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Tomohisa Takimi ( 基研 )

A non-perturbative study of Supersymmetric Lattice Gauge Theories. Tomohisa Takimi ( 基研 ). Introduction. Lattice regularization is a non-perturbative tool . But practical application of lattice gauge theory

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Tomohisa Takimi ( 基研 )

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

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

  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. Models utilizing nilpotent SUSY • CKKU models (Cohen-Kaplan-Katz-Unsal) • 2-dN=(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. 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. How do we non-perturbatively study?

  7. Criteria The desired continuum limit includes a topological field theory as a subsector. So if the theories recover the desired target theory,topological field theory and its property must be recovered Extended Supersymmetric gauge theory Supersymmetric lattice gauge theory continuum limit a 0 Topological property (BRST cohomology) Topological field theory Must be realized in a 0

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

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

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

  12. BRST cohomology in the continuum theory (E.Witten, Commun. Math. Phys. 117 (1988) 353) 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 homology 1-cycle

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

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

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

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

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

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

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

  21. 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. 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.

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

  24. 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)

  25. 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

  26. 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 topological field theory Hilbert space of extended super Yang-Mills

  27. 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

  28. 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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