Transcript Recent developments in the lattice construction of SUSY

Lattice Formulation of
Two Dimensional
Topological Field Theory
Tomohisa Takimi (理研、川合理論物理)
K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2
[hep-lat /0611011] （Hepにのってない話も含む）
平成19年5月18日 於 筑波大学セミナー
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Contents
1. Introduction
(review, fine-tuning problem, recent development)
2. Our proposal for non-perturbative criteria
3. Topological property in the continuum theory
3.1 BRST exact form of the model
3.2 BRST cohomology (BPS state)
4. Topological property on the lattice
my work
(4.0 Construction of SUSY on lattice)
K. Ohta, T.T
4.1 BRST exact form of the model
4.2 BRST cohomology (BPS state)
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5. Summary
1. Introduction
Supersymmetric gauge theory
One solution of hierarchy problem
Dynamical SUSY breaking
Lattice study may help to get deeper understanding
but lattice construction of SUSY field theory is difficult.
SUSY algebra includes infinitesimal
translation which is broken on the lattice.
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Fine-tuning problem in present approach
Standard action
Plaquette gauge action + Wilson or Overlap fermion action
Violation of SUSY for finite lattice spacing.
Many SUSY breaking terms appear;
Fine-tuning is required to recover SUSY in continuum.
Time for computation becomes huge.
Difficult to perform numerical analysis
ex. N=1 SUSY with matter fields
gaugino mass,
scalar mass
scalar quartic coupling
fermion mass
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Lattice formulations free from fine-tuning
Exact supercharge on the lattice
for a nilpotent (BRST-like) supercharge
in Extended SUSY
We call
as BRST charge
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Twist in Extended SUSY
(E.Witten, Commun. Math. Phys. 117 (1988) 353, N.Marcus, Nucl. Phys. B431 (1994) 3-77
Redefine the Lorentz algebra by a diagonal
subgroup of the Lorentz and the R-symmetry
in the extended SUSY
ex. d=2, N=2
d=4, N=4
There are some scalar supercharges under this
diagonal subgroup. If we pick up the charges, they
become nilpotent supersymmetry generator which
does not include infinitesimal translation in their
algebra.
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After the twist, we can reinterpret the extended
supersymmetric gauge theory action as an
equivalent topological field theory action
Nilpotent scalar supercharge
is extracted from spinor
supercharges
Extended
Supersymmetric
gauge theory action
Twisting
Supersymmetric Lattice
Topological Field
Gauge Theory action
Theory action
is preserved
lattice
regularization
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Models utilizing nilpotent SUSY from Twisting
CKKU models (Cohen-Kaplan-Katz-Unsal)
•
2-d N=(4,4),N=(2,2),N=(8,8),3-d N=4,N=8, 4-d N=4
super Yang-Mills theories
( JHEP 08 (2003) 024, JHEP 12 (2003) 031, JHEP 09 (2005) 042)
Sugino models
2 -d N=(2,2),N=(4,4),N=(8,8),3-dN=4,N=8, 4-d N=4
super Yang-Mills
(JHEP 01 (2004) 015, JHEP 03 (2004) 067, JHEP
01 (2005) 016 Phys.Lett. B635 (2006) 218-224）
Catterall models
2 -d N=(2,2),（杉野さん模型の改良版？実は・・。）
(JHEP 11 (2004) 006, JHEP 06 (2005) 031)
We will treat 2-d N=(4,4) CKKU’s model
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Do they really have the desired continuum
theory with full supersymmetry ?
Perturbative investigation
They have the desired continuum limit
CKKU JHEP 08 (2003) 024, JHEP 12 (2003) 031,
Onogi, T.T Phys.Rev. D72 (2005) 074504
Non-perturbative investigation
Sufficient investigation has not been done !
Our main purpose
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2. Our proposal for the
non-perturbative study
-
( Topological Study ) -
10
We look at that the lattice model actions are lattice
regularization of topological field theory action
equivalent to the target continuum action
Extended
Supersymmetric
gauge theory action
Supersymmetric Lattice
continuum
limit a 0
Topological Field
Theory action
Gauge Theory action
lattice
regularization
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And the target continuum theory includes a topological
field theory as a subsector.
So if the theories recover the desired target theory,
even including quantum effect,
topological field theory and its property must be recovered
continuum
Supersymmetric
lattice gauge theory
Witten index
BPS states
limit a 0
Must be
realized
in a 0
Extended
Supersymmetric
gauge theory
Topological field
theory
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Topological property (action
)
BRST cohomology (BPS state)
these are independent of gauge coupling
Because
We can obtain this value non-perturbatively
in the semi-classical limit.
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The aim
A non-perturbative study
whether the lattice theories have
the desired continuum limit or not
through the study of
topological property on the lattice
We investigate it in 2-d N=(4,4)
CKKU model.
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3. Topological property in the
continuum theories
-
(Review)
In the 2 dimesional N = (4,4) super
Yang-Mills theory
3.1 BRST exact action
3.2 BRST cohomology
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3.1 BRST exact form of the action
(Dijkgraaf and Moore, Commun. Math. Phys. 185 (1997) 411)
Equivalent topological field theory action
: covariant derivative
(adjoint representation) : gauge
field
(Set of Fields)
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BRST transformation
BRST partner sets
If
is set
of homogeneous
linear function of
def
is homogeneous
transformation of
Let’s consider
(
is just the coefficient)
（甲） Is BRST transformation is homogeneous ?
（乙） Does
change the gauge transformation laws?
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（甲）What is homogeneous ?
ex) For function
We
homogeneous
as follows
Wedefine
treat the as
coefficient of
for discussion
of
homogeneous of
homogeneous
We define the homogeneous of not homogeneous
as follows
We treat as coefficient for discussion of
homogeneous
homogeneous of
not homogeneous
Homogeneous
linear
not homogeneous
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Answer for （甲）、（乙）
（乙）
BRST transformation
change the gauge
transformation law
BRST
（甲）BRST transformation is not homogeneous of
: homogeneous function of
: not homogeneous of
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3.2 BRST cohomology in the continuum theory
(E.Witten, Commun. Math. Phys. 117 (1988) 353)
The following set of k –form operators,
(k=0,1,2)
satisfies so-called
descent relation
Integration of
over k-homology cycle ( on torus)
becomes BRST-closed
homology 1-cycle
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(Polynomial of
(
BRST cohomology )
Although
) is trivially
(k=1,2) are formally BRST exact
not BRST exact !
,
and
are not gauge invariant
This is because BRST transformation change the
gauge transformation law （（乙）（オツ）な性質による）
are BRST cohomology composed by
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4.Topological property on the lattice
（K.Ohta，T.T (2007))
4.1 BRST exact action
4.2 BRST cohomology
We investigate in the 2 dimensional N = (4,4)
CKKU supersymmetric lattice gauge theory
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- 4.0 The model
2 dimensional N=(4,4) CKKU model (Cohen-Kaplan-Katz-Unsal JHEP 12 (2003) 031)
Dimensional reduction of 6 dimensional
super Yang-Mills theory
Orbifolding by
in global symmetry
2-dimensional lattice structure
in the field degrees of freedom
Deconstruction
kinetic term in 2-dim
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4.1 BRST exact form of the lattice action
（K.Ohta，T.T (2007))
N=(4,4) CKKU action
Set of Fields
as BRST exact form
.
Boson
Fermion
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BRST transformation on the lattice
are
homogeneous
functions of
BRST partner sets
（甲）Homogeneous transformation of
In continuum theory,
（甲）Not Homogeneous transformation of
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comment on homogeneous property(1)
Due to homogeneous property of
can be written as tangent vector
If we introduce fermionic operator
They Compose the number operator
as
which counts the number of fields within
commute with the number operator
is homogeneous transformation
since
which does not change the number of fields in
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comment on homogeneous property(2)
Any term in a general function of fields
has a definite number of fields in
Ex)
A general function
can be written as
:Polynomial of
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（乙）Gauge symmetry under
fields
and the location of
* BRST partners sit on same links or sites
* （乙）Gauge transformation laws do not change under
BRST transformation
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4.2 BRST cohomology on the lattice theory
(K.Ohta, T.T (2007))
BRST cohomology
cannot be realized!
The BRST closed operators on the
N=(4,4) CKKU lattice model
must be the BRST exact
except for the polynomial of
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Proof (page 1)
Consider
Only
have BRST cohomology
From
(end of proof)
【1】 for
with
commute with gauge transformation
【2】
: gauge invariant
:
: gauge invariant
must be BRST exact .
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Essence of the No-go result
（甲）Lattice BRST transformation is homogeneous
about
We can define the number operator
of
by using another fermionic transformation
（乙）Lattice BRST transformation does not change
the representation under the gauge transformation
We cannot construct the gauge invariant
BRST cohomology by the BRST
transformation of gauge variant value
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on the lattice
BRST cohomology must be composed only by
disagree with each other
BRST cohomology are composed by
in the continuum theory
* BRST cohomology on the lattice
Not realized in continuum limit !
* BRST cohomology in the continuum theory
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Really ?
continuum
Supersymmetric
lattice gauge theory
Topological field
theory on the lattice
limit a 0
Extended
Supersymmetric
gauge theory action
Topological field
theory
33
One might think the No-go theorem (A) has
Even
in
case
(B),
we
cannot
realize
not forbidden the realization of BRST
the
observables
in continuum
the continuum
cohomology
in the
limit inlimit
the
case (B)
continuum
Supersymmetric
lattice gauge theory
limit a 0
(B)
Topological field
theory
Extended
Supersymmetric
gauge theory
Topological field
theory
(A)
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The discussion via the path (B)
Topological observable in the continuum limit
via path (B)
Representation of
on the lattice
These satisfy following property
(
lattice spacing )
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Also in this case,
We transformation
cannot realizeis homogeneous,
Since the BRST
the
topological
property
We can expand
as
via path (B)
And in,
it can be written as
So the expectation value of this becomes
!
since
Since
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Caution!
In the torus, there might not be
non-zero topological observable
even in target continuum theory
In supersymmetric BF theory
Blau, Thompson (hep-th/930144)
Witten(J.Geom.Phys.9:303-368,1992 )
We have to confirm whether this
situation stands also on our target gauge
theories or not ?
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The 2-d N=(4,4) CKKU lattice model
cannot realize the topological property
in the continuum limit!
continuum
Supersymmetric
lattice gauge theory
limit a 0
Extended
Supersymmetric
gauge theory
(B)
Topological field
theory
Topological field
theory
(A)
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5. Summary
•
We have proposed that
the topological property
(like as BRST cohomology)
should be used as
a non-perturbative criteria to judge
whether supersymmetic lattice theories
which preserve BRST charge on it
have the desired continuum limit or not.
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We apply the criteria to N= (4,4) CKKU model
There is a possibility that
topological property cannot be realized.
The target continuum limit
might not be realized
by including non-perturbative effect.
It can be a powerful criteria.
if we confirm the target theory include
non-zero topological observable
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