Transcript Relations among Supersymmetric Lattice Gauge Theories So Matsuura @ Niels Bohr Institute based on the works arXiv:0704.2696 arXiv:0706.3007 arXiv:0708.4129 arXiv:0709.4193 with P.H.Damgaard 9/10/2007 Isaac Newton Institute.

Relations among Supersymmetric
Lattice Gauge Theories
So Matsuura
@ Niels Bohr Institute
based on the works
arXiv:0704.2696
arXiv:0706.3007
arXiv:0708.4129
arXiv:0709.4193
with P.H.Damgaard
9/10/2007
Isaac Newton Institute
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Introduction
Lattice Gauge Theory
 Constructive definition of a gauge theory
 Non-perturbative analysis by numerical simulations
If supersymmetric gauge theories are constructed on a lattice,
 It gives a “definition” of the theory.
 We can compute any physical observable even if it is not restricted
by the SUSY algebra.
 We can compare results in strong coupling region directly with,
say, the AdS/CFT correspondence.
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Difficulty
It seems impossible to construct a SUSY invariant theory on a lattice.
SUSY invariant action in continuum space-time
Suppose an action is written as
; superfield
Essentially, a SUSY generator can be represented as
Variation of the action
Leibniz rule
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continuum theory
differential operator
lattice theory
difference operator
Leibniz rule
deformed Leibniz rule
It seems impossible to keep all SUSY on a lattice.
QUESTION
Can we keep a part of SUSY on a lattice?
Yes!
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Map of lattice theories with SUSY on a lattice
A.Cohen, E.Katz, D.Kaplan, M.Unsal,
Orbifold lattice theories
S. Catterall
Catterall's lattice theories
an extension
P.H.Damgaard, S.M.
equivalent
P.H.Damgaard, S.M.
Lattice theories by
the “link approach”
T.Takimi
Sugino’s lattice theories
A. D'Adda, I. Kanamori, N. Kawamoto, K. Nagata
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some reduction
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F.Sugino
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Contents
1.
Introduction
2.
Classification of Orbifold Lattice Gauge Theories
3.
Exact Vacuum Energy of Orbifold Theory
4.
Relation with Catterall’s Supersymmetric Lattice Gauge Theory
5.
Equivalence between the Orbifolding and the Link Approach
6.
Conclusion
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Classification of Orbifold Lattice Gauge Theories
OUT LINE
Mother Theory
A supersymmetric Yang-Mills matrix theory
STEP 1
orbifold projection
Orbifolded Matrix Theory
A matrix theory with “scalar supercharges”
(a lattice formulation without kinetic terms)
STEP 2
deconstruction
Orbifold Lattice Theory
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A lattice theory with scalar supercharges
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Yang-Mills matrix with 4 SUSY
Yang-Mills matrix with 8 SUSY
mother theory 1
mother theory 2
・・・・・
lattice 1
1
latticelattice
2
・・・・・
lattice ∞?
lattice 1
lattice 8
lattice 2
Yang-Mills matrix with 16 SUSY
Unknown
mother theory 4
mother theory 3
・・・・・
・・・・・
lattice 1
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lattice 2
lattice ??
lattice 1
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lattice 2
lattice 1024
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[1] construction of Q=4 orbifold lattice theory
STEP 0
Mother theory with 4 SUSY
A matrix theory that is obtained by dimensional reduction of
Euclidean 4D N=1 SYM theory with a gauge group
.
: four hermitian matrices
: a Majorana spinor
Symmetries
maximal U(1) subgroup
1) global symmetry
2) gauge symmetry
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Equivalent expression in which the U(1) symmetries are manifest:
U(1) charges
We can take any
linear combination.
where
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supersymmetry
with
The variation of the action is zero if and only if the SUSY parameters are trivial;
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STEP1
Orbifold projection
We consider a
where
transformation generated by
and
: clock matrix
We keep only components that are invariant under this transformation.
simple example
projection of a matrix
z(1)
with U(1) charge 1
z(2)
projection by
z(3)
z(4)
each block is an MxM matrix
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Orbifolded action
Substituting the projected field,
1
1
we obtain
Projection of the supersymmetry
The supersymmetry parameters have definite U(1) charges:
They become non-trivial
after orbifolding.
The only preserved supersymmetry is the one corresponding to k.
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STEP2
Deconstruction
We introduce kinetic terms and a lattice spacing by
Finally, we get an action:
where
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Classification of the theories
The lattice action depends on
 two vectors
and
.
 two real numbers
and
.
How the theory depends on them?
a physical interpretation
a space-time lattice
an abstract lattice
: linear mapping
Impose
The continuum theory should be Lorentz invariant
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・・・☆
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The kinetic terms in the continuum limit
The condition ☆ determines the linear mapping f as
The lattice theory is unique and on a square lattice.
the continuum theory
2D N=(2,2) SYM theory with the gauge group U(M)
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[2] construction of Q=8 orbifold lattice theory
Mother theory with 8 SUSY
A matrix theory that is obtained by dimensional reduction of
Euclidean 6D N=1 SYM theory.
: six hermitian matrices
: independent four-component spinors
Symmetries
maximal U(1) subgroup
1) global symmetry
2) gauge symmetry
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Equivalent expression in which the U(1) symmetries are manifest:
U(1) charges
linear combinations of
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1) Orbifold projection
2) Deconstruction
the lattice action:
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Classification of the theories
generates the lattice
Dimensionality of the lattice
the dimensionality of the lattice
the number of linearly independent
vectors in
Preserved supersymmetry on the lattice
The supercharges corresponding to scalar fermions are preserved.
 At least one SUSY corresponding to h is preserved.
 SUSY enhances if another U(1) charges of a fermion becomes zero.
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(1) 3D lattice with 1 SUSY (three linearly independent
continuum
limit
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3D SUSY Yang-Mills theory
with 8 SUSY
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(2-1) 2D lattice with 2 SUSY
(2-3) 2D lattice with 2 SUSY
(2-2) 2D lattice with 2 SUSY
(3) 2D lattice with 1 SUSY (an example)
The common continuum limit is 2D N=(4,4) SUSY Yang-Mills theory.
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N.B.
 There are additional three kinds of 2D lattice theories obtained
by shifting only two bosons, say,
and
as
The continuum theory is the same.
 We can construct 4D, 3D and 2D lattice theories from the mother theory
with sixteen supercharges (IKKT matrix theory).
In particular, the 4D theory is a lattice formulation of the 4D N=4 SYM theory.
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Exact Vacuum Energy of the Orbifold Theories
The classical moduli space of the orbifold lattice theories are parametrized by
the vacuum expectation values of
the potential terms
up to gauge transformations.
QUESTION
Can we estimate quantum corrections to the vacuum energy?
It seems non-trivial since the supersymmetry is almost broken.
 contributions from higher-loops
 non-perturbative contributions
We can estimate the exact vacuum energy in this case!
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key points
 The orbifold theories have a BRST symmetry on a lattice.
 The actions can be written in Q-exact forms.
The partition function does not depend on the coupling constant;
The vacuum energy estimated in the 1-loop level is exact.
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The second order actions
For both the case it is easy to show that
the 1-loop contribution to the partition function is equal to 1.
The vacuum energy of the orbifold lattice theories constructed from
the mother theories with 4 and 8 SUSY never receive quantum corrections.
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Relation with Catterall’s Supersymmetric
Lattice Gauge Theory
Brief review of Catterall’s formulation
Starting with the topologically twisted 2D N=(2,2) SYM,
where Q is a BRST charge acting the fields as
nilpotent up to a gauge transformation
The lattice theory is obtained by the following three steps.
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STEP 1
The theory is latticised by putting fields on a lattice corresponding
to the tensor structures;
tensors
vectors
scalars
STEP 2
Complexified fields are introduced to make the action real;
tensors
vectors
STEP 3
The field strength and the covariant derivatives are replaced by
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Then we obtain a lattice action,
 A BRST symmetry is preserved on a lattice.
 The path-integral is carried out along the real line,
 The other supersymmetries are shown to be
restored in the continuum limit by numerical
simulations. S.Catterall 2006
 By restricting the complexified fields in a different way,
we obtain Sugino’s lattice formulation.
F.Sugino 2004
T.Takimi 2007
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Claim
This prescription is automatically reproduced by orbifolding.
The only assumption is a complexification.
Let us consider the mother theory with 4 SUSY in a Q-exact form;
where Q acts on the fields as
We complexified the matrices and extend the action as
where d is a auxiliary field and we have also doubled h as
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By this extension, the theory acquires extra U(1) symmetries,
and the charge assignment for the fields is
Then we can define the corresponding orbifold projection.
The orbifold projected action is obtained by substituting
into the action.
Furthermore, instead of shifting
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and
, we replace them as
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Then we obtain a lattice action,
This is equivalent with Catterall’s formulation.
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N.B.
 The BRST symmetry is enhanced by the complexification;
with
which satisfies
.
 This method can be applied to other SUSY gauge theories.
(ex) 4D N=2 SYM theory, etc...
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Equivalence between the
Orbifolding and the Link Approach
Recall the U(1) charges of the fields in the mother theory with 4 SUSY
new U(1) charges
three-component vectors
obtained from
We can carry out the orbifold projection using these U(1) charges.
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The result is
A typical example
This action is completely the same with
the one obtained by the link approach.
A. D'Adda, I. Kanamori, N. Kawamoto, K. Nagata (2005)
The lattice action given by the link approach
is obtained by orbifolding procedure.
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Supersymmetry of this theory
The SUSY is completely broken because of the discussion given above.
but
They claim that all the supercharges are preserved on a lattice
in a deformed sense.
deformed SUSY in the mother theory
Consider the “supersymmetry transformation” in the mother theory
with non-trivial
Instead of the usual Leibniz rule,
a matrix made from
the shift matrix.
let us impose a modified Leibniz rule by hand,
for each of
and
They satisfy
F.Bruckmann, S.Catterall,
M.de Kok (2006)
although there is some discussion in whether this is consistent or not….
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Conclusion
• We classified the lattice theories constructed from the mother
theories with 4 and 8 supercharges.
• We showed that the vacuum energy of the orbifold lattice theories
does not receive any quantum correction.
• We showed that the formulation given by Catterall can be
understood in terms of the orbifolding procedure.
• We showed that the SUSY lattice theories obtained by the link
approach are equivalent to the orbifold lattice theories.
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Future Problems
• Lattice theories constructed from the mother theory with
16 supercharges (IKKT matrix theory)
 classification of the theories
 structure of the quantum vacuum
 including 4D N=4 SYM theory
 AdS/CFT correspondence in terms of lattice theories?
• Numerical simulations
 recovering of the supersymmetries in the continuum limit
 comparison with exact results
 non-BPS operators
• Connection to the superstring theory
orbifold
D-instantons
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