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

2 Contents 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 A non-perturbative criterion whether the recent strategy (model) solve the problem Judge Difficulty to take the desired continuum limit Perturbative-level discussion

1. Introduction Supersymmetric gauge theory One solution of hierarchy problem Dark Matter, AdS/CFT correspondence Important issue for particle physics 3 *Dynamical SUSY breaking. *Study of AdS/CFT Non-perturbative study is important

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

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

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

7 Example). N=1 SUSY with matter fields gaugino mass,scalar massfermion mass scalar quartic coupling Computation time grows as the power of the number of the relevant parameters By standard lattice action. (Plaquette gauge action + Wilson or Overlap fermion action) too many4 parameters (2) SUSY case No preserved subgroup

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

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

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

11 CKKU models (Cohen-Kaplan-Katz-Unsal) 2-d N=(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) ） Geometrical approach Catterall (JHEP 11 (2004) 006, JHEP 06 (2005) 031) (Relationship between them: SUSY lattice gauge models with the T.T (JHEP 07 (2007) 010))

12 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) Non-perturbative investigation Sufficient investigation has not been done ! Our main purpose Do they have the desired target continuum limit with full supersymmetry ?

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

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

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

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

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

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

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

21 (I) What is homogeneous ? 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) BRST transformation change the gauge transformation law BRST (I)BRST transformation is not homogeneous of : homogeneous function of : not homogeneous of (II)

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

24 not BRST exact ! not gauge invariant formally BRST exact change the gauge transformation law(II) Due to (II) can be BRST cohomology BRST exact(gauge invariant quantity)

25 4.Topological Field theory 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

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

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

28 tangent vector can be written as tangent vector Due to homogeneous property of 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 comment on homogeneous property(1)

29 has a definite number of fields in can be written as Any term in a general function of fields Ex) A general function :Polynomial of comment on homogeneous property(2) :Eigenvelue 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 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))

32 【1】【1】 【 2 】 commute with gauge transformation : gauge invariant for Proof with Consider From : gauge invariant : must be BRST exact must be BRST exact. Only have BRST cohomology (end of proof)

33 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 !

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

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

36 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 (

37 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, since ! We cannot realize the topological property via path (B)

38 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! The 2-d N=(4,4) CKKU lattice model would not have the desired continuum limit!

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) Homogeneous property of BRST transformation on the lattice. (II) BRST transformation does not change the gauge transformation laws. (I)and (II) plays the crucial role. These relate with the gauge transformation law on the lattice. Gauge parameters are defined on each sites as the independent parameters. VnVn+i topology

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

43 What is the continuum limit ? Matrix model without space-time (Polynomial of )0-formAll right * IR effects and the topological quantity * The destruction of lattice structure soft susy breaking mass term is required Non-trivial IR effect Only the consideration of UV artifact not sufficient. not sufficient. Dynamical lattice spacing by the deconstruction which can fluctuateLattice spacing infinity