Current Status of Exact Supersymmetryon the LatticeNoboru Kawamoto (Hokkaido University)
離散的手法・・研究会2013
Spt.30 KEK
D’adda, Feo, Kanamori, Nagata, Saito,Asaka, Kondo, Giguere

Ｗｈｙ Lattice SUSY ？
Practical reason:

discovery of SUSY particles ?
lattice SUSY
lattice QCD
N=D=4 lattice super Yang-Mills (string)

What makes exact lattice SUSY regularization difficult ?
exact lattice SUSY regularization possible ?
connection with lattice chiral fermion problem ?
clue for SUSY breaking mechanism ?
super gravity ?

Exact SUSY on the Lattice

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 operator
Kato, Sakamoto and So (2008)

New approaches for exact SUSY on the latticeA) 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)

10 years of Sapporo-Torino collaboration

Exactness ofsuper charge (Kaplan, Sugino..)Part of super charges of extended SUSYExact supersymmetry for all superchageson the latticeTwo major difficulties for lattice SUSY
Let’s consider the simplest lattice SUSY algebra:(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difference operator
symmetric

cancelation
Link nature
breakdown of Leibniz rule
Modified Leibniz ruleforward(2) Species doublers of lattice chiral fermion copies appear:
unbalance of d.o.f. between bosons and fermions

Massless fermion

species doublers

0

doubling of fermionsContinuum:How do we solve these two
fundamental problems ?
Our proposals

A) Link Approach:
twisted SUSY,
shifted Leibniz rule for super charges
Dirac-Kaehler (Ivanenko-Landau) fermionsB) Super doubler approach:lattice momentum conservation
Leibniz rule is satisfied under productnon-local field theory

doublers = super partners for A) and B)
No chiral fermion problem

A) Link Approach:
N=D=2 SUSY
Dirac-Kaehler Twist

Dirac-Kaehler fermion

N=D=2 Twisted SUSY

Continuum

Lattice:on a Lattice

N=D=2 SUSY
Dirac-Kaehler Twist
J’ = J + R

Dirac-Kaehler fermioni : flavour ?

Extended SUSY suffix

super charges in d-dim.

y2x2-dim. N=23-dim. N=44-dim. N=4

Dirac-Kaehler twisting#boson = #fermion

New Ansatz:

We need a modified Leibniz rule forCompatibility of Shifts

too !

Cond. for Twisted N=D=2Symm. ChoiceSolutions

Asymm. Choice

Twisted N=D=2
Lattice SUSY Algebra

Equivalent to orbifold
construction:by Kaplan et.al.

This solution has three dimensional nature even if
it is two dimensional lattice formulation.

N=D=2 Twisted Super Yang-Mills
Introduce Bosonic & Fermionic Link variables

Gauge trans.

Scalar fields
in SYM multiplet

N=D=2 Twisted Lattice SUSY Algebra for SYM“Shifted” Anti-commutator…

Jacobi Identities…

Define fermionic link components…

Auxiliary Field

Twisted N=2 Lattice SUSY Transformation Shifts of Fields

Twisted SUSY Algebra closes off-shellTwisted D=N=2 Super Yang-Mills Action
Action has twisted SUSY exact form.

Off-shell SUSY invariancefor all twisted super charges.

Bosonic part of the Action

Fermionic part of the Action
… (1)

… (2)

(1)

(2)

Higher dimensional extension is possible:
Twisted D=3,N=4 Super Yang-Mills Action

Higer dimensional extension is possible:

3 dimensions

3-dim. N=4 super Yang-Mills

Noncommutativity needed for link approach

Bruckmann
Kok

When

“inconsistency ?”but if we introduce the following “mild non-commutativity”:

thenIn general

Algebraic consistency of Link Approach1) Modified Leibniz rule:2) Shifted anti-commutators

3) non-commutativity

Hopf algebraic consistency(D’Adda, N.K., Saito, 2009)

Lattice super algebra as Hopf Algebra
Anzats:How do we look at modified Leibniz rule ?

multiplication
operation
co derivativebraiding

Summary of Link Approach1) For D=N=2, three dimensiona space is suggested
for the lattice space. In fact fermionic links are notin the space.
2) Totally different derivation of Kaplan’s exact lattice
SUSY (a=0). No orbifold condition used.
3) Hopf algebraic exact lattice SUSY invariance isrealized.

Remaining one problem in this approach
After
operation to a gauge invariant quantity,gauge variant terms appear ?
gauge variant ?

No-gauge invariance in the fermionic direction“since it is the extra dimention ? “B) Super doubler approach
Difficulties(１) No Leibniz rulein coordinate spaceSolutions
algebraic construction
with lattice momentum

new ＊ productLeibniz rule on ＊product(２) doublers of chiral fermion

Doublers as
super partners

No chiral fermion problem !

Basic Idea

The simplest example (D=N=1)
translation generator ofhalf translation generatorrole of supercoordinate

D=1 N=2 Lattice SUSY

alternating sign

N=2 lattice SUSY algebra

species doubler

Lattice super derivative

Chiral lattice SUSY algebra (D=1,N=2)

No influence to the cont. limit

Chiral conditions

rescaled field !
meaning ?

chiralanti-chiraltruncation ofspecies doub. d.o.f.

The meaning ofis shiftedfromin the coordinateExact Lattice SUSY action for N=2 D=1Super charge exact form

exact lattice SUSY invariant

lattice momentum conservation

In the continuum only these terms appear !

New＊product and Leibniz rule(coordinate rep.)
New star ＊ product

Leibniz rule in lattice momentum space

Leibniz rule on＊product (coordinate rep.)N=2 Wess-Zumino modelin two dimensions
N=D=2 algebra:

Light cone coordinate

2-dim. = (1 dim. ) x (1 dim.)has 4 species doublers
truncation neededchiral conditions

rescaling of fields

D=N=2 lattice SUSY transformation

ChiralAnti-chiralWess-Zumino action in two dimensions
Super charge exact form
Kinetic term

Interaction term

exact lattice SUSY inv.

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 !

1) Is the SUSY realized in the quantum level ?

2) How does the non-local nature influence to thephysical quantities ?
Check of the Ward-Takahashi identities

3) Does the translational invariance recover in thecontinuum limit ?

Numerical check !

Exact lattice SUSY at the quantum level
Asaka, D’Adda, N.K. Kondo (2013)

One loop Ward-Takahashi Identity:
Example:loop W.I.

tree W.I.

2 points functions for 2d Wess-Zumino< 𝜙(0)𝜙(𝑑) >E. Giguere, N.K. 2013

0.095

0.075
g No
= 0 interaction
g Weak= 10m interaction
g Strong= 20m interaction

0.055

analytical
Analytical

0.035

0.015

-0.005

00.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

𝑑Ward-Takashi Identity for 2d Wess-Zumino
0.25

0.2

I1
I2
I3

0.15

0.10.05

0-1
-0.05

-0.1

-0.15

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

𝑝0.8

1

Investigations of Ward-Takahashi identities ofWess-Zumino models in one and two dimensions
by cut-off model has also been studies by
Kado and Suzuki, Kamada and Suzuki..

Translational variance of value of three point function
with fixed lattice distance

0.00090.00080.00070.00060.00050.00040.0003

系列46
系列41
系列36
系列31
系列26
系列21
系列16
系列110.0002

0.0001
01 3 57 9 11
13 15

17 19 21

23 25 27

29 31 33系列635 37 39

41 43 45

系列147

Recovery of translational invariance(E. Giguere, N.K. 2013)

0-1.6

-1.4

-1.2

-1-0.8

-0.6

-0.4

-0.2

0-2-4-6-8-10-12

Can we generalize this formulation tosuper Yang-Mills ?
• Breakdown of associativity:

(2)
(3)

Non-locality does not seem to cause problem.
Translational invariance is recoveredin the continuum limit.
Even though associativity is broken, well establishedproduct is enough to define Wess-Zumino models
since SUSY transformation is linear.
Immediate extension to gauge theory is not possible
if associativity is broken.

Gauge invariance will be broken by the breakdown ofassociativity since gauge transformation is non-linear.

However we can recover the associativity.

Non-local

non associative

associative
Continuum to lattice projection

has nice derivative structure !

generalized bolocking transformationof Ginzparg-Wilson type

A proposal foran alternative solution to chiral fermion problem
1) Introduce half lattice ….. introduction of
double d.o.f. per dimension
2) Truncate fields d.o.f. into half by identificationof species doublers for bosons and fermions:same chirality for species doublers3) Replace the momentum conservation by

non-local field theory

2 dim.

Because of boundary in the first quadrant regiontranslational invariance is lost but recovered in the limitSummary for Exact Lattice SUSYA) Link Approach:
Hopf algebraic exact SUSY invarianceNon-commutative super Yang-Mills theoryB) Super doubler approach:Exact lattice SUSY on a new star productNon-local field theory

No chiral fermion problem:
Species doublers are super partners.
Higer dimensions, gauge extension of B)