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Lattice Formulation of Two Dimensional Topological Field Theory

Tomohisa Takimi (RIKEN,Japan)

K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [hep-lat /0611011] (and more)

August 3rd 2007 　 Lattice 2007 @Regensburg, Deutschland

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1. Introduction1. Introduction

Lattice construction of SUSY gauge theory is difficult.

Fine-tuning problem SUSY breaking

Difficult* taking continuum limit

* numerical study

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Candidate to solve fine-tuning problem

A lattice model of Extended SUSY

preserving a partial SUSY

: does not include the translation

(BRST charge of TFT (topological field theory))

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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 the

ories ( 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))

(other studies D’Adda, Kanamori, Kawamoto, Nagata (arXiv 0707:3533,Phys.Lett.B633:645-652,2006),F.Bruckmann, M.de Kok (Phys.Rev.D73:074511,2006)M.Harada, S.Pinsky(Phys.Rev. D71 (2005) 065013 ))

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Do they really recover the target continuum theory ?

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continuumlimit a 0

Lattice

Target continuum theory

Perturbative studies

All right!All right!

CKKU JHEP 08 (2003) 024, JHEP 12 (2003) 031,

Sugino JHEP 01 (2004) 015, Onogi, T.T Phys.Rev. D72 (2005) 074504, etc

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Non-perturbative study

Analytic investigationby the study of

Topological Field Theory

No sufficient result

S.Catterall JHEP 0704:015,2007. H.Suzuki arXiv:0706.

1392 J.Giedt hep-lat/0405021. hep-lat/0312020 etc

numerical

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Topological fieldtheory

Must be realized

Non-perturbative quantity

Non-perturbative study

Lattice

Target continuum theory

BRST-cohomology

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

2-d N=(4,4) CKKU

Topological fieldtheory Forbidden

Imply

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2.1 The target continuum theory (2-d N=(4,4))

: 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?

Questions

　　　　 is set of homogeneous linear function of

　　 is homogeneous transformation of

def

( : coefficient)

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Answer for (I) and (II)

change the gauge transformation law

BRST

(I) is not homogeneous : not homogeneous of

(II)

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2.2 BRST cohomology in the continuum theory

satisfyingdescent

relation

Integration of over k-homology cycle

BRST-cohomology

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

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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BRST exact form .

3.1 Two dimensional N=(4,4) CKKU action　（ K.Ohta ， T.T (2007))

Set of Fields

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BRST transformation on the on the latticelattice

(I)Homogeneous transformation of(I)Homogeneous transformation of BRST partner sets

In continuum theory,

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

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tangent vectortangent vector

with Number operator as

counts the number of fields in

closed term including the field of exact

form 　

Homogeneous property

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(II)Gauge symmetry under on the lattice

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

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BRST cohomology BRST cohomology cannot be realizcannot be realized!ed!

Only the polynomial ofcan be BRST cohomology

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

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Essence of the proof of the result

closed terms including the fields in

exact form

(II) (II) does not change gauge does not change gauge transformationtransformation

: gauge invariant

: gauge invariant

must be BRST exact .

(I) (I) Homogeneous property ofHomogeneous property of

Only polynomial of can be BRST cohomology

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N=(4,4) CKKU model Target theory

Topological field theory

BRST cohomology must be composed only by

BRST cohomology are composed by

N=(4,4) CKKU model without mass term wouldnot recover the target theory non-perturbatively

Topological fieldtheory

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5. Summary5. Summary

The topological property (like as BRST cohomology) could be used as a non-perturbative criteria to judge wheth

er 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 without mass term

The target continuum limit would not be realized

It can be a powerful criteria.

Implication by an Implication by an explicit form. explicit form.

Perturbative studydid not show it

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The realization is difficult due to the independence of gauge parameters

BRST cohomology

Topological quantity defined by the inner product of homology and the cohomology

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

Vn Vn+i

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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