agt 関係式 (3) 一般化に向け て (string advanced lectures no.20)...
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AGT 関係式 (3) 一般化に向け
て(String Advanced Lectures No.20)
高エネルギー加速器研究機構 (KEK)
素粒子原子核研究所 (IPNS)
柴 正太郎
2010 年 6 月 23 日(水) 12:30-14:30
Contents
1. AGT relation for SU(2) quiver theory
2. Partition function of SU(N) quiver theory
3. Toda theory and W-algebra
4. Generalized AGT relation for SU(N) case
5. Towards AdS/CFT duality of AGT relation
AGT relation for SU(2) quiver
We now consider only the linear quiver gauge theories in AGT relation.We now consider only the linear quiver gauge theories in AGT relation.
Gaiotto’s discussion
An example : SW curve is a sphere with multiple punctures.An example : SW curve is a sphere with multiple punctures.
The Seiberg-Witten curve in this case corresponds to
4-dim N=2 linear quiver SU(2) gauge theory.
Nekrasov instanton partition function
where equals to the conformal block of
Virasoro algebra with for the vertex operators which are
inserted at z=
Liouville correlation function (corresponding n+3-point function)
where is Nekrasov’s full partition function.
(↑ including 1-loop part)
U(1) part
[Alday-Gaiotto-Tachikawa ’09]AGT relation : SU(2) gauge theory AGT relation : SU(2) gauge theory Liouville theory Liouville theory !!
Gauge theory Liouville theory
coupling const. position of punctures
VEV of gauge fields internal momenta
mass of matter fields external momenta
1-loop part DOZZ factors
instanton part conformal blocks
deformation parameters Liouville parameters
4-dim theory : SU(2) quiver gauge theory
2-dim theory : Liouville (A1 Toda) field theory
In this case, the 4-dim theory’s partition function Z and the 2-dim
theory’s correlation function correspond to each other :
central charge :
Now we calculate Nekrasov’s partition function of 4-dim SU(N) quiver
gauge theory as the quantity of interest.
SU(2) case : We consider only SU(2)×…×SU(2) quiver gauge
theories.
SU(N) case : According to Gaiotto’s discussion, we consider, in
general, the
cases of SU(d1) x SU(d2) x … x SU(N) x … x SU(N) x … x SU(d’2) x
SU(d’1) group,
where is non-negative.
SU(N) partition function
Nekrasov’s partition function of Nekrasov’s partition function of 4-dim gauge theory4-dim gauge theory
xx xxx
*
… …x
*
…
…
d’3 – d’2d’2 – d’1d’1
… ………
…
d3 – d2
d2 – d1
d1… ………
1-loop part 1-loop part of partition function of 4-dim quiver gauge theoryof partition function of 4-dim quiver gauge theory
We can obtain it of the analytic
form :
where each factor is defined as
: each factor is a product of double Gamma function!
,
gauge antifund. bifund. fund.
mass massmassflavor symm. of bifund. is U(1)
VEV# of d.o.f. depends on dk
deformation parameters
We obtain it of the expansion form of instanton
number :
where : coupling const. and
and
Instanton part Instanton part of partition function of 4-dim quiver gauge theoryof partition function of 4-dim quiver gauge theory
Young tableau
< Young tableau >
instanton # = # of boxes
leg
arm
Naive assumption is 2-dim AN-1 Toda theory, since Liouville theory is
nothing but A1 Toda theory. This means that the generalized AGT
relation seems
Difference from SU(2) case…
• VEV’s in 4-dim theory and momenta in 2-dim theory have more than
one d.o.f.
In fact, the latter corresponds to the fact that the punctures are
classified with more than one kinds of N-box Young tableaux :
< full-type > < simple-type > < other types >
(cf. In SU(2) case, all these Young tableaux become ones of the same type .)
• In general, we don’t know how to calculate the conformal blocks of
Toda theory.
……
…
…
………
What kind of 2-dim CFT corresponds to 4-dim SU(N) quiver gauge theory?What kind of 2-dim CFT corresponds to 4-dim SU(N) quiver gauge theory?
Action :
Toda field with :
It parametrizes the Cartan subspace of AN-1 algebra.
simple root of AN-1 algebra :
Weyl vector of AN-1 algebra :
metric and Ricci scalar of 2-dim surface
interaction parameters : b (real) and
central charge :
Toda theory and W-algebra
What is AWhat is AN-1N-1 Toda theory? : some extension of Liouville theory Toda theory? : some extension of Liouville theory
• In this theory, there are energy-momentum tensor and higher spin fields
as Noether currents.
• The symmetry algebra of this theory is called W-algebra.
• For the simplest example, in the case of N=3, the generators are defined as
And, their commutation relation is as follows:
which can be regarded as the extension of Virasoro algebra, and where
,
What is AWhat is AN-1N-1 Toda field theory? Toda field theory? (continued)(continued)
We ignore Toda potential
(interaction) at this stage.
• The primary fields are defined as ( is called
‘momentum’) .
• The descendant fields are composed by acting / on the
primary fields as uppering / lowering operators.
• First, we define the highest weight state as usual :
Then we act lowering operators on this state, and obtain various
descendant fields as .
• However, some linear combinations of descendant fields accidentally
satisfy the highest weight condition. They are called null states. For
example, the null states in level-1 descendants are
• As we will see next, we found the fact that these null states in W-
algebra are closely related to the singular behavior of Seiberg-Witten
curve near the punctures. That is, Toda fields whose existence is
predicted by AGT relation may in fact describe the form (or behavior)
of Seiberg-Witten curve.
As usual, we compose the primary, descendant, and null fields.As usual, we compose the primary, descendant, and null fields.
• As we saw, Seiberg-Witten curve is generally represented as
and Laurent expansion near z=z0 of the coefficient function is
generally
• This form is similar to Laurent expansion of W-current (i.e. W-
generators)
• Also, the coefficients satisfy similar equations, except full-type
puncture’s case
This correspondence becomes exact, in some kind of ‘classical’ limit:(which is related to Dijkgraaf-Vafa’s discussion on free fermion’s system?)
• This fact strongly suggests that vertex operators corresponding non-
full-type punctures must be the primary fields which has null states in
their descendants.
The singular behavior of SW curve is related to the null fields of W-algebra.The singular behavior of SW curve is related to the null fields of W-algebra.[Kanno-Matsuo-SS-Tachikawa ’09]
null condition
~ direction of D4 ~ direction of NS5
• If we believe this suggestion, we can conjecture the form of
momentum of Toda field in vertex operators
, which corresponds to each kind of punctures.
• To find the form of vertex operators which have the level-1 null state,
it is useful to consider the screening operator (a special type of vertex
operator)
• We can show that the state satisfies the highest
weight condition, since the screening operator commutes with all the
W-generators.(Note a strange form of a ket, since the screening operator itself has non-zero
momentum.)
• This state doesn’t vanish, if the momentum satisfies
for some j. In this case, the vertex operator has a null state at level
.
The punctures on SW curve corresponds to the ‘degenerate’ fields!The punctures on SW curve corresponds to the ‘degenerate’ fields![Kanno-Matsuo-SS-Tachikawa ’09]
• Therefore, the condition of level-1 null state becomes for
some j.
• It means that the general form of mometum of Toda fields
satisfying this null state condition is
.
Note that this form naturally corresponds to Young tableaux
.
• More generally, the null state condition can be written as
(The factors are abbreviated, since they are only the images under Weyl
transformation.)
• Moreover, from physical state condition (i.e. energy-momentum is
real), we need to choose , instead of naive
generalization:
Here, is the same form of β,
is Weyl vector,
and .
The punctures on SW curve corresponds to the ‘degenerate’ fields!The punctures on SW curve corresponds to the ‘degenerate’ fields!
Generalized AGT relation
Natural form : former’s partition function and latter’s correlation
function
Problems and solutions for its proof
• correspondence between each kind of punctures and vertices:
we can conjecture it, using level-1 null state condition.
< full-type > < simple-type > < other types >
• difficulty for calculation of conformal blocks: null state condition
resolves it again!
[Wyllard ’09][Kanno-Matsuo-SS-Tachikawa ’09]
……
…
…
………
Correspondence : 4-dim SU(N) quiver gauge and 2-dim ACorrespondence : 4-dim SU(N) quiver gauge and 2-dim AN-1N-1 Toda theoryToda theory
• We put the (primary) vertex operators at punctures, and
consider the correlation functions of them:
• In general, the following expansion is valid:
where
and for level-1 descendants,
: Shapovalov matrix
• It means that all correlation functions consist of 3-point functions and
inverse Shapovalov matrices (propagator), where the intermediate
states (descendants) can be classified by Young tableaux.
On calculation of correlation functions of 2-dim AOn calculation of correlation functions of 2-dim AN-1N-1 Toda theory Toda theory
descendants
primaries
In fact, we can obtain it of the factorization form of 3-point functions
and inverse Shapovalov matrices :
3-point function : We can obtain it, if one entry has a null state in
level-1!
wherehighest weight~ simple punc.
On calculation of correlation functions of 2-dim AOn calculation of correlation functions of 2-dim AN-1N-1 Toda theory Toda theory
’
Case of SU(3) quiver gauge theory
SU(3) : already checked successfully. [Wyllard ’09] [Mironov-Morozov ’09]
SU(3) x … x SU(3) : We have checked successfully. [Kanno-Matsuo-SS ’10]
SU(3) x SU(2) : We could check it, but only for restricted cases. [Kanno-Matsuo-
SS ’10]
Case of SU(4) quiver gauge theory
• In this case, there are punctures which are not full-type nor simple-type.
• So we must discuss in order to check our conjucture (of the simplest
example).
• The calculation is complicated because of W4 algebra, but is mostly
streightforward.
Case of SU(∞) quiver gauge theory
• In this case, we consider the system of infinitely many M5-branes, which may
relate to AdS dual system of 11-dim supergravity.
• AdS dual system is already discussed using LLM’s droplet ansatz, which is
also governed by Toda equation. [Gaiotto-Maldacena ’09] →
subject of next talk…
Our plans of current and future research on generalized AGT relationOur plans of current and future research on generalized AGT relation
Towards AdS/CFT of AGT
CFT side : 4-dim SU(N) quiver gauge theory and 2-dim AN-1Toda
theory
• 4-dim theory is conformal.
• The system preserves eight supersymmetries.
AdS side : the system with AdS5 and S2 factor and eight
supersymmetries
• This is nothing but the analytic continuation of LLM’s system in M-
theory.
• Moreover, when we concentrate on the neighborhood of punctures
on Seiberg-Witten curve, the system gets the
additional S1 ~ U(1) symmetry.
• According to LLM’s discussion, such system can
be analyzed using 3-dim electricity system:
[Lin-Lunin-Maldacena ’04]
[Gaiotto-Maldacena ’09]