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Exact Supersymmetry on the Lattice
Noboru Kawamoto
(Hokkaido University)
CFT and Integrability
In memory of Alexei Zamolodchikov
Dec.18, Seoul
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A bit of history:More than 30 years unsuccessful: Dondi&Nicolai (1977) Many theoretical and numerical investigations No realization of exact SUSY on the lattice untill 2003
Later developments: Exact lattice SUSY was realized only for nilpotent super charge: Kaplan, Katz, Unsal, Cohen (2003), Sugino, Catterall….
No-Go theorem for Leibniz rule of difference operatorKato, Sakamoto and So (2008)
A) Link approach: noncommutative D’Adda, Kanamori, N.K. Nagata.(2005,6,8)
Hopf algebra invariance : D’adda, N.K. Saito (2010)
B) Super doubler approach: nonlocal D’Adda, Feo, Kanamori, N.K. Saito (2011,12)
New approaches for exact SUSY on the lattice
10 years of Sapporo-Torino collaboration
Exact SUSY on the Lattice
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Two major difficulties for lattice SUSY
(0) Loss of Poincare invariance: discrete invariance ?
(1) Difference operator does not satisfy Leibniz rule.
(2) Species doublers of lattice chiral fermion copies appear: unbalance of d.o.f. between bosons and fermions
Let’s consider the simplest lattice SUSY algebra:
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breakdown of Leibniz rule
cancelation
difference operator
Modified Leibniz rule
symmetric
forward
Link nature
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Massless fermion species doublers
0
doubling of fermions
Continuum:
(2) Species doublers of lattice chiral fermion copies appear: unbalance of d.o.f. between bosons and fermions
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How do we solve these two fundamental problems ?
A) Link Approach:
twisted SUSY,
shifted Leibniz rule for super charges
Dirac-Kaehler (Ivanenko-Landau) fermions
Hopf algebra invariance
B) Super doubler approach:
lattice momentum conservation
Leibniz rule is satisfied under product
non-local field theory
doublers = super partners for A) and B) No chiral fermion problem
Our proposals
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A) Link Approach:N=D=2 SUSY
Dirac-Kaehler Twist
Continuum Lattice:
on a Lattice
Dirac-Kaehler fermion
N=D=2 Twisted SUSY
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Compatibility of Shifts
We need a modified Leibniz rule for too !
New Ansatz:
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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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N=D=2 Twisted Super Yang-Mills
Gauge trans.
Scalar fieldsin SYM multiplet
Introduce Bosonic & Fermionic Link variables
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Jacobi Identities
…
Define fermionic link components
…
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
3 dimensions
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Algebraic consistency of Link Approach
1) Modified Leibniz rule:
2) Shifted anti-commutators
3) non-commutativity
Hopf algebraic consistency(D’Adda, N.K., Saito, 2009)
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Difficulties
( 1 ) No Leibniz rule in coordinate space
( 2 ) doublers of chiral fermion
Solutions
algebraic construction with lattice momentum
new * product Leibniz rule on * product
Doublers as super partners
B) Super doubler approach
No chiral fermion problem !
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SUSY transformation from dimensional analyses
1-dim.
How can we realize this structure naturally on a lattice ?
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translation generator of
half translation generator
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single lattice constant translation generator
symmetric difference operator Hermiticity
SUSY transformation: half lattice translation generator
We need fields defined on lattice !
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translation generator of
half translation generator
role of supercoordinate
The simplest example (D=N=1) Basic Idea
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alternating sign species doubler
The meaning of alternating sign in momentum space
Brillioun zone
space time pointsextra dimension ?
We have increased d.o.f. twice !
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N=2 lattice SUSY algebra
alternating sign species doubler
D=1 N=2 Lattice SUSY
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Lattice super derivative
Chiral lattice SUSY algebra (D=1,N=2)
No influence to the cont. limit
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Chiral conditions
rescaled field ! meaning ?
both on a/4 lattice
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chiral
anti-chiral
Chiral conditions truncation of species doub. d.o.f.
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Exact Lattice SUSY action for N=2 D=1
lattice momentum conservation
Super charge exact form exact lattice SUSY invariant
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New * product and Leibniz rule(coordinate rep.)
New star * product
Leibniz rule in lattice momentum space
Leibniz rule on * product (coordinate rep.)
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N=2 Wess-Zumino model in two dimensions
N=D=2 algebra:
Light cone coordinate
2-dim. = (1 dim. ) x (1 dim.)
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has 4 species doublers
truncation needed
chiral conditions
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rescaling of fields
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D=N=2 lattice SUSY transformation
Chiral
Anti-chiral
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Wess-Zumino action in two dimensions
Kinetic term
Interaction term
Super charge exact form exact lattice SUSY inv.
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N=2 Wess-Zumino actions in coordinate
product actions in two dimensions
Kinetic term
Interaction term
SUSY algebra with Leibniz rule is satisfied on product !
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Can we generalize this formulation to
super Yang-Mills ?
• Breakdown of associativity:
(A)
(B)
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Even though the associativity is broken the following product is well defined.
In the formulation of lattice super Yang-Mills the breakdown of the associativity is problem.
non-gauge lattice SUSY has exact symmetry
SUSY transformation is linear in fields while gauge transformation is non-linear.
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Associative now
Recovery of associativity
Exact lattice SUSY with associative but nonlocal equivalent to continuum theory
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Summary for Exact Lattice SUSY
A) Link Approach: Hopf algebraic exact SUSY invariance Non-commutative super Yang-Mills theory
B) Super doubler approach: Exact lattice SUSY on a new star product Non-local field theory
No chiral fermion problem: Species doublers are super partners.
Higer dimensions, gauge extension possible