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### Relations among Supersymmetric Lattice Gauge Theories

So Matsuura

@ Niels Bohr Institute

based on the works

arXiv:0704.2696

arXiv:0706.3007

arXiv:0708.4129

arXiv:0709.4193

with P.H.Damgaard

Isaac Newton Institute

Introduction

Lattice Gauge Theory

- Constructive definition of a gauge theory

- Non-perturbative analysis by numerical simulations

If supersymmetric gauge theories are constructed on a lattice,

- It gives a “definition” of the theory.

- We can compute any physical observable even if it is not restricted

by the SUSY algebra.

- We can compare results in strong coupling region directly with,

say, the AdS/CFT correspondence.

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It seems impossible to construct a SUSY invariant theory on a lattice.

SUSY invariant action in continuum space-time

Suppose an action is written as

; superfield

Essentially, a SUSY generator can be represented as

Variation of the action

Leibniz rule

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lattice theory

difference operator

differential operator

deformed Leibniz rule

Leibniz rule

It seems impossible to keep all SUSY on a lattice.

QUESTION

Can we keep a part of SUSY on a lattice?

Yes!

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Map of lattice theories with SUSY on a lattice

A.Cohen, E.Katz, D.Kaplan, M.Unsal,

S. Catterall

Orbifold lattice theories

Catterall\'s lattice theories

an extension

P.H.Damgaard, S.M.

equivalent

some reduction

P.H.Damgaard, S.M.

T.Takimi

Lattice theories by

the “link approach”

Sugino’s lattice theories

F.Sugino

A. D\'Adda, I. Kanamori, N. Kawamoto, K. Nagata

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Contents

- Introduction
- Classification of Orbifold Lattice Gauge Theories
- Exact Vacuum Energy of Orbifold Theory
- Relation with Catterall’s Supersymmetric Lattice Gauge Theory
- Equivalence between the Orbifolding and the Link Approach
- Conclusion

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Classification of Orbifold Lattice Gauge Theories

OUT LINE

Mother Theory

A supersymmetric Yang-Mills matrix theory

STEP 1

orbifold projection

Orbifolded Matrix Theory

A matrix theory with “scalar supercharges”

(a lattice formulation without kinetic terms)

STEP 2

deconstruction

Orbifold Lattice Theory

A lattice theory with scalar supercharges

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Yang-Mills matrix with 8 SUSY

mother theory 1

mother theory 2

・・・・・

・・・・・

lattice ∞?

lattice 8

lattice 1

lattice 1

lattice 1

lattice 2

lattice 2

Yang-Mills matrix with 16 SUSY

Unknown

mother theory 4

mother theory 3

・・・・・

・・・・・

lattice 1024

lattice 1

lattice 2

lattice ??

lattice 1

lattice 2

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[1] construction of Q=4 orbifold lattice theory

STEP 0

Mother theory with 4 SUSY

A matrix theory that is obtained by dimensional reduction of

Euclidean 4D N=1 SYM theory with a gauge group .

: four hermitian matrices

: a Majorana spinor

Symmetries

maximal U(1) subgroup

1) global symmetry

2) gauge symmetry

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Equivalent expression in which the U(1) symmetries are manifest:

U(1) charges

We can take any

linear combination.

where

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with

The variation of the action is zero if and only if the SUSY parameters are trivial;

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STEP1

We consider a transformation generated by

where and

: clock matrix

We keep only components that are invariant under this transformation.

simple example

z(1)

projection of a matrix with U(1) charge 1

z(2)

projection by

z(3)

z(4)

each block is an MxM matrix

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1

Orbifolded action

Substituting the projected field,

we obtain

Projection of the supersymmetry

The supersymmetry parameters have definite U(1) charges:

They become non-trivial

after orbifolding.

The only preserved supersymmetry is the one corresponding to k.

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STEP2

We introduce kinetic terms and a lattice spacing by

Finally, we get an action:

where

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Classification of the theories

The lattice action depends on

- two vectors and .
- two real numbers and .

How the theory depends on them?

a physical interpretation

a space-time lattice

an abstract lattice

: linear mapping

Impose

・・・☆

The continuum theory should be Lorentz invariant

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The kinetic terms in the continuum limit

The condition ☆ determines the linear mapping f as

The lattice theory is unique and on a square lattice.

the continuum theory

2D N=(2,2) SYM theory with the gauge group U(M)

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[2] construction of Q=8 orbifold lattice theory

Mother theory with 8 SUSY

A matrix theory that is obtained by dimensional reduction of

Euclidean 6D N=1 SYM theory.

: six hermitian matrices

: independent four-component spinors

Symmetries

maximal U(1) subgroup

1) global symmetry

2) gauge symmetry

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Equivalent expression in which the U(1) symmetries are manifest:

U(1) charges

linear combinations of

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Classification of the theories

generates the lattice

Dimensionality of the lattice

the dimensionality of the lattice

the number of linearly independent

vectors in

Preserved supersymmetry on the lattice

The supercharges corresponding to scalar fermions are preserved.

- At least one SUSY corresponding to his preserved.
- SUSY enhances if another U(1) charges of a fermion becomes zero.

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(1) 3D lattice with 1 SUSY (three linearly independent )

continuum

limit

3D SUSY Yang-Mills theory

with 8 SUSY

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(2-2) 2D lattice with 2 SUSY

(2-3) 2D lattice with 2 SUSY

(3) 2D lattice with 1 SUSY (an example)

The common continuum limit is 2D N=(4,4) SUSY Yang-Mills theory.

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- There are additional three kinds of 2D lattice theories obtained

by shifting only two bosons, say, and as

The continuum theory is the same.

- We can construct 4D, 3D and 2D lattice theories from the mother theory

with sixteen supercharges (IKKT matrix theory).

In particular, the 4D theory is a lattice formulation of the 4D N=4 SYM theory.

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Exact Vacuum Energy of the Orbifold Theories

The classical moduli space of the orbifold lattice theories are parametrized by

the vacuum expectation values of

the potential terms

up to gauge transformations.

QUESTION

Can we estimate quantum corrections to the vacuum energy?

It seems non-trivial since the supersymmetry is almost broken.

- contributions from higher-loops
- non-perturbative contributions

We can estimate the exact vacuum energy in this case!

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- The orbifold theories have a BRST symmetry on a lattice.
- The actions can be written in Q-exact forms.

The partition function does not depend on the coupling constant;

The vacuum energy estimated in the 1-loop level is exact.

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For both the case it is easy to show that

the 1-loop contribution to the partition function is equal to 1.

The vacuum energy of the orbifold lattice theories constructed from

the mother theories with 4 and 8 SUSY never receive quantum corrections.

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Relation with Catterall’s Supersymmetric Lattice Gauge Theory

Brief review of Catterall’s formulation

Starting with the topologically twisted 2D N=(2,2) SYM,

where Q is a BRST charge acting the fields as

nilpotent up to a gauge transformation

The lattice theory is obtained by the following three steps.

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The theory is latticised by putting fields on a lattice corresponding

to the tensor structures;

tensors

vectors

scalars

STEP 2

Complexified fields are introduced to make the action real;

tensors

vectors

STEP 3

The field strength and the covariant derivatives are replaced by

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Then we obtain a lattice action,

- A BRST symmetry is preserved on a lattice.
- The path-integral is carried out along the real line,
- The other supersymmetries are shown to be

restored in the continuum limit by numerical

simulations.

- By restricting the complexified fields in a different way,

we obtain Sugino’s lattice formulation.

S.Catterall 2006

F.Sugino 2004

T.Takimi 2007

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This prescription is automatically reproduced by orbifolding.

The only assumption is a complexification.

Let us consider the mother theory with 4 SUSY in a Q-exact form;

where Q acts on the fields as

We complexified the matrices and extend the action as

where d is a auxiliary field and we have also doubled h as .

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By this extension, the theory acquires extra U(1) symmetries,

and the charge assignment for the fields is

Then we can define the corresponding orbifold projection.

The orbifold projected action is obtained by substituting

into the action.

Furthermore, instead of shifting and , we replace them as

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Then we obtain a lattice action,

This is equivalent with Catterall’s formulation.

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- The BRST symmetry is enhanced by the complexification;

with

which satisfies .

- This method can be applied to other SUSY gauge theories.

(ex) 4D N=2 SYM theory, etc...

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Equivalence between the Orbifolding and the Link Approach

Recall the U(1) charges of the fields in the mother theory with 4 SUSY

new U(1) charges

three-component vectors

obtained from

We can carry out the orbifold projection using these U(1) charges.

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A typical example

This action is completely the same with

the one obtained by the link approach.

A. D\'Adda, I. Kanamori, N. Kawamoto, K. Nagata (2005)

The lattice action given by the link approach

is obtained by orbifolding procedure.

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The SUSY is completely broken because of the discussion given above.

but

They claim that all the supercharges are preserved on a lattice

in adeformed sense.

deformed SUSY in the mother theory

Consider the “supersymmetry transformation” in the mother theory

with non-trivial

Instead of the usual Leibniz rule,

a matrix made from

the shift matrix.

let us impose a modified Leibniz rule by hand,

for each of and

F.Bruckmann, S.Catterall,

M.de Kok (2006)

They satisfy

although there is some discussion in whether this is consistent or not….

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Conclusion

- We classified the lattice theories constructed from the mother theories with 4 and 8 supercharges.
- We showed that the vacuum energy of the orbifold lattice theories does not receive any quantum correction.
- We showed that the formulation given by Catterall can be understood in terms of the orbifolding procedure.
- We showed that the SUSY lattice theories obtained by the link approach are equivalent to the orbifold lattice theories.

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Future Problems

- Lattice theories constructed from the mother theory with 16 supercharges (IKKT matrix theory)
- Numerical simulations
- Connection to the superstring theory

- classification of the theories
- structure of the quantum vacuum
- including 4D N=4 SYM theory
- AdS/CFT correspondence in terms of lattice theories?

- recovering of the supersymmetries in the continuum limit
- comparison with exact results
- non-BPS operators

orbifold

D-instantons

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