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A non-perturbative study ofSupersymmetric Lattice Gauge Theories

Tomohisa Takimi ( 基研 )

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IntroductionIntroduction

SUSY algebra includes infinitesimal translation which is broken on the lattice.

Lattice regularization is a non-perturbative tool.

But practical application of lattice gauge theory

for supersymmetric gauge theory is difficult since

Fine-tuning problem occurs to realize the desired continuum limit huge simulation time

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Exact supercharge on the lattice for a nilpotent supercharge

which do not include translationin Extended SUSY

Strategy to solve the fine-tuning problem

We call as BRST charge

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Models utilizing nilpotent SUSY• CKKU models (Cohen-Kaplan-Katz-Unsal) 2-d N=(4,4),N=(2,2),N=(8,8),3-d N=4,N=8, 4-d N=4 super Yang-Mills theories ( JHEP 08 (2003) 024, JHEP 12 (2003) 031, JHEP 09 (2005) 042)• Catterall models (Catterall) 2 -d N=(2,2),4-d N=4 super Yang-Mills (JHEP 11 (2004) 006, JHEP 06 (2005) 031)

Sugino models 2 -d N=(2,2),N=(4,4),N=(8,8),3-dN=4,N=8, 4-d N=4 super Yang-Mills (JHEP 01 (2004) 015, JHEP 03 (2004) 0

67, JHEP 01 (2005) 016 Phys.Lett. B635 (2006) 218-224)

We will treat 2-d N=(4,4) CKKU’s model

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Do they really have the desired continuum limit with full supercharge ?

Is fine-tuning problem solved ? Non-perturbative investigation

Sufficient investigation has not been done ! Our main purpose

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How do we non-perturbatively study?

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The desired continuum limit includes a topological field theory as a subsector.

Extended Supersymmetric gauge theory

Supersymmetric lattice gauge theory

Topological field theory

continuumlimit a 0

Must be realizedin a 0

So if the theories recover the desired target theory,topological field theory and its property must be recovered

Topological property

(BRST cohomology)

Criteria

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Topological property (action )

We can obtain this value non-perturbatively in the semi-classical limit.

this is independent of gauge coupling

Because

BRST cohomology

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The aim of my doctor thesis

A non-perturbative studywhether the lattice theories havethe desired continuum limit or not

through the study of topological property on the lattice

We will study on 2 dimensional N=(4,4) CKKU model.

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Topological field theory in 2-d N= (4,4) continuum action

: covariant derivative(adjoint representatio

n)

: gauge field

(Dijkgraaf and Moore, Commun. Math. Phys. 185 (1997) 411)

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BRST transformation BRST transformation change the gauge transformation law

BRST

BRST transformation is not homogeneous of : linear function

of : not linear function of

BRST partner sets

If 　　　　 is homogeneous linear function of

def 　　　　

homogeneous of

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BRST cohomology in the continuum theory

The following set of k –form operators, (k=0,1,2) satisfies so-

called descent

relationIntegration of over k-homology cycle ( on torus)

becomes BRST-closed

(E.Witten, Commun. Math. Phys. 117 (1988) 353)

homology 1-cycle

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not BRST exact !

, and are not gauge invariant

are BRST cohomology composed by

Although (k=1,2) are formally BRST exact

This is because BRST transformation change the gauge transformation law

(Polynomial of ( ) is trivially BRST cohomology )

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N=(4,4) CKKU lattice action 　

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BRST transformation

Fermionic field

Bosonic field

is not included in

If we split the field content as

Homogeneous transformation ofHomogeneous transformation of

on the latticeIn continuum theory, it is not homogeneous transformation ofBRST partner sets

16* Gauge transformation law does not change under BRST

BRST partners sit on same links or sites

gauge transformation law is same as BRST partner

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The BRST closed operators on the N=(4,4) CKKU lattice model

must be the BRST exactexcept for the polynomial of

BRST cohomology on the lattice theory(K.Ohta, T.T (2007))

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The reason Lattice BRST transformation is homogeneous about

We can define the number operator of by using another fermionic transformation

Lattice BRST transformation does not change the representation under the gauge transformation

We cannot construct the gauge invariant BRST cohomology by the BRST transformation of gauge variant value

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BRST cohomology must be composed only by

BRST cohomology are composed by

in the continuum theory

　　　　　　　　　　　　 on the lattice

disagree with each otherdisagree with each other

* BRST cohomology on the lattice

* BRST cohomology in the continuum theory

Not realized in continuum limit !

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Result of topological study on the lattice

Supersymmetric lattice gauge theory

continuumlimit a 0 Extended Supersymmetr

ic gauge theory action

Topological field theory

Really ?Really ?

We have found a problem in the 2 dimensional N=(4,4) CKKU model

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SummarySummary

• We have proposed that the topological property (like as partition function, BRST cohomology) should be used as a non-perturbative criteria to judge wheth

er supersymmetic lattice theories which preserve BRST charge on it have the desired continuum limit or not.

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We apply the criteria to N= (4,4) CKKU model *The model can be written as BRST exact form. *BRST transformation becomes homogeneous transfo

rmation on the lattice. *The No-go theorem in the BRST cohomology on the l

attice.

It becomes clear that there is possibility that N=(4,4) CKKU model does not work well !

This becomes clear by using this criteria. (We do not know this in perturbative level.)

It is shown that the criteria is powerful tool.

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h: number of genus

h-independent constant depend on

Parameter of regularization

Parameter which decide the additional BRST exact termWeyl group

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ProspectsApplying the criteria to other models

(for example Sugino models ) to judge whether they work as supersym

metric lattice theories or not.

Clarifying the origin of impossibility to define the BRST cohomology on N=(4,4) CKKU model to construct the model which have desired continuum limit.

(Idea: to study the deconstrution)

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Since N=(2,2) Catterall model can be obtained from N=(4,4) CKKU model, it would be judged by utilizing the topological analysis in N=(4,4) CKKU model

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Hilbert spaceHilbert space of extended super Yang-Mills: Hilbert space of topological field theory:

Topological field theory is obtained from extended super Yang-Mills as a subsector

Hilbert space of extended super Yang-Mills

Hilbert space of topological field theory

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Possible virtue of this construction

We might be able to analyze the topological property of N=(2,2) Catterall model by utilizing that of topological property on N=(4,4) CKKU

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• If the theory lead to desired continuum limit, continuum limit must permit the realization of topological field theory

• There we pick up the topological property on the lattice which enable us non-perturbative investigation.

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