sees right away that the relations in IM as described in (B.1, B.2, B.3, B.4, B.5)give raise to a Π compatible with the given order. Furthermore, notice that this order isthe Manin ordering (see [16]) in two dimensions when restricted to each of the sets txiju,tyklu, ttrsu.

As one can readily see, the fact that Π is compatible with the given order ensures thatany reordering procedure terminates.

Theorem B.1.6. Let OqpPlq � Cqxxij, ykl, tilyLIPl

be the algebra corresponding to thequantum Poincaré group. Then, the monomials in the order:

t32 ¡ t31 ¡ t42 ¡ t41 ¡ x11 ¡ x12 ¡ x21 ¡ x22 ¡ y33 ¡ y34 ¡ y43 ¡ y44.

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B.1. A BASIS FOR THE POINCARÉ QUANTUM GROUP

are a basis for OqpPlq.Proof .By the Diamond Lemma B.1.5 we only need to show that all ambiguities are resolvable.We notice that when two generators a, b, q-commute, that is ab � qsba, they behave, asfar the reordering is concerned, exactly as commutative indeterminates. Hence we onlytake into consideration ambiguities where no q-commuting relations appear. The proofconsists in checking directly that all such ambiguities are resolvable.Let us see, as an example of the procedure to follow, how to show that the ambiguityx22x11t32 is resolvable. All the other cases follow the same pattern since the relationshave essentially the same form as far as the reordering procedure is concerned.We shall indicate the application of a reduction with an arrow, as it is customary to do.

px22x11qt32 ÝÑ px11x22 � pq�1 � qqx12x21qt32 ÝÑ x11pq�1t32x22�

�pq�1 � qqt31x12q � pq�1 � qqrx12pq�1t32x21 � pq�1 � qqt31x11qs

ÝÑ q�1t32x11x22 � q�1pq�1 � qqt31x11x12�

�q�1pq�1 � qqt32x12x21 � qpq�1 � qqt31x11x12 �

� q�1t32x11x22 � q�1pq�1 � qqt32x12x21 � p1� q2qt31x11x12.

Similarly

x22px11t32q ÝÑ x22t32x11 ÝÑ pq�1t32x22 � pq�1 � qqt32x12qx11

ÝÑ q�1t32px11x22 � pq�1 � qqx12x21q � p1� q2qt31x11x12.

As one can see the two expressions are the same and reduced, hence we obtain that thisambiguity is resolvable. �

Remark B.1.7. We end the discussion by noticing that the Theorem B.1.6 holds alsofor the order:

x11 ¡ x12 ¡ x21 ¡ x22 ¡ y33 ¡ y34 ¡ y43 ¡ y44 ¡ t32 ¡ t31 ¡ t42 ¡ t41

the proof being the same. �

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APPENDIX B. APPENDIX 2

114