1 Lattice Formulation of Two Dimensional Topological Field Theory Tomohisa Takimi (RIKEN,Japan) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [hep-lat / ] (and more) August 3rd 2007 Lattice Deutschland

2 1. Introduction Lattice construction of SUSY gauge theory is difficult. Fine-tuning problemSUSY breaking Difficult * taking continuum limit * numerical study

3 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))

4 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) ） 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: ,2006), F.Bruckmann, M.de Kok (Phys.Rev.D73:074511,2006) M.Harada, S.Pinsky(Phys.Rev. D71 (2005) ))

5 Do they really recover the target continuum theory ?

6 continuum limit a  0 Lattice Target continuum theory Perturbative studies All right! CKKU JHEP 08 (2003) 024, JHEP 12 (2003) 031, Sugino JHEP 01 (2004) 015, Onogi, T.T Phys.Rev. D72 (2005) , etc

7 Non-perturbative study Analytic investigation by the study of Topological Field Theory No sufficient result S.Catterall JHEP 0704:015,2007. H.Suzuki arXiv: J.Giedt hep-lat/ hep-lat/ etc numerical

8 Topological field theory 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 field theory Forbidden Imply

9 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)

10 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)

11 Answer for (I) and (II) change the gauge transformation law BRST (I) is not homogeneous : not homogeneous of (II)

BRST cohomology in the continuum theory satisfying descent relation Integration of over k-homology cycle BRST-cohomology (E.Witten, Commun. Math. Phys. 117 (1988) 353) are BRST cohomology composed by

13 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)

14 BRST exact form. 3.1 Two dimensional N=(4,4) CKKU action （ K.Ohta ， T.T (2007)) Set of Fields

15 on the lattice BRST transformation on the lattice (I)Homogeneous transformation of BRST partner sets In continuum theory, (I)Not Homogeneous transformation of

16 tangent vector with Number operator as counts the number of fields in closed term including the field of exact form Homogeneous property

17 (II)Gauge symmetry under on the lattice * (II) Gauge transformation laws do not change under BRST transformation

18 BRST cohomology cannot be realized! cannot be realized! Only the polynomial of can be BRST cohomology 3.2 BRST cohomology on the lattice theory (K.Ohta, T.T (2007))

19 Essence of the proof of the result closed terms including the fields in exact form (II) does not change gauge transformation : gauge invariant must be BRST exact. (I) Homogeneous property of Only polynomial of can be BRST cohomology

20 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 would not recover the target theory non-perturbatively Topological field theory

21 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.

22 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 explicit form. Perturbative study did not show it

23 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 VnVn+i

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27 What is the continuum limit ? Matrix model without space-time (Polynomial of )0-formAll right * IR effects and the topological quantity * The destruction of lattice structure soft susy breaking mass term is required Non-trivial IR effect Only the consideration of UV artifact not sufficient. not sufficient. Dynamical lattice spacing by the deconstruction which can fluctuateLattice spacing infinity