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Lattice Formulation of Topological Field theory

Lattice Formulation of Topological Field theory. Ref) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [hep-lat /0611011] (Too simple). Tomohisa Takimi (NCTU). Ref) K. Ohta , T.T Prog.Theor . Phys. 117 (2007) No2 [ arXiv:0710.0438 ] (more correct).

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Lattice Formulation of Topological Field theory

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  1. Lattice Formulation ofTopological Field theory Ref) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [hep-lat /0611011] (Too simple) Tomohisa Takimi (NCTU) Ref) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 [arXiv:0710.0438] (more correct) 7th Dec 2007 at (NCTS)

  2. A non-perturbative criterion whether the recent strategy (model) solve the problem Contents Difficulty to take the desired continuum limit 1. Introduction (review, fine-tuning problems, recent strategy for them ) 2. Motivation of topological study on the lattice 3. Topological property in the continuum theory 3.1 About the continuum theory 3.2 BRST cohomology (BPS state) 4. Topological property on the lattice 4.1 About the model 4.2 BRST cohomology (BPS state) 5. Summary Perturbative-level discussion Judge

  3. 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

  4. Lattice: A non-perturbative method Application to SUSY theory difficult. SUSY breaking Fine-tuning problem * taking continuum limit Difficult * numerical study

  5. Fine-tuning problem To take the desired continuum limit. Whole symmetry must be recovered at the limit SUSY case • in the standard action. • (Plaquette gauge action + Wilson or Overlap fermion action) • Violation is too hard to repair the symmetry at the limit. • Many SUSY breaking counter terms appear; • prevents the restoration of the symmetry Fine-tuning is required. • (To suppress the breaking term effects) Tuning of the too many parameters. Time for computation becomes huge. Difficult to perform numerical analysis

  6. How is the situation terrible ? Let us compare with the Lorentz symmetry case. (1)Lorentz symmetry in 4-d theory Lorentz symmetry is also broken on the lattice Subgroup (90o rotation) is still preserved - Symmetry breaking term Relevant counter terms are forbidden by the subgroup !

  7. (2)SUSY case No preserved subgroup Example). N=1 SUSY with matter fields By standard lattice action. • (Plaquette gauge action + Wilson or Overlap fermion action) gaugino mass, scalar mass fermion mass scalar quartic coupling 4 parameters too many Computation time grows as the power of the number of the relevant parameters

  8. 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

  9. 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

  10. 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

  11. 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))

  12. 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

  13. Our proposal for the • non-perturbative study - ( Topological Study ) -

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

  15. 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

  16. 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.

  17. 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.

  18. 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

  19. 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)

  20. 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?

  21. (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

  22. 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

  23. 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

  24. 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)

  25. 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

  26. 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

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

  28. comment on homogeneous property(1) 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 commute with the number operator since is homogeneous transformation which does not change the number of fields in

  29. comment on homogeneous property(2) 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

  30. （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

  31. 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

  32. 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 . :

  33. 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

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

  35. 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)

  36. 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 ) (

  37. 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

  38. 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)

  39. 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.

  40. 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.

  41. 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

  42. 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

  43. 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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