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A non-perturbative study ofSupersymmetric Lattice Gauge Theories

Tomohisa Takimi ( 基研 )

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ContentsContents1. Introduction (our purposal)2. Our proposal for non-perturbative study3. Topological property in the continuum theory 3.1 BRST exact form of the model 3.2 Partition function (Witten index) 3.3 BRST cohomology (BPS state) 4. Topological property on the lattice 4.1 BRST exact form of the model 4.2 Partition function (Witten index) 4.3 BRST cohomology (BPS state) 5. Construction of new class of model 6. Summary

K.Ohta, T.T(To appear in Prog.Theor.PhysVol.117, No2)

(K.Ohta, T.T(2007))

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1. Introduction1. Introduction

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

Supersymmetric gauge theoryOne solution of hierarchy problem Dynamical SUSY breaking

Lattice study may help to get deeper understandingbut lattice construction of SUSY field theory is diffic

ult.

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Fine-tuning problem in present approach

Standard actionPlaquette gauge action + Wilson or Overlap fermion action

Violation of SUSY for finite lattice spacing.Many SUSY breaking terms appear;Fine-tuning is required to recover SUSY in continuu

m.Time for computation becomes huge.

ex. N=1 SUSY with matter fields

gaugino mass, scalar mass

fermion massscalar quartic coupling

Difficult to perform numerical analysis

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Lattice formulations free from fine-tuning

Exact supercharge on the lattice for a nilpotent (BRST-like) supercharge

in Extended SUSY

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Twist in Extended SUSY

Redefine the Lorentz algebra by a diagonal subgroup of the Lorentz and the R-symmetry

in the extended SUSY ex. d=2, N=2 d=4, N=4There are some scalar supercharges under this diagonal subgroup. If we pick up the charges, they become nilpotent supersymmetry generator which does not include infinitesimal translation in their algebra.

(E.Witten, Commun. Math. Phys. 117 (1988) 353, N.Marcus, Nucl. Phys. B431

(1994) 3-77

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After the twist, we can reinterpret the extendedsupersymmetric gauge theory action as an equivalent topological field theory action

Extended Supersymmetric gauge theory action

Topological Field Theory

action Supersymmetric Lattice

Gauge Theory action latticeregularization

Twisting

Nilpotent scalar supercharge is extracted from spin

or supercharges

is preserved

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Models utilizing nilpotent SUSY from Twisting• 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 and 2-d N=(2,2) Catterall’s model among these.

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Do they really have the desired continuum theory with full supersymmetry ?

Perturbative investigation They have the desired continuum limit

CKKU JHEP 08 (2003) 024, JHEP 12 (2003) 031, Onogi, T.T Phys.Rev. D72 (2005) 074504

Non-perturbative investigation Sufficient investigation has not been done ! Our main purpose

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2.2. Our proposal for the Our proposal for the non-perturbative studynon-perturbative study

-

( Topological Study ) -

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We look at that the lattice model actions are lattice regularization of topological field theory action equivalent to the target continuum action

Extended Supersymmetric gauge theory action

Topological Field Theory

action Supersymmetric Lattice

Gauge Theory action limit a 0continuum

latticeregularization

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And the target continuum theory 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,even including quantum effect,topological field theory and its property must be recovered

Witten indexBPS states

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Topological property (action )

Partition function( Witten index)

BRST cohomology(BPS state)

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

these are independent of gauge coupling

Because

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The aim of this thesis

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

through the study of topological property on the lattice

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In the 2 dimesional N = (4,4) super Yang-Mills theory

3. Topological property in the 3. Topological property in the continuum theoriescontinuum theories -

3.1 BRST exact action3.2 Partition function3.3 BRST cohomology

(Review)

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Equivalent topological field theory action

3.1 BRST exact form of the 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

: linear function of : not linear function of

BRST partner sets

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• 3.2 Partition function (Witten index)

||

It should be checked whether the partition function of lattice theory realizes this in the continuum limit

:Partition function of continuum theory

explicit form

(Gerasimov and Shatasvli hep-lat/0609024)

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3.3BRST cohomology in the continuum theory

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

satisfies so-called descent relation

Integration 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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They are not BRST exact

Although they are BRST transformation of something

, and are not gauge invariant

due to the inhomogeneous term in

gauge transformationThey are BRST cohomology composed by

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4.Topological property on the lattice4.Topological property on the lattice

We investigate in the 2 dimensional = (4,4) CKKU supersymmetric lattice gauge theory

（ K.Ohta ， T.T (2007))

4.1 BRST exact action4.2 Partition function4.3 BRST cohomology

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Dimensional reduction of 6 dimensional super Yang-Mills theory

Orbifolding by in global symmetry

2-dimensional lattice structure in the field degrees of freedom

Deconstruction kinetic term in 2-dim

(Cohen-Kaplan-Katz-Unsal JHEP 12 (2003) 031)

- 4.0 The model2 dimensional N=(4,4) CKKU model -

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To investigate the topological properties we rewrite the N=(4,4) CKKU action as BRST exact form .

4.1 BRST exact form of the lattice action 　（ K.Ohta ， T.T (2007))

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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

So the transformation can be written as tangent vectortangent vector

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

They are Linear functions of partner

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Location on the Lattice

* BRST partners sit on same links or sites

* Gauge transformation law does not change under BRST

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4.2 Partition function (Witten index) (K.Ohta, T.T (2007))

We will compare this with that of the target continuum theory

:Partition function of continuum theory

(1)

Problem: How do we carry out the path integral (1) ?

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Exact integration by Nicolai Map

(1) is exactly obtained by the semi-classical limit.

action can be simplified in semi-classical limit

as

By the change of variables

( )

()Integration over becomes

Gaussian integration overthis is first time to discover the Nicolai map in su

persymmetric lattice gauge theory

(Nicolai Map)

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Then we can simplify the (1) as

We only have to perform the last integral and compare with continuum results

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No-go theorem

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

must be the BRST exactexcept for the polynomial of

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

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proof

【 1】 the BRST transformation :

and following fermionic operator

Compose the number operator as counting the number of fields

within 【 2】 commute with the number operator sinc

e is homogeneous about

【 3】 Any field function

can be written as

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From

【 4】【5】

, from 【 1】 ,

in 【2】 ,

【 6 】 transformation commute with gauge transformation

: gauge invariant

: gauge invariant

From 【 5】 【6】 ,

BRST closed eigenfunction

:

must be BRST exactmust be BRST exact . )

【 7】

BRST closed function including the field in Must be BRST exact

for

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BRST cohomology in BRST closed function in 【 4 】 must come from zero eigenstates

namely a term composed only of can be BRST cohomology (End of proof)

which does not contain any field in

From 【 7】 ,

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Essence of the No-go theorem 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

Topological field theory on the lattice

Really ?Really ?

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

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5. Construction of new class of mod5. Construction of new class of model el

From N =

(4,4)

N = (2,2)

Truncati

ng Half degree of

freedom of

an

d And their BRST their BRST

partnerpartner

In the continuum theory we can obtain N=(2,2)

from N =(4,4)

Is the N=(2,2) supersymmetric lattice model obtained from N=(4,4) lattice model by using analogous method ?

(K.Ohta, T.T (2007))

Non-trivial

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* N= (2,2) lattice model can be obtained by the suitable truncation of fields in N=(4,4) CKKU lattice model. The N=(2,2) model can preserve same BRST charge.

Since we find the BRST exact form of the N=(4,4) CKKU action, we can utilize analogous method inthe lattice theory.

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* We find that the N= (2,2) lattice model is equivalent to N=(2,2) lattice model proposed by Catterall. (JHEP 11 (2004) 006) It is not expected since Catterall model does not originally use the matrix model construction

Since Catterall model is obtained from N=(4,4) CKKU model,Topological analysis on N=(4,4) would be utilized in N=(2,2) Catterall model to judge whether the Catterall model work well.(future work)

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6. Summary6. Summary• 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 transfor

mation on the lattice. *We discover Nicolai Map and calculate the partition fu

nction to compare with the continuum result. *The No-go theorem in the BRST cohomology on the la

ttice.

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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