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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. 1. Introduction. - PowerPoint PPT Presentation
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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. Introduction1. Introduction
Lattice construction of SUSY gauge theory is difficult.
Fine-tuning problem SUSY breaking
Difficult* taking continuum limit
* numerical study
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Candidate to solve fine-tuning problem
A lattice model of Extended SUSY preserving a partial SUSY
: does not include the translation
(BRST charge of TFT (topological field theory))
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CKKU models (Cohen-Kaplan-Katz-Unsal) 2-d N=(4,4),3-d N=4, 4-d N=4 etc. super Yang-Mills th
eories ( JHEP 08 (2003) 024, JHEP 12 (2003) 031, JHEP 09 (2005) 042)
Sugino models (JHEP 01 (2004) 015, JHEP 03 (2004) 067, JHEP 01 (2005) 016
Phys.Lett. B635 (2006) 218-224 ) Catterall models
(JHEP 11 (2004) 006, JHEP 06 (2005) 031)
(Relationship between them:
Candidate Models
T.T (JHEP 07 (2007) 010))
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Do they really recover the target continuum theory ?
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continuumlimit a 0
Lattice
Target continuum theory
Perturbative studies
All right!All right!
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Non-perturbative studies
I have done itwith the study of
Topological Field Theory
No sufficient study
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Topological fieldtheory
Must be realized
Non-perturbative quantity
Non-perturbative study
Lattice
Target continuum theory
BRST-cohomology
For 2-d N=(4,4) CKKU models
2-d N=(4,4) CKKU
Topological fieldtheory Forbidden
Imply
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2.1 The target continuum theory (2-d N=(4,4))
: gauge field
(Dijkgraaf and Moore, Commun. Math. Phys. 185 (1997) 411)
(Set of Fields)
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BRST transformation BRST partner sets
(I) Is BRST transformation homogeneous ? (II) Does change the gauge transformation laws?
Questions
is set of homogeneous linear function of
is homogeneous transformation of
def
( : coefficient)
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Answer for (I) and (II) change the
gauge transformation law
BRST
(I) is not homogeneous : not homogeneous of
(II)
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2.2 BRST cohomology in the continuum theory
satisfyingdescent
relation
Integration of over k-homology cycle
BRST-cohomology
(E.Witten, Commun. Math. Phys. 117 (1988) 353)
are BRST cohomology composed by
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not BRST exact !
not gauge invariant
formally BRST exact
change the gauge transformation law(II)
Due to (II) can be BRST cohomology
BRST exact (gauge invariant quantity)
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BRST exact form .
3.1 Two dimensional N=(4,4) CKKU action ( K.Ohta , T.T (2007))
Set of Fields
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BRST transformation on the on the latticelattice
(I)Homogeneous transformation of(I)Homogeneous transformation of BRST partner sets
In continuum theory,
(I)Not Homogeneous transformation of(I)Not Homogeneous transformation of
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tangent vectortangent vectorwith
Number operator as
counts the number of fields in
closed term including the field of exact
form
Homogeneous property
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(II)Gauge symmetry under on the lattice
* (II) Gauge transformation laws do not change under BRST transformation
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BRST cohomology BRST cohomology cannot be realizcannot be realized!ed!
Only the polynomial ofcan be BRST cohomology
3.2 BRST cohomology on the lattice theory(K.Ohta, T.T (2007))
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Essence of the proof of the result
closed terms including the fields in
exact form
(II) (II) does not change gauge does not change gauge transformationtransformation : gauge
invariant: gauge invariant
must be BRST exact .
(I) (I) Homogeneous property ofHomogeneous property of Only polynomial of can be BRST cohomology
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N=(4,4) CKKU model Target theory
Topological field theory
BRST cohomology must be composed only by
BRST cohomology are composed by
N=(4,4) CKKU model without mass term wouldnot recover the target theory non-perturbatively
Topological fieldtheory
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5. Summary5. Summary The topological property (like as BRST cohomology) could be used as a non-perturbative criteria to judge wheth
er 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 would not be realized
It can be a powerful criteria.
Implication by an Implication by an explicit form. explicit form.
Perturbative studydid not show it
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The realization is difficult due to the independence of gauge parameters
BRST cohomology
Topological quantity defined by the inner product of homology and the cohomology
(Singular gauge transformation)Admissibility condition etc. would be needed
Vn Vn+i