1 Lattice Formulation of Two Dimensional Topological Field Theory Tomohisa Takimi ( 基研、理研 ) K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2 hep-lat （ Hep にのってない話も含む）

2 Contents 1. Introduction (our proposal) 2. Our proposal for non-perturbative study 3. Topological property in the continuum theory 3.1 BRST exact form of the model 3.2 Partition function (Witten index) 3.3 BRST cohomology (BPS state) 4. Topological property on the lattice (4.0 Construction of SUSY on lattice) 4.1 BRST exact form of the model 4.2 Partition function (Witten index) 4.3 BRST cohomology (BPS state) 5. Summary my work K. Ohta, T.T

3 1. Introduction SUSY algebra includes infinitesimal translation which is broken on the lattice. Supersymmetric gauge theory One solution of hierarchy problem Dynamical SUSY breaking Lattice study may help to get deeper understanding but lattice construction of SUSY field theory is difficult.

4 Fine-tuning problem in present approach Standard action Plaquette gauge action + Wilson or Overlap fermion action Violation of SUSY for finite lattice spacing. Many SUSY breaking terms appear; Fine-tuning is required to recover SUSY in continuum. Time for computation becomes huge. ex. N=1 SUSY with matter fields gaugino mass,scalar massfermion mass scalar quartic coupling Difficult to perform numerical analysis

5 Lattice formulations free from fine-tuning Exact supercharge on the lattice for a nilpotent (BRST-like) supercharge in Extended SUSY We call as BRST charge

6 Twist in Extended SUSY Redefine the Lorentz algebra by a diagonal subgroup of the Lorentz and the R-symmetry in the extended SUSY ex. d=2, N=2 d=4, N=4 There are some scalar supercharges under this diagonal subgroup. If we pick up the charges, they become nilpotent supersymmetry generator which does not include infinitesimal translation in their algebra. (E.Witten, Commun. Math. Phys. 117 (1988) 353, N.Marcus, Nucl. Phys. B431 (1994) 3-77

7 After the twist, we can reinterpret the extended supersymmetric gauge theory action as an equivalent topological field theory action Extended Supersymmetric gauge theory action Topological Field Theory action Supersymmetric Lattice Gauge Theory action lattice regularization Twisting Nilpotent scalar supercharge is extracted from spinor supercharges is preserved

8 Models utilizing nilpotent SUSY from Twisting 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) ) We will treat 2-d N=(4,4) CKKU’s model

9 Do they really have the desired continuum theory with full supersymmetry ? 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

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

11 We look at that the lattice model actions are lattice regularization of topological field theory action equivalent to the target continuum action Extended Supersymmetric gauge theory action Topological Field Theory action Supersymmetric Lattice Gauge Theory action limit a  0 continuum lattice regularization

12 And the target continuum theory 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, even including quantum effect, topological field theory and its property must be recovered Witten index BPS states

13 Topological property (action ) Partition function( Witten index) BRST cohomology(BPS state) We can obtain these value non-perturbatively in the semi-classical limit. these are independent of gauge coupling Because

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 In the 2 dimesional N = (4,4) super Yang-Mills theory 3. Topological property in the continuum theories continuum theories BRST exact action 3.2 Partition function 3.3 BRST cohomology (Review)

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

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

Partition function (Witten index) || It should be checked whether the partition function of lattice theory realizes this in the continuum limit : Partition function of continuum theory explicit form ( Gerasimov and Shatasvli hep-lat/ )

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

20 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 )

21 4.Topological property 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 Partition function 4.3 BRST cohomology

22 Dimensional reduction of 6 dimensional super Yang-Mills theory Orbifolding by in global symmetry 2-dimensional lattice structure in the field degrees of freedom Deconstruction kinetic term in 2-dim (Cohen-Kaplan-Katz-Unsal JHEP 12 (2003) 031) The model 2 dimensional N=(4,4) CKKU model -

23 To investigate the topological properties we rewrite the N=(4,4) CKKU action as BRST exact form. 4.1 BRST exact form of the lattice action （ K.Ohta ， T.T (2007))

24 BRST transformation Fermionic field Bosonic field is not included in If we split the field content as Homogeneous transformation of tangent vector So the transformation can be written as tangent vector on the lattice In continuum theory, it is not homogeneous transformation of BRST partner sets They are Linear functions of

25 Location on the Lattice * BRST partners sit on same links or sites * Gauge transformation law does not change under BRST

Partition function (Witten index) (K.Ohta, T.T (2007)) We will compare this with that of the target continuum theory : Partition function of continuum theory (1) Problem: How do we carry out the path integral (1) ?

27 Exact integration by Nicolai Map (1) is exactly obtained by the semi-classical limit. action can be simplified in semi-classical limit as By the change of variables () () Integration over becomes Gaussian integration over this is first time to discover the Nicolai map in supersymmetric lattice gauge theory (Nicolai Map)

28 Then we can simplify the (1) as We only have to perform the last integral and compare with continuum results

29 No-go theorem The BRST closed operators on the N=(4,4) CKKU lattice model must be the BRST exact except for the polynomial of 4.3 BRST cohomology on the lattice theory (K.Ohta, T.T (2007))

30 proof 【 1 】 the BRST transformation : and following fermionic operator Compose the number operator as counting the number of fields within 【 2 】 commute with the number operator sinceis homogeneous about 【 3 】 Any field function can be written as

31 From 【4】【4】 【5】【5】, from 【 1 】, in 【 2 】, 【 6 】 transformation commute with gauge transformation : gauge invariant From 【 5 】 【 6 】, BRST closed eigenfunction : must be BRST exact must be BRST exact. ) 【7】【7】 BRST closed function including the field in Must be BRST exact for

32 BRST cohomology in BRST closed function in 【 4 】 must come from zero eigenstates namely a term composed only of can be BRST cohomology (End of proof) which does not contain any field in From 【7】, 【7】,

33 Essence of the No-go theorem 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

34 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 !

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

36 5. 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.

37 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. *We discover Nicolai Map and calculate the partition function to compare with the continuum result. *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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39 5. Construction of new class of model From N = (4,4) N = (2,2) Truncating Half degree of freedom of and their BRST partner And their BRST partner In the continuum theory we can obtain N=(2,2) from N =(4,4) Is the N=(2,2) supersymmetric lattice model obtained from N=(4,4) lattice model by using analogous method ? (K.Ohta, T.T (2007)) Non-trivial

40 * N= (2,2) lattice model can be obtained by the suitable truncation of fields in N=(4,4) CKKU lattice model. The N=(2,2) model can preserve same BRST charge. Since we find the BRST exact form of the N=(4,4) CKKU action, we can utilize analogous method in the lattice theory.

41 * We find that the N= (2,2) lattice model is equivalent to N=(2,2) lattice model proposed by Catterall. ( JHEP 11 (2004) 006) It is not expected since Catterall model does not originally use the matrix model construction Since Catterall model is obtained from N=(4,4) CKKU model, Topological analysis on N=(4,4) would be utilized in N=(2,2) Catterall model to judge whether the Catterall model work well. (future work)

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

44 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)

45 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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48 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

49 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

50 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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