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Tomohisa Takimi (NCTU)

A non-perturbative analytic study of the supersymmetric lattice gauge theory. Tomohisa Takimi (NCTU). Ref) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [hep-lat /0611011] (Too simple). Ref) K. Ohta , T.T Prog.Theor . Phys. 117 (2007) No2

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Tomohisa Takimi (NCTU)

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  1. A non-perturbative analytic study ofthe supersymmetric lattice gauge theory • Tomohisa Takimi (NCTU) Ref) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [hep-lat /0611011] (Too simple) Ref) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [arXiv:0710.0438] (more correct) 14th March 2008 at (NTU) 1

  2. 1. Introduction • Supersymmetric gauge theory • One solution of hierarchy problem • Dark Matter, AdS/CFT correspondence • Important issue for particle physics *Dynamical SUSY breaking. *Study of AdS/CFT Non-perturbative study is important 2

  3. Lattice: A non-perturbative method lattice construction of SUSY field theory is difficult. SUSY breaking Fine-tuning problem * taking continuum limit Difficult * numerical study

  4. Fine-tuning problem • Hard SUSY breaking generates • Many relevant SUSY breaking counter terms Computation time becomes huge ex). N=1 SUSY with matter fields • (with standard lattice action (Plaquette gauge action + Wilson or Overlap fermion)) gaugino mass, scalar mass fermion mass scalar quartic coupling too many! 4 parameters Computation time becomes huge (proportional to power of # of the relevant parameters)

  5. A lattice model of Extended SUSY preserving a partial SUSY Lattice formulations free from fine-tuning : does not include the translation P We call as BRST charge _ { ,Q}=P Q

  6. Twist in the Extended SUSY (E.Witten, Commun. Math. Phys. 117 (1988) 353, N.Marcus, Nucl. Phys. B431 (1994) 3-77 • Redefine the Lorentz algebra by a diagonal subgroup of (Lorentz) (R-symmetry) Ex) d=2, N=2 d=4, N=4 Scalar supercharges under , . BRST charge they do not include in their algebra

  7. Extended Supersymmetric gauge theory action BRST charge is extracted from spinor charges Twisting equivalent Topological Field Theory action Supersymmetric Lattice Gauge Theory action lattice regularization is preserved

  8. 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)) Damgaard, Matsuura (JHEP 08(2007)087)

  9. Do they have the desired target continuum limit with full supersymmetry ? Do they really solve fine-tuning problem? • 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. Our proposal for the • non-perturbative study - ( Topological Study ) -

  11. Extended Supersymmetric gauge theory action Topological Field Theory action Supersymmetric Lattice Gauge Theory action continuum limit a 0 lattice regularization

  12. How to perform the Non-perturbative investigation The target continuum theory includes a topological field theory as a subsector. For 2-d N=(4,4) CKKU models BRST-cohomology Imply Target continuum theory 2-d N=(4,4) Judge CKKU Lattice Topological fieldtheory Must be realized Forbidden Non-perturbative quantity

  13. Why it is non-perturbaitve? (action ) Hilbert space of topological field theory: BRST cohomology (BPS state) these are independent of gauge coupling Because • We can obtain this valuenon-perturbatively • in the semi-classical limit.

  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. 3. Topological field theory in the continuum theories - 3.1 About the continuum theory 3.2 BRST cohomology In the 2 dimensional N = (4,4) super Yang-Mills theory

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

  17. BRST transformation BRST partner sets If is set of homogeneous linear function of def ishomogeneous transformation of ( is just the coefficient) Let’s consider (I) Is BRST transformation homogeneous ? (II) Does change the gauge transformation laws?

  18. (I) What is homogeneous ? ex) For function ex) For function We define the homogeneous of as follows homogeneous not homogeneous We treat as coefficient for discussion of homogeneous of

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

  20. 3.2 BRST cohomology in the continuum theory (E.Witten, Commun. Math. Phys. 117 (1988) 353) satisfies so-called descent relation Integration of over k-homology cycle ( on torus) BRST-cohomology are BRST cohomology composed by homology 1-cycle

  21. 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)

  22. 4.Topological Field theory on the lattice （K.Ohta，T.T (2007)) 4.1 BRST exact action 4.2 BRST cohomology We investigatein the 2 dimensionalN = (4,4) CKKU supersymmetric lattice gauge theory

  23. 4.1 BRST exact form of the lattice action （K.Ohta，T.T (2007)) N=(4,4) CKKU action as BRST exact form . Boson Set of Fields Fermion

  24. BRST transformation on the lattice are homogeneous functions of BRST partner sets (I)Homogeneous transformation of In continuum theory, (I)Not Homogeneous transformation of

  25. homogeneous property and Due to homogeneous property of can be written as tangent vector If we introduce fermionic operator They Compose the number operator as which counts the number of fields within

  26. Operation of the number operator :Eigenvelue of

  27. Eigenfunction decomposition under Any term in a general function of fields has a definite number of fields in Ex) :Eigenvelue of A general function can be written as :Polynomial of

  28. Comment of (2) since is homogeneous transformation which does not change the number of fields in 28

  29. （II）Gauge symmetry under and the location of fields *BRST partners sit on same links or sites *(II)Gauge transformation laws do not change under BRST transformation

  30. 4.2 BRST cohomology on the lattice theory (K.Ohta, T.T (2007)) BRST cohomology 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

  31. Proof Consider Only have BRST cohomology (end of proof) From 【1】 for with 【2】 commute with gauge transformation : gauge invariant : gauge invariant must be BRST exact . :

  32. BRST cohomology must be composed only by Target theory N=(4,4) CKKU model Imply Topological field theory Topological fieldtheory BRST cohomology are composed by

  33. Even in case (B), we cannot realize the observables in the continuum limit One might think the No-go result (A) has not forbidden the realization of BRST cohomology in the continuum limit in the case (B) Extended Supersymmetric gauge theory Supersymmetric lattice gauge theory continuum limit a 0 (B) Topological field theory Topological field theory (A)

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

  35. Also in this case, We cannot realize the topological property via path (B) Since the BRST transformation is homogeneous, We can expand as And in, it can be written as So the expectation value of this becomes ! since Since

  36. The 2-d N=(4,4) CKKU lattice model cannot realize the topological property in the continuum limit! The 2-d N=(4,4) CKKU lattice model would not have the desired continuum limit! Extended Supersymmetric gauge theory Supersymmetric lattice gauge theory continuum limit a 0 (B) Topological field theory Topological field theory (A)

  37. 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 • have the desired continuum limit or not.

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

  39. Discussion on the No-go result (I)and (II) plays the crucial role. (I) Homogeneous property ofBRST transformation on the lattice. (II) BRST transformation does not change the gauge transformation laws. These relate with the gauge transformation law on the lattice. Gauge parameters are defined on each sites as the independent parameters. topology Vn Vn+i

  40. BRST cohomology Topological quantity (Intersection number)= 1 The realization is difficult due to the independence of gauge parameters Vn Vn+i (Singular gauge transformation) Admissibility condition etc. would be needed

  41. 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.

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