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21pYD-3. 位相的場の理論の格子正則化. 瀧見 知久 ( 理研 ). K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [hep-lat /0611011] (and more) 平成 19 年  9 月 21 日  日本物理学会第 62 回年次大会 於 北海道大学. 1. Introduction. Numerical study using the Lattice regularization of SUSY gauge theory is difficult. SUSY breaking.

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slide1

21pYD-3

位相的場の理論の格子正則化

瀧見 知久(理研)

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

[hep-lat /0611011] (and more)

平成19年 9月21日  日本物理学会第62回年次大会

於 北海道大学

slide2

1. Introduction

Numerical study using the

Lattice regularization of SUSY gauge theory is difficult.

SUSY breaking

Fine-tuning problem

* taking continuum limit

Difficult

* numerical study

slide3

Candidate to solve fine-tuning problem

A lattice model of Extended SUSY

preserving a partial SUSY

generalized by the topological twist

: does not include the translation

(BRST charge of TFT (topological field theory))

slide4

SUSY lattice gauge models with the

    • CKKU models(Cohen-Kaplan-Katz-Unsal)
    • 2-dN=(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:

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 ))

slide5

Do they really recover the

target continuum theory ?

slide6

Perturbative studies

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

Sugino JHEP 01 (2004) 015,

Onogi, T.T Phys.Rev. D72 (2005) 074504, etc

continuum

Target continuum theory

limit a 0

Lattice

All right!

slide7

Non-perturbative study

S.Catterall JHEP 0704:015,2007.H.Suzuki arXiv:0706.1392

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

numerical

No sufficient result

Analytic investigation

by the study of

Topological Field Theory

slide8

Non-perturbative study

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

BRST-cohomology

Imply

Target continuum theory

2-d N=(4,4)

CKKU

Lattice

Topological fieldtheory

Topological fieldtheory

Must be

realized

Forbidden

Non-perturbative quantity

slide9

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

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

: gauge field

(Set of Fields)

slide10

BRST transformation

BRST partner sets

is set of homogeneous linear function of

def

ishomogeneous transformation of

( : coefficient)

Questions

(I) Is BRST transformation homogeneous ?

(II) Does change the gauge transformation laws?

slide11

Answer for (I) and (II)

(II)

change the gauge transformation law

BRST

(I) is not homogeneous

: not homogeneous of

slide12

2.2 BRST cohomology in the continuum theory

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

satisfying

descent relation

Integration of over k-homology cycle

BRST-cohomology

are BRST cohomology composed by

slide13

Due to (II)

can be BRST cohomology

formally BRST exact

not BRST exact !

not gauge invariant

BRST exact

(gauge invariant quantity)

change the gauge transformation law(II)

slide14

3.1 Two dimensional N=(4,4) CKKU action

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

BRST exact form .

Set of Fields

slide15

BRST transformation on the lattice

BRST partner sets

(I)Homogeneous transformation of

In continuum theory,

(I)Not Homogeneous transformation of

slide16

Homogeneous property

tangent vector

with

Number operator as

counts the number of fields in

closed term including the field of

exact form

slide17

(II)Gauge symmetry under on the lattice

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

slide18

3.2 BRST cohomology on the lattice theory

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

BRST cohomology

cannot be realized!

Only the polynomial of

can be BRST cohomology

slide19

Essence of the proof of the result

closed terms including the fields in

must be BRST exact .

(I) Homogeneous property of

exact form

Only polynomial of

can be BRST cohomology

(II) does not change gauge transformation

: gauge invariant

: gauge invariant

slide20

N=(4,4) CKKU model without mass term would

not recover the target theory non-perturbatively

BRST cohomology must be composed only by

Target theory

N=(4,4) CKKU model

Topological field theory

Topological fieldtheory

BRST cohomology are composed by

slide21

5. Summary

The topological property

(like as BRST cohomology)

could be used as

a non-perturbative criteria to judge whether supersymmetic lattice theories

(which preserve BRST charge)

have the desired continuum limit or not.

slide22

We apply the criteria to N= (4,4) CKKU model without mass term

The target continuum limit

would not be realized

Implication by an explicit form.

Perturbative study

did not show it

It could be a powerful criteria.

slide23

BRST cohomology

Topological quantitydefined by the inner product of homology and the cohomology

The realization is difficult

due to the independence

of gauge parameters

Vn

Vn+i

(Singular gauge transformation)

Admissibility condition etc. would be needed

slide27

What is the continuum limit ?

Matrix model without space-time

Dynamical lattice spacing by the deconstruction

which can fluctuate

Lattice spacing infinity

0-form

(Polynomial of )

All right

* The destruction of lattice structure

Non-trivial IR effect

soft susy breaking mass term is required

* IR effects and the topological quantity

Only the consideration of UV artifact

not sufficient.