Current Status of Link Approach for Twisted Lattice SUSY Noboru Kawamoto Hokkaido University, Sapporo, Japan
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1 Current Status of Link Approach for Twisted Lattice SUSY Noboru Kawamoto Hokkaido University, Sapporo, Japan In collaboration with S.Arianos, A.D adda, A.Feo, I.Kanamori, K.Nagata, J.Saito
2 Plan of talk 1. Introduction 2. Summary of Link approach 3. Claimed inconsistency 4. Essence of difficulties 5. A new proposal
3 Success of Lattice Regularization Lattice QCD Exact gauge symmetry Broken Lorentz invariance 2-dim. quantum gravity Random lattice ( dynamical triangulation, matrix model) Lattice gauge theory + Gravity on random lattice Regularization of fermion on a lattice Lattice SUSY What is the guiding principle?
4 SUSY algebra on a Lattice Difficulties No Poincare invariance on lattice No small fermionic parameter (difference operator) No Leibniz rule 3) Chiral fermion on lattice Doubling of fermions Unbalance of degrees of freedom between #boson #fermion
5 Continuum A Clue to a possible solution to 3) Kato, Miyake, N.K., Tsukioka, Uchida (Two dimensional Abelian BF) Auxiliary field Off-shell invariance Nilpotency of BRS charge s N=D=2 Twisted SUSY Kato,N.K.&Uchida
6 N=D=2 SUSY Dirac-Kaehler Twist Dirac-Kaehler fermion N=D=2 Twisted SUSY Continuum Lattice: on a Lattice
7 3) N=D=2 SUSY Dirac-Kaehler Twist Dirac-Kaehler fermion i : flavour? Extended SUSY suffix super charges in d-dim. y 2x 2-dim. N=2 3-dim. N=4 4-dim. N=4 Dirac-Kaehler twisting #boson = #fermion
8 Lattice Fermions Free Dirac Staggered Naïve (Kluberg-Stern et.al. Gliozzi) (N.K. & J.Smit) Kogut-Susskind (N.K. & I.Kanamori) Dirac-Kaehler Ivanenko&Landau 28 y 2x i : flavour?
9 (2) Difference Operator No Leibniz rule Modified Leibniz rule left-right shift symmetric
10 To prove exact translational invariance of lattice action we need modified Leibniz rule Translational shift modified Leibniz Matrix representation: left-right shift symmetric
11 We need a modified Leibniz rule for too! Compatibility of Shifts
12 Cond. for Twisted N=D=2 Symm. Choice Solutions Asymm. Choice Twisted N=D=2 Lattice SUSY Algebra Equivalent to orbifold construction: by Kaplan et.al.
13 3) Massless fermion propagator on a lattice (poles) doubling of fermions dual Dirac-Kaehler fermion
14 Lattice SUSY Difficulties 1) No Poincare invariance 2) No Leibniz rule 3) (Chiral) fermion species doublers Our Solution 1) Discrete SUSY algebra 2) A non-commutativity for the difference operator B modified Leibniz rule 3) Dirac-Kaehler twist SUSY algebra Extended SUSY d.o.f. BRST charge = scalar twisted SUSY
15 1), 2) Difference Operator Shifted Leibniz rule (A) Non-commutative (B) Shifted Leibniz rule
16 N=D=2 Twisted Super Yang-Mills Introduce Bosonic & Fermionic Link variables Gauge trans. Scalar fields in SYM multiplet
17 N=D=2 SUSY Algebra (continuum) Dirac-KaehlerTwist: (Ghost > matter fermion)
18 Lattice Formulation of N=D=2 Twisted SYM c.f. N=D=2 Twisted SYM multiplet in continuum spacetime : Covariant expression w.r.t.
19 N=D=2 Twisted Lattice SUSY Algebra for SYM Shifted Anti-commutator
20 Jacobi Identities Define fermionic link components Auxiliary Field
21 Fermionic Link Fields Auxiliary Field
22 Twisted N=2 Lattice SUSY Transformation Shifts of Fields Twisted SUSY Algebra closes off-shell
23 Twisted N=2 Super Yang-Mills Action Action has twisted SUSY exact form. Off-shell SUSY invariance for all twisted super charges.
24 Bosonic part of the Action
25 Fermionic part of the Action (1) (2) (1) (2)
26 Higer dimensional extension is possible: 3-dim. N=4 super Yang-Mills
27 Origin of inconsistency (non-gauge case) Bruckmann Kok Presentation by N=D=1 model with superfield Superfield: Matrix representation Super charge: (one step translation) Boson: Fermion:
28 and can be taken diagonal SUSY transformation Similarly, diagonal:
29 Superfields product similarly inconsistency while since
30 This equality is accidental because of the choice of diagonal. In general may not be diagonal matrix. For example, if the scalar fields are defined not only on sites but also on links then, there will be one off-diagonal elements and thus. In general superfields are not commutative on the lattice and this is the natural consequence of it. In the products of component fields the product order should be arranged to inherit the original superfields order. If the order is properly kept:
31 Possible origin for the problem SUSY transformation is asymmetric between left and right operation, while translation is left-right symmetric. Translation: SUSY transformation: This generates ordering ambiguity!
32 Exact SUSY invariance? We need modified Leibniz rule for super charges too. With the modified Leibniz rule we can construct twisted SUSY algebra representation on a lattice. We can construct exact twisted SUSY algebra on a lattice. Then we can construct twisted supercharge exact action parallel to continuum formulation. Example of action: twisted SUSY algebra SUSY transformation of the action Exact SUSY invariance with the modified Leibniz rule, if we keep the proper order of the component fields.
33 Exact SUSY on a Lattice?? Twisted lattice SUSY algebra is a modification of ordinary algebra with shifted (anti-)commutator on the lattice. Shifted commutator is needed to show exact translational invariance of lattice actions. This modification of algebra reminds us of the Modification of lattice chiral transformation for Ginzberg-Wilson relation.
34 However, for super Yang-Mills another problem: Link hole by SUSY operation operation generates a link hole to the action: no gauge invariance? The gauge invariance is lost, since the edges of the links are gauge dependent. ( Bruckmann, Kok, Catterall)
35 But, there is, however, one free parameter which corresponds to the shift of and can be taken to be 0. Thus the corresponding supercharge s doesn t create link holes and thus SUSY and gauge invariance are assured. (Bruckmann, Kok, Catterall, Damgaard, Matsuura) for asymmetric choice of the parameters corresponds to the orbifold construction by Kaplan et. al. for symmetric choice of the parameters are not SUSY invariant!
36 A possible solution We claim: if there is covariantly constant super parameter which has opposite shift of and commutes with all the super covariant derivatives: lattice SUSY and gauge invariant! compensates the link holes. We need to prove the existence of covariantly constant super parameter.
37 A trial We need to find covariantly constant super parameter We need to introduce non-commutative structure. This can be realized by the following observation:
38 We introduce further matrix structure:
39 Gauge transformation After taking off the shift operators, loose a Link nature and space-time points shrink to a point. Reduced model No solution yet!
40 for asymmetric choice of the parameters corresponds to the orbifold construction by Kaplan et. al. for symmetric choice of the parameters are not SUSY invariant! Our Claim There is one free parameter in the formulation and physical quantities may not depend on it. The symmetric choice of takes into account the dual lattice as well and geometrically most symmetric and may lead to the continuum limit quickly (Unsal).
41 A New Proposal for the first problem This part will be presented later when it s ready.
42 Summary of the Proposal Half translation = Lattice SUSY transformation Lattice SUSY exact half translation inv. broken naïve statistics Lattice QCD exact gauge inv. broken Lorentz inv.
43 Summary Discrete Twisted SUSY algebra is realized on the Lattice with modified Leibniz rule. Ordering ambiguity for the product of component fields is cleared up and the product order should inherits the original order of the superfields for the asymmetric definition of SUSY. We need covariantly constant super parameter for lattice super Yang-Mills. No proof yet! We propose that lattice SUSY transformation can be identified as half-translation with new super parameters.
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