Current Status of Link Approach for Twisted Lattice SUSY€¦ · Current Status of Link Approach...
Transcript of Current Status of Link Approach for Twisted Lattice SUSY€¦ · Current Status of Link Approach...
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Current Status ofLink 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
TexPoint fonts used in EMF.Read the TexPoint manual before you delete this box.: A
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Plan of talk
1. Introduction 2. Summary of Link approach3. Claimed “inconsistency”4. Essence of difficulties 5. A new proposal
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Success of Lattice Regularization
Lattice QCD Exact gauge symmetry Broken Lorentz invariance
2-dim. quantum gravityRandom 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 ?
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SUSY algebra on a Lattice
1) No Poincare invariance on lattice
2)(difference operator)
No Leibniz rule
3) Chiral fermion on lattice Doubling of fermions
Unbalance of degrees of freedom between #boson #fermion
Difficulties
No small fermionic parameter
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A Clue to a possible solution to 3)
(Two dimensional Abelian BF)
Nilpotency of BRS charge s
Auxiliary fieldOff-shell invariance
Kato,N.K.&Uchida
Continuum
N=D=2 Twisted SUSY
Kato, Miyake, N.K., Tsukioka, Uchida
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N=D=2 Twisted SUSY
N=D=2 SUSY
Dirac-Kaehler Twist
Continuum Lattice:
on a Lattice
Dirac-Kaehler fermion
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N=D=2 SUSY
Dirac-Kaehler Twist
Dirac-Kaehler fermion
i : flavour ? Extended SUSYsuffix
y 2x
2-dim. N=2 3-dim. N=44-dim. N=4
#boson = #fermion
3)
super charges in d-dim.
Dirac-Kaehler twisting
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Lattice FermionsFree Dirac
Naïve Staggered
Kogut-Susskind
Dirac-Kaehler
(N.K. & J.Smit)
(N.K. & I.Kanamori)
(Kluberg-Stern et.al.&Gliozzi)
y 2x
Ivanenko&Landau ‘28
i : flavour ?
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Difference Operator No Leibniz rule ? (2)
Modified Leibniz rule
left-right shift symmetric
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To prove exact translational invariance of lattice action we need modified Leibniz rule
Matrix representation:
left-right shift symmetric
Translational shift modified Leibniz
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Compatibility of Shifts
We need a modified Leibniz rule for too !
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Symm. Choice
Asymm. Choice
Twisted N=D=2 Lattice SUSY Algebra
Cond. for Twisted N=D=2
Solutions
Equivalent to orbifold construction: by Kaplan et.al.
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3) Massless fermion propagator on a lattice
(poles)
doubling of fermions
dual
Dirac-Kaehler fermion
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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
Lattice SUSY
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Difference Operator Shifted Leibniz rule
(A) Non-commutative
(B) Shifted Leibniz rule
1), 2)
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N=D=2 Twisted Super Yang-Mills
Gauge trans.
Scalar fieldsin SYM multiplet
Introduce Bosonic & Fermionic Link variables
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N=D=2 SUSY Algebra
Dirac-KaehlerTwist:
(Ghost > matter fermion)
(continuum)
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Lattice Formulation of N=D=2 Twisted SYMc.f. N=D=2 Twisted SYM multiplet in continuum spacetime
: Covariant expression w.r.t.
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N=D=2 Twisted Lattice SUSY Algebra for SYM
“Shifted” Anti-commutator
…
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Jacobi Identities
…
Define fermionic link components
…
Auxiliary Field
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Fermionic Link Fields
Auxiliary Field
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Twisted N=2 Lattice SUSY Transformation Shifts of Fields
Twisted SUSY Algebra closes off-shell
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Twisted N=2 Super Yang-Mills Action
Action has twisted SUSY exact form. Off-shell SUSY invariancefor all twisted super charges.
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Bosonic part of the Action
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Fermionic part of the Action
…
…
(1)
(2)
(1) (2)
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Higer dimensional extension is possible:
3-dim. N=4 super Yang-Mills
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Origin of “inconsistency”
Presentation by N=D=1 model with superfield
Superfield:
Super charge: (one step translation)
Boson:
Fermion:
Matrix representation
BruckmannKok
(non-gauge case)
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and can be taken diagonal
SUSY transformation
diagonal:Similarly,
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Superfields product
similarly
while since
“inconsistency”
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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:
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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 !
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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.
Exact SUSY invariance ?
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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.
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However, for super Yang-Mills another problem:
The gauge invariance is lost, since the edges of the links are gauge dependent. ( Bruckmann, Kok, Catterall)
Link hole by SUSY operation
operation generates a link hole to the action:
no gauge invariance ?
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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)
But,
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 !
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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:
We need to prove the existence of covariantly constant super parameter.
compensates the link holes.
lattice SUSY and gauge invariant !
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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:
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We introduce further matrix structure:
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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 !
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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).
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A New Proposal for the first problem
This part will be presented later when it’s ready.
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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.
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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.