1. 1 A non-perturbative study of Supersymmetric Lattice Gauge Theories Tomohisa Takimi ( 基研 )

2. 2 Introduction SUSY algebra includes infinitesimal translation which is broken on the lattice. Lattice regularization is a non-perturbative tool. But practical application of lattice gauge theory for supersymmetric gauge theory is difficult since Fine-tuning problem occurs to realize the desired continuum limit huge simulation time

3. 3 Exact supercharge on the lattice for a nilpotent supercharge which do not include translation in Extended SUSY Strategy to solve the fine-tuning problem We call as BRST charge

4. 4 Models utilizing nilpotent SUSY 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) Catterall models (Catterall) 2 -d N=(2,2),4-d N=4 super Yang-Mills ( JHEP 11 (2004) 006, JHEP 06 (2005) 031) 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) We will treat 2-d N=(4,4) CKKU’s model

5. 5 Do they really have the desired continuum limit with full supercharge ? Is fine-tuning problem solved ? Non-perturbative investigation Sufficient investigation has not been done ! Our main purpose

6. 6 How do we non-perturbatively study?

7. 7 The desired continuum limit 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, topological field theory and its property must be recovered Topological property (BRST cohomology) Criteria

8. 8 Topological property (action ) We can obtain this value non-perturbatively in the semi-classical limit. this is independent of gauge coupling Because BRST cohomology

9. 9 The aim of my doctor thesis 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 will study on 2 dimensional N=(4,4) CKKU model.

10. 10 Topological field theory in 2-d N= (4,4) continuum action : covariant derivative (adjoint representation) : gauge field (Dijkgraaf and Moore, Commun. Math. Phys. 185 (1997) 411)

11. 11 BRST transformation BRST transformation change the gauge transformation law BRST BRST transformation is not homogeneous of : linear function of : not linear function of BRST partner sets If is homogeneous linear function of def homogeneous of

12. 12 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

13. 13 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 )

14. 14 N=(4,4) CKKU lattice action

15. 15 BRST transformation Fermionic field Bosonic field is not included in If we split the field content as Homogeneous transformation of on the lattice In continuum theory, it is not homogeneous transformation of BRST partner sets

16. 16 * Gauge transformation law does not change under BRST BRST partners sit on same links or sites gauge transformation law is same as BRST partner

17. 17 The BRST closed operators on the N=(4,4) CKKU lattice model must be the BRST exact except for the polynomial of BRST cohomology on the lattice theory (K.Ohta, T.T (2007))

18. 18 The reason 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

19. 19 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 !

20. 20 Result of topological study on the lattice Supersymmetric lattice gauge theory continuum limit a  0 Extended Supersymmetric gauge theory action Topological field theory Really ? We have found a problem in the 2 dimensional N=(4,4) CKKU model

21. 21 Summary Summary We have proposed that the topological property (like as partition function, 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.

22. 22 We apply the criteria to N= (4,4) CKKU model * The model can be written as BRST exact form. *BRST transformation becomes homogeneous transformation on the lattice. *The No-go theorem in the BRST cohomology on the lattice. It becomes clear that there is possibility that N=(4,4) CKKU model does not work well ! This becomes clear by using this criteria. (We do not know this in perturbative level.) It is shown that the criteria is powerful tool.

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24. 24 h: number of genus h-independent constant depend on Parameter of regularization Parameter which decide the additional BRST exact term Weyl group

25. 25 Prospects Applying the criteria to other models (for example Sugino models ) to judge whether they work as supersymmetric lattice theories or not. Clarifying the origin of impossibility to define the BRST cohomology on N=(4,4) CKKU model to construct the model which have desired continuum limit. (Idea: to study the deconstrution)

26. 26 Since N=(2,2) Catterall model can be obtained from N=(4,4) CKKU model, it would be judged by utilizing the topological analysis in N=(4,4) CKKU model

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29. 29 Hilbert space Hilbert space of extended super Yang-Mills: Hilbert space of topological field theory: Topological field theory is obtained from extended super Yang-Mills as a subsector Hilbert space of extended super Yang-Mills Hilbert space of topological field theory

30. 30 Possible virtue of this construction We might be able to analyze the topological property of N=(2,2) Catterall model by utilizing that of topological property on N=(4,4) CKKU

31. 31 If the theory lead to desired continuum limit, continuum limit must permit the realization of topological field theory There we pick up the topological property on the lattice which enable us non-perturbative investigation.

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