1. 1 Lattice Formulation of Two Dimensional Topological Field Theory Tomohisa Takimi ( 理研、川合理論物 理 ) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [hep-lat /0611011] （ Hep にのってない話も含む） 平成 19 年 4 月 27 日 於 金沢大学セミナー

2. 2 Contents 1. Introduction (review, fine-tuning problem, recent development) 2. Our proposal for non-perturbative criteria 3. Topological property in the continuum theory 3.1 BRST exact form of the model 3.2 BRST cohomology (BPS state) 4. Topological property on the lattice (4.0 Construction of SUSY on lattice) 4.1 BRST exact form of the model 4.2 BRST cohomology (BPS state) 5. Summary my work K. Ohta, T.T

3. 3 1. Introduction SUSY algebra includes infinitesimal translation which is broken on the lattice. Supersymmetric gauge theory One solution of hierarchy problem Dynamical SUSY breaking Lattice study may help to get deeper understanding but lattice construction of SUSY field theory is difficult.

4. 4 Fine-tuning problem in present approach Standard action Plaquette 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 continuum. Time for computation becomes huge. ex. N=1 SUSY with matter fields gaugino mass,scalar massfermion mass scalar quartic coupling Difficult to perform numerical analysis

5. 5 Lattice formulations free from fine-tuning Exact supercharge on the lattice for a nilpotent (BRST-like) supercharge in Extended SUSY We call as BRST charge

6. 6 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=4 There 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

7. 7 After the twist, we can reinterpret the extended supersymmetric gauge theory action as an equivalent topological field theory action Extended Supersymmetric gauge theory action Topological Field Theory action Supersymmetric Lattice Gauge Theory action lattice regularization Twisting Nilpotent scalar supercharge is extracted from spinor supercharges is preserved

8. 8 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) 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) 067, JHEP 01 (2005) 016 Phys.Lett. B635 (2006) 218-224 ） Catterall models 2 -d N=(2,2), （杉野さん模型の改良版？実は・・。 ） (JHEP 11 (2004) 006, JHEP 06 (2005) 031) We will treat 2-d N=(4,4) CKKU’s model

9. 9 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

10. 10 2. Our proposal for the non-perturbative study non-perturbative study - ( Topological Study ) -

11. 11 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  0 continuum lattice regularization

12. 12 And the target continuum theory includes a topological field theory as a subsector. Extended Supersymmetric gauge theory Supersymmetric lattice gauge theory Topological field theory continuum limit a  0 Must be realized in 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 index BPS states

13. 13 Topological property (action ) BRST cohomology (BPS state) We can obtain this value non-perturbatively in the semi-classical limit. these are independent of gauge coupling Because

14. 14 The aim A non-perturbative study whether the lattice theories have the 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.

15. 15 In the 2 dimesional N = (4,4) super Yang-Mills theory 3. Topological property in the continuum theories continuum theories - 3.1 BRST exact action 3.2 BRST cohomology (Review)

16. 16 Equivalent topological field theory action 3.1 BRST exact form of the action : covariant derivative (adjoint representation) : gauge field (Dijkgraaf and Moore, Commun. Math. Phys. 185 (1997) 411)

17. 17 BRST transformation BRST transformation change the gauge transformation law BRST BRST partner sets If is homogeneous linear function of def is homogeneous of BRST transformation is not homogeneous of : homogeneous function of : not homogeneous of Here we treat as just the coefficient Homogeneous linear on x Not Homogeneous linear on x

18. 18 3.2 BRST 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

19. 19 not BRST exact !, and are not gauge invariant are BRST cohomology composed by Although (k=1,2) are formally BRST exact This is because BRST transformation change the gauge transformation law (Polynomial of ( ) is trivially BRST cohomology )

20. 20 4.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 action 4.2 BRST cohomology

21. 21 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 model 2 dimensional N=(4,4) CKKU model -

22. 22 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))

23. 23 BRST transformation Fermionic field Bosonic field is not included in If we split the field content as Homogeneous transformation of tangent vector So the transformation can be written as tangent vector on the lattice In continuum theory, it is not homogeneous transformation BRST partner sets They are homogeneous functions of

24. 24 Location on the Lattice * BRST partners sit on same links or sites * Gauge transformation law does not change under BRST

25. 25 BRST cohomology cannot be realized! cannot be realized! The BRST closed operators on the N=(4,4) CKKU lattice model must be the BRST exact except for the polynomial of 4.2 BRST cohomology on the lattice theory (K.Ohta, T.T (2007))

26. 26 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 sinceis homogeneous about 【 3 】 Any field function can be written as

27. 27 From 【4】【4】 【5】【5】, from 【 1 】, in 【 2 】, 【 6 】 transformation commute with gauge transformation : gauge invariant From 【 5 】 【 6 】, BRST closed eigenfunction : must be BRST exact must be BRST exact. ) 【7】【7】 BRST closed function including the field in Must be BRST exact for

28. 28 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】, 【7】,

29. 29 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

30. 30 BRST cohomology must be composed only by BRST cohomology are composed by in the continuum theory on the lattice disagree with each other * BRST cohomology on the lattice * BRST cohomology in the continuum theory Not realized in continuum limit !

31. 31 Supersymmetric lattice gauge theory continuum limit a  0 Extended Supersymmetric gauge theory action Topological field theory Topological field theory on the lattice Really ?

32. 32 Topological field theory continuum limit a  0 Extended Supersymmetric gauge theory Supersymmetric lattice gauge theory Topological field theory One might think the No-go theorem (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

33. 33 lattice spacing ) The discussion via the path (B) Topological observable in the continuum limit via path (B) Representation of on the lattice These satisfy following property (

34. 34 We can expand as And in, it can be written as So the expectation value of this becomes Since Also in this case, Since the BRST transformation is homogeneous,

35. 35 Ward identity There is not Jacobian factor since Since Therefore We cannot realize non-zero with We cannot realize the topological property via path (B)

36. 36 Caution! Caution! In the torus, there might not be non-zero topological observable even in target continuum theory We have to confirm whether this situation stands also on our target gauge theories or not ? Blau, Thompson (hep-th/930144) Witten( Blau, Thompson (hep-th/930144) Witten(J.Geom.Phys.9:303-368,1992 ) In supersymmetric BF theory

37. 37 Topological field theory continuum limit 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!

38. 38 5. 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 on it have the desired continuum limit or not.

39. 39 We apply the criteria to N= (4,4) CKKU model It can be a powerful criteria. if we confirm the target theory include non-zero topological observable There is a possibility that topological property cannot be realized. The target continuum limit might not be realized by including non-perturbative effect.