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Lattice Formulation of　　　　　 Topological Field theory

Tomohisa Takimi (NCTU)Ref) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2

[hep-lat /0611011] (Too simple)

Ref) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2

[arXiv:0710.0438] (more correct)

7th Dec 2007 at (NCTS)

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ContentsContents1. Introduction (review, fine-tuning problems, recent strategy

for them )2. Motivation of topological study on the

lattice 3. Topological property in the continuum

theory 3.1 About the continuum theory 3.2 BRST cohomology (BPS state) 4. Topological property on the lattice 　　 4.1 About the model 4.2 BRST cohomology (BPS state) 5. Summary

A non-perturbative criterion whether the recent strategy (model) solve the problem

Judge

Difficulty to take the desired continuum limit

Perturbative-level discussion

1. Introduction1. Introduction Supersymmetric gauge theory

One solution of hierarchy problem Dark Matter, AdS/CFT correspondence

Important issue for particle physics

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*Dynamical SUSY breaking. *Study of AdS/CFT

Non-perturbative study is important

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Lattice: Lattice: A non-perturbative methodApplication to SUSY theory difficultApplication to SUSY theory difficult..

Fine-tuning problem

SUSY breaking Difficult

* taking continuum limit* numerical study

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Fine-tuning problem

Difficult to perform numerical analysisTime for computation becomes huge.

To take the desired continuum limit.

SUSY SUSY casecaseViolation is too hard to repair the symmetry at the limit.

in the standard action. (Plaquette gauge action + Wilson or Overlap fermion action)

Many SUSY breaking counter terms appear;

is required.

prevents the restoration of the symmetry

Fine-Fine-tuningtuningTuning of the too many parameters.

(To suppress the breaking term effects)

Whole symmetry must be recovered at the limit

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(1) 　 Lorentz symmetry in 4-d theoryLorentz symmetry is also broken on

the lattice

Relevant counter terms are forbidden by the subgroup !

Subgroup (90o

rotation) is still preserved -

Symmetry breaking term

How is the situation terrible ?

Let us compare with the Lorentz symmetry case.

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Example). N=1 SUSY with matter fields

gaugino mass, scalar mass

fermion massscalar quartic coupling

Computation time grows as the power of the number of the relevant parameters

By standard lattice action.(Plaquette gauge action + Wilson or Overlap fermion action)

too many4 parameters

(2) 　 SUSY case

No preserved subgroup

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Lattice formulations free from fine-tuning

We call as BRST charge

{ ,Q}=P_

P

Q

A lattice model of Extended SUSY

preserving a partial SUSY

: does not include the translation

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Twist in the Extended SUSY

Redefine the Lorentz algebra

.

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

Phys. B431 (1994) 3-77

by a diagonal subgroup of (Lorentz) (R-symmetry)

Ex) d=2, N=2

d=4, N=4

they do not include in their algebra

Scalar supercharges under , BRST

charge

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Extended Supersymmetric gauge theory action

Topological Field

Theory action Supersymmetric Lattice Gauge

Theory action latticeregularization

Twisting

BRST charge is extracted from spinor

charges

is preserved

equivalent

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CKKU models (Cohen-Kaplan-Katz-Unsal)

2-d N=(4,4),3-d N=4, 4-d N=4 etc. super Yang-Mills theories

( JHEP 08 (2003) 024, JHEP 12 (2003) 031, JHEP 09 (2005) 042)

Sugino models (JHEP 01 (2004) 015, JHEP 03 (2004) 067, JHEP 01

(2005) 016 Phys.Lett. B635 (2006) 218-224 ） 　　 Geometrical approach 　　

Catterall 　 (JHEP 11 (2004) 006, JHEP 06 (2005) 031)

(Relationship between them:

SUSY lattice gauge models with the

T.T (JHEP 07 (2007) 010))

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Do they really solve fine-tuning problem?

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

Do they have the desired target continuum limit with full supersymmetry ?

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( Topological Study ) -

2.2. Our proposal for the Our proposal for the non-non-

perturbative studyperturbative study -

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Extended Supersymmetric gauge theory action

Topological Field

Theory action Supersymmetric Lattice Gauge

Theory action

limit a 0continuum

latticeregularization

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Topological fieldtheory

Must be realized

Non-perturbative

quantity

How to perform the Non-perturbative investigation

Lattice

Target continuum theory

BRST-cohomol

ogy

For 2-d N=(4,4) CKKU models

2-d N=(4,4)

CKKU

Forbidden

Imply

The target continuum theory includes a topological field theory as a subsector.

Judge

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Why it is non-perturbaitve? (action )

BRST cohomology (BPS state)

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

these are independent of gauge coupling

Because

Hilbert space of topological field theory:

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The aim

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

through the study of topological property on the lattice

We investigate it in 2-d N=(4,4) CKKU model.

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

3. Topological field theory 3. Topological field theory in the in the continuum theoriescontinuum theories -

3.1 About the continuum theory

3.2 BRST cohomology

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Equivalent topological field theory action

3.1 About the continuum theory

: covariant derivative(adjoint

representation) : gauge field

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

(Set of Fields)

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BRST transformation BRST partner sets

(I) Is BRST transformation homogeneous ?

(II) Does change the gauge transformation laws?

Let’s consider

If 　　　　 is set of homogeneous linear function of

　　　 is homogeneous transformation of

def

( is just the coefficient)

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(I) What is homogeneous ?

ex) For function ex) For function

We define the homogeneous of as follows

homogeneous

not homogeneous

We treat as coefficient for discussion of homogeneous of

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Answer for (I) and (II) BRST

transformation change the gauge transformation law

BRST

(I)BRST transformation is not homogeneous of : homogeneous

function of : not homogeneous of

(II)

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3.2 BRST cohomology in the continuum theory

satisfies so-called

descent relation

Integration of over k-homology cycle ( on torus)

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

homology 1-cycle

BRST-cohomology

are BRST cohomology composed by

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not BRST exact !

not gauge invariant

formally BRST exact

change the gauge transformation law(II)

Due to (II) can be BRST cohomology

BRST exact (gauge invariant quantity)

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4.Topological Field theory on the lattice4.Topological Field theory 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 BRST cohomology

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N=(4,4) CKKU action as BRST exact form .

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

Fermion

Boson

Set of Fields

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BRST transformation on the on the latticelattice

(I)Homogeneous (I)Homogeneous transformation oftransformation of

BRST partner sets

are homogeneous functions of

In continuum theory,

(I)Not Homogeneous (I)Not Homogeneous transformation oftransformation of

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can be written as tangent vectortangent vectorDue to homogeneous property of

If we introduce fermionic operator

They Compose the number operator as which counts the number of fields within

commute with the number operator sinc

e is homogeneous transformationwhich does not change the number

of fields in

comment on homogeneous property(1)

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has a definite number of fields in

can be written as

Any term in a general function of fields

Ex)

A general function

:Polynomial of

comment on homogeneous property(2)

:Eigenvelue of

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（ II ） Gauge symmetry under and the location of fields

* BRST partners sit on same links or sites

* (II)Gauge transformation laws do not change under BRST transformation

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BRST cohomology BRST cohomology cannot be cannot be realized!realized!

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

must be the BRST exactexcept for the polynomial of

4.2 BRST cohomology on the lattice theory

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

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【1】

【 2 】　　　　　　　　　　　 commute with gauge transformation : gauge

invariant　

for

Proof

with

Consider

From

: gauge invariant

　　:

　must be BRST must be BRST exactexact .

Only have BRST cohomology(end of proof)

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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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Supersymmetric lattice gauge theory

continuumlimit a 0

Extended Supersymmetric gauge theory action

Topological field theory

Topological field theory on the lattice

Really Really ??

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Topological field theory

continuumlimit a 0 Extended

Supersymmetric gauge theory

Supersymmetric lattice gauge theory

Topological field theory

One might think the No-go result (A) has not forbidden the realization of BRST cohomology in the continuum limit in the case (B)

(A)

(B)

Even in case (B), we cannot realize the observables in the continuum limit

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lattice spacing )

The discussion via the path The discussion via the path (B) (B) Topological observable in the

continuum limit via path (B)Representation of on the lattice

These satisfy following property

(

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We can expand as

And in, it can be written as

So the expectation value of this becomes

Sinc

e

Also in this case, Since the BRST transformation is

homogeneous,

since

!

We cannot realize the topological property

via path (B)

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Topological field theory

continuumlimit a 0 Extended

Supersymmetric gauge theory

Supersymmetric lattice gauge theory

Topological field theory

(A)

(B)

The 2-d N=(4,4) CKKU lattice model cannot realize the topological property in the continuum limit!

The 2-d N=(4,4) CKKU lattice model would not have the desired continuum limit!

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5. Summary5. Summary

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

judge whether supersymmetic lattice theories

which preserve BRST charge

have the desired continuum limit or not.

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We apply the criteria to N= (4,4) CKKU model

There is a possibility that topological property cannot be realized.

The target continuum limit might not be realized by including non-perturbative effect.

It can be a powerful criteria.

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Discussion on the No-go result

(I) Homogeneous property of BRST transformation

on the lattice. (II) BRST transformation does not change

the gauge transformation laws.

(I)and (II) plays the crucial role.

These relate with the gauge transformation law on the lattice.

Gauge parameters are defined on each sites as the independent parameters.

Vn Vn+itopology

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The realization is difficult due to the independence of gauge parameters

BRST cohomology

Topological quantity

(Singular gauge transformation)Admissibility condition etc. would be needed

Vn Vn+i

(Intersection number)= 1

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What is the continuum limit ? Matrix model without

space-time(Polynomial of

)0-form All

right

* IR effects and the topological quantity

* The destruction of lattice structure

soft susy breaking mass term is requiredNon-trivial IR Non-trivial IR

effecteffect

Only the consideration of UV artifact Only the consideration of UV artifact

not sufficient.not sufficient.

Dynamical lattice spacing by the deconstructionwhich can fluctuate

Lattice spacing infinity