Transcript Recent developments in the lattice construction of SUSY

Lattice Formulation of
Two Dimensional
Topological Field Theory
Tomohisa Takimi (RIKEN,Japan)
K. Ohta, T.T Prog.Theor. Phys. 117 (2007) No2
[hep-lat /0611011] (and more)
July 26th 2007 SUSY 2007 @Karlsruhe, Deutschland
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1. Introduction
Lattice construction of SUSY gauge theory is difficult.
SUSY breaking
Fine-tuning problem
(Difficulty for taking continuum
limit and numerical study)
Candidate to solve fine-tuning problem
Exact supercharge on the lattice
for a nilpotent (BRST-like) supercharge
in Extended SUSY
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Models utilizing nilpotent SUSY
•
CKKU models (Cohen-Kaplan-Katz-Unsal)
2-d N=(4,4),3-d N=4, 4-d N=4 etc. super Yang-Mills theories
( JHEP 08 (2003) 024, JHEP 12 (2003) 031, JHEP 09 (2005) 042)
Sugino models
2-d N=(2,2),3-dN=4, 4-d N=4,etc. 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), 4-d N=4
(JHEP 11 (2004) 006, JHEP 06 (2005) 031)
(We will treat 2-d N=(4,4) CKKU’s model as a target model later on)
(Relationship between them: T.T [0705.3831(hep-lat)])
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The way to construct such formulations
Lattice regularization of TFT action
Nilpotent scalar supercharge
is extracted from spinor
Extended
Supersymmetric
gauge theory action
supercharges
Twisting
Supersymmetric Lattice
continuum
limit a 0
Gauge Theory action
is preserved
lattice
regularization
-exact
Topological Field
Theory action (TFT)
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The purposes of discussion of the
topological field theory on the lattice
Deeper understanding how to treat the topology
on the lattice
Non-perturbative investigation whether such
We
can treatrecover
the topological
field theory
formulation
the target continuum
theory
(Do they on
reallythe
solvelattice
fine-tuning problem non-perturbatively?)
(Perturbative investigation has been done )
Cohen-Kaplan-Katz-Unsal JHEP 08 (2003) 024
Sugino JHEP 01 (2004) 015
Onogi, T.T Phys.Rev. D72 (2005) 074504
Our main purpose
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The ground of non-perturbative study using the
topological field theory
If the theories recover the desired target theory
including the topological field theory as a subsector
its property must be recovered
continuum
Supersymmetric
lattice gauge theory
limit a 0
Must be
realized
in a 0
BRSTExtended
cohomology
Supersymmetric
gauge theory
Topological field
theory
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Non-perturbative quantity
The aim
A non-perturbative study
whether the lattice theories have
the desired continuum limit or not
through the study of
BRST cohomology on the lattice
We investigate it in 2-d N=(4,4)
CKKU model without mass term.
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2.1 Two dimensional N = (4,4) target continuum theory
(Dijkgraaf and Moore, Commun. Math. Phys. 185 (1997) 411)
Equivalent topological field theory action
: gauge
field
(adjoint representation)
(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)
(I) Is BRST transformation homogeneous ?
(II) Does
Homogeneous of
change the gauge transformation
laws?
Not homogeneous
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Answer for (I) and (II)
(II)
BRST transformation
change the gauge
transformation law
BRST
(I) BRST transformation is not homogeneous
: not homogeneous of
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2.2 BRST cohomology in the continuum theory
(E.Witten, Commun. Math. Phys. 117 (1988) 353)
(k=0,1,2)
From the set of k –form operators,
satisfies so-called
descent relation
Integration of
over k-homology cycle
BRST-cohomology
are BRST cohomology composed by
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Properties (II) play a crucial role to refuse that
the given
become BRST exact
(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 (property (II))
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3.1 Two dimensional N=(4,4) CKKU action
（K.Ohta，T.T (2007))
CKKU action can be written as BRST exact form
Set of Fields
.
Boson
Fermion
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BRST transformation on the lattice
are
homogeneous
functions of
BRST partner sets
(I)Homogeneous transformation of
In continuum theory,
(I)Not Homogeneous transformation of
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comment on homogeneous property
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
closed term including the component of
are written by the
exact form
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（II）Gauge symmetry under
on the lattice
* (II) Gauge
transformation laws do not
change under BRST transformation
(II) BRST transformation
change the gauge
transformation law
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3.2 BRST cohomology from the lattice theory
(K.Ohta, T.T (2007))
BRST cohomology
cannot be realized!
The BRST closed operators realized
from N=(4,4) CKKU lattice model
must be the BRST exact
except for the polynomial of
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Essence of the proof of the result
(I)Homogeneous property of
closed terms including the component of
are written as
exact form
(II)
does not change gauge transformation
: gauge invariant
: gauge invariant
must be BRST exact .
Only polynomial not including
Only polynomial
of
can be BRST
cohomology
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N=(4,4)
CKKUmust
model
without mass
BRST
cohomology
be composed
only byterm
cannot recover the target theory non-perturbatively
Really ?
continuum
Supersymmetric
lattice gauge theory
limit a 0
Extended
Supersymmetric
gauge theory action
Topological field
theory
BRST cohomology are composed by
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5. Summary
•
We have proposed that
the topological property
(like as BRST cohomology)
can be used as
a non-perturbative criteria to judge
whether supersymmetic lattice theories
which preserve BRST charge
have the desired continuum limit or not.
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We apply the criteria to N= (4,4) CKKU model
without mass term
The target continuum limit
cannot be realized
by including non-perturbative (IR) effect.
The criteria showed it by an explicit form.
(Perturbative investigation did not show)
Perturbative level
recover the target continuum theory
It can be a powerful criteria.
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Discussion on the No-go result
(I)and (II) plays the crucial role.
(I) Homogeneous property of BRST transformation on the
lattice.
(II) BRST transformation does not change the gauge
transformation laws.
These relate with the gauge
transformation law on the lattice.
Gauge parameters are defined on each
sites as the independent parameters.
topology
Vn
Vn+i
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BRST cohomology in the continuum theory
the integration over the k-homology cycle
Topological quantity defined by the inner
product of homology and the cohomology
The realization of topological quantities on the
lattice is difficult due to the independence of
gauge parameters
Singular gauge transformation which makes
the topology ill-defined is admitted
Admissibility condition etc. would be
needed to the realization.
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