Maximal super Yang-Mills theories on curved background with off-shell supercharges総合研究大学院大学藤塚 理史

共同研究者： 吉田 豊 氏 (KEK), 本多 正純 氏 ( 総研大 /KEK)

based on M. F, M. Honda and Y. Yoshida, arxiv:1209.4320[hep-th]

2012. 10.24. String Advanced Lectures (SAL) at KEK

Our motivationGauge/Gravity correspondence

SUGRA solution for N Dp-branes

[Maldacena ‘97]

The well known example:

dual !?

Different views of low energy effective theory on D-branes (or M-branes)

(p+1)-dim. U(N) SYM

・ N D3-branes

at near horizon

In non-conformal field theory, the correspondence is also expected.

ex.)

[Itzhaki-Maldacena-Sonnenschein-Yankielowicz ‘98]

Our motivationGauge/Gravity correspondence

SUGRA solution for N Dp-branes

[Maldacena ‘97]

dual !?

Different views of low energy effective theory on D-branes (or M-branes)

(p+1)-dim. U(N) SYMat near horizon

This duality is a strong/weak duality:

strong coupling region weak coupling region

“Localization” method

Localization

In these days it has been performed various exact calculations using “ Localization ” method in SUSY gauge theories.

・ a Supercharge Q such that

・ V such that

Deformation of the expectation value:

Then we note

Localization

In these days it has been performed various exact calculations using “ Localization ” method in SUSY gauge theories.

・ a Supercharge Q such that

Then we note

Off-shell

Localization

In these days it has been performed various exact calculations using “ Localization ” method in SUSY gauge theories.

・ a Supercharge Q such that

Off-shell

If we consider it on flat space,

“infrared effect” and “flat directions” divergence !

compact space mass terms

Localization

If we consider it on flat space,

“infrared effect” and “flat directions” divergence !

compact space mass terms

Ex) 4D N=4 SYM on [Pestun ‘07]

etc…

[Pestun ‘07]

[Kapustin-Willet-Yaakov ‘09]

[Hosomichi –Seong-Terashima ‘12]

[Hama-Hosomichi-Lee ‘11], [Imamura-Yokoyama ‘11]

[Hama-Hosomichi ‘12]

[Imamura ‘12]

[Gang ‘09]

[Hama-Hosomichi-Lee ‘10], [Jafferis ‘10]

[Imamura-Yokoyama ‘12]

Many off-shell SUSY theories on curved space have been studied in these days:

Round sphere

Squashed sphere

Others

4D N=1 [Festuccia-Seiberg ‘11][Dumitrescu-Festuccia-Seiberg ‘12]

The more the number of SUSY and dimension grows, the more difficult we construct the off-shell SUSY theories on curved space generally.

It has been constructed partially by Berkovits. [Berkovits ‘93]

However its general formalism has not been known.

Off-shell SUSY

ex.) off-shell maximal SYM on flat space

Rigid SUSY on curved space

ex.)

We can construct off-shell maximal SYM on curved space on which a Killing spinor exists.

Our research purpose

Maximal SYM It’s important for gauge/gravity duality.

Off-shell formulation on curved space

Localization

Main result

Off-shell maximal SYM on flat sp.[Berkovits ‘93]

Contents

1. Off-shell maximal SYM on flat sp.

2. Off-shell maximal SYM on curved sp.

3. Some examples

4. Summary and discussions

1. Off-shell maximal SYM on flat sp.

SUSY tr.

Notation 　

where

Berkovits method [Berkovits ‘93]

on-shell SYM on flat space:

where is a 16 components Majorana-Weyl spinor, and .

maximal SYMdred

Charge conjugation matrix:

Note　

where is a constant bosonic spinor.

Notation 　

where

Berkovits method [Berkovits ‘93]

on-shell SYM on flat space:

where is a 16 components Majorana-Weyl spinor, and .

Charge conjugation matrix:

In off-shell,

7-(bosonic) auxiliary fields

where is a (bosonic) pure imaginary auxiliary field.

where depends on , and is (bosonic) spinor satisfying

Off-shell maximal SYM on flat space

SUSY tr.

For any nonzero , there exist which satisfy above constraint.

solution

The number of off-shell supercharges

[Berkovits ‘93][Evans ‘94]

Given any , we can construct which solves the constraints.

linear in

conventional SUSY

d.o.f of the number of off-shell supercharges

We can construct the solution which has 9 off-shell supercharges at least.

more than 9 ??

16-components

The number of off-shell supercharges

We impose the restriction to as

(1). 8 off-shell supercharges

By using the ,

Reduce “d.o.f of “ to 1/2 8 off-shell supercharges

Next in the case of 9 off-shell charges solution…

We have to introduce concrete notation.

16-components

the solution of 8 the solution of 9

eigenspinors of

In this representation,

the solution with 8 off-shell charges:

where is the anti-symmetric matrix satisfying

Notation

eigenspinors of

In this representation,

the solution with 8 off-shell charges:

(2). 9 off-shell supercharges

Note in 8 off-shell charges,

We can construct a solution in which is nonzero:

(2). 9 off-shell supercharges

Note in 8 off-shell charges,

We can construct a solution in which is nonzero:

Introduce a matrix:

Then,

9 off-shell supercharges

2. Off-shell maximal SYM on curved sp.

On curved space

SUST tr.Same as the flat one

Constant spinor doesn’t exist on the curved space in general.

Note

On curved space

SUST tr.Same as the flat one

Constant spinor doesn’t exist on the curved space in general.

Note

Then,

On curved space

Then,

The condition for invariance is

Parallel spinors

The condition for the invariance is

Existence of the above spinors can be characterized by the holonomy group.

[Hitchin ‘74][Wang ‘89]

For example these don’t include spheres.

is the odd product of internal gamma matrices.

Killing spinors extension

where is a constant that depends on a space, and

The above eq. implies

Examples of spaces

We take ,

We take ,

where is satisfied by .

[Hijazi ‘86]

These have been also classified:

Next we consider whether SUSY theories can be constructed on curved space on which Killing spinor exists.

On curved space

Then,

The condition for invariance is

So the action is not invariant.

Deformation of the action and SUSY tr.

Class 1 (d=4)

We modify the action and transformation in the following way,

SUSY tr.

Using the Killing spinor eq.

Note that this is the equivalent to the well known theory on conformally flat space.

[Pestun ‘07] etc.ex.)

Thus, the action

is invariant under the transformation

SUSY algebra

We consider the square of the SUSY tr. of the each field,

Class 2 ( )

We modify the action and transformation in the following way,

SUSY tr.

There is a unique nontrivial solution,

Using the Killing spinor eq.

SUSY algebra

We consider the square of the SUSY tr. of the each field,

The dilatation vanishes automatically in this class because of anti-symmetry of .

Thus, the action

is invariant under the transformation

3. Some examples

BMN matrix model [Berenstein-Maldacena-Nastase ‘02]

If we integrate out, this is the on-shell BMN matrix model.

We take d=1 and in class 2, then

SUSY tr.

BMN matrix model [Berenstein-Maldacena-Nastase ‘02]

β-function and Wilson loop of

・ Non-perturbative formulation of

[Ishiki-Shimasaki-Tsuchiya ‘11]

conformal map

[Ishii-Ishiki-Shimasaki-Tsuchiya ‘08]

・ Gravity dual corresponding to theory around each vacuum [Lin-Maldacena ‘06]

flat direction

Large-N equivalence

6D N=(1,1) SYM on

We take d=6 and in class 2.

SUSY tr.

3D N=8 SYM on

There are 2 ways of constructing this theory:

(1). Applying to the class 2 directly

i.e. we take d=3 and in class 2.

(2). Dimensional reduction of the class 1 on to

These theories are different!

main difference

R-symmetry:

reduction from 4D.

(1) (2)

4. Summary and discussions

・ We have constructed off-shell maximal SYM on curved space on which a Killing spinor exists.

・ This class of the space contains and so on.

・ We have also constructed the different maximal SYM with same number of supercharges on same space.

Ex.) d=3, N=8 SYM on

Summary

Future work

・ Gauge/Gravity duality

ex.)

・ Localization of BMN matrix model

Non-perturbative verification of the large-N equivalence

・ Extending to more larger class of curved space

ex.) spaces which include a connection and so on.

Supplements

The rewriting of Killing spinor eq.

Killing spinor eq.

We can decompose as

Then we can rewrite the Killing spinor eq.

where D is the Dirac op.

Here we take

and we define

Then,

Therefore the existence of the Killing spinors can be characterized by the holonomy group similarly:

Killing spinors

Killing spinor eq.

Here we introduce “cone” over

Then the Killing spinor eq. can be rewritten as

where is covariant derivative on the cone.

-action: subgroup of isometry .

Since Killing spinors on are constants along , so the former is also the Killing spinors on .

Therefore the number of off-shell supercharges is 4 at least.

Also there are orbifolds in which exist Killing spinors :

Also the solution of the Killing spinor eq. is

where is any constant spinor.

The solution of the Killing spinor eq. is

The solution for Killing spinor eq.

We give the metric:

Also the solution of the Killing spinor eq. is

where is any constant spinor.

The solution of the Killing spinor eq. is

We give the metric:

Furthermore we can construct another class of maximal SYM on with off-shell SUSY :

d=3, N=8 SYM on

dred

The action of the class 2 is

Remarks

・ In the case of d=2 and R<0 ( ), the mass terms of scalar fields become negative.

・ In the case of d≠4, the action becomes non-hermitian because of the 3-pt. term of scalar fields and fermionic mass-term.

Integrability condition

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