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A lattice formulation of 4 dimensional N=2 supersymmetric Yang-Mills theories
Tomohisa Takimi (TIFR)
Ref) Tomohisa Takimi arXiv:1205.7038 [hep-lat]
19th July 2012 Free Meson Seminar
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1. Introduction1. Introduction
Supersymmetric gauge theoryOne solution of hierarchy problem of SM. Dark Matter, AdS/CFT correspondence
Important issue for particle physics
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*Dynamical SUSY breaking. *Study of AdS/CFT
Non-perturbative study is important
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Lattice: Lattice: A non-perturbative method
lattice construction of SUSY field theory is difficultlattice construction of SUSY field theory is difficult..
Fine-tuning problem
SUSY breaking Difficult
* taking continuum limit* numerical study
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Lattice: Lattice: A non-perturbative method
lattice construction of SUSY field theory is difficultlattice construction of SUSY field theory is difficult..
Fine-tuning problem
SUSY breaking Difficult
* taking continuum limit* numerical study
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Lattice: Lattice: A non-perturbative method
lattice construction of SUSY field theory is difficultlattice construction of SUSY field theory is difficult..
Fine-tuning problem
SUSY breaking Difficult
* taking continuum limit* numerical study
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Lattice: Lattice: A non-perturbative method
lattice construction of SUSY field theory is difficultlattice construction of SUSY field theory is difficult..
Fine-tuning problem
SUSY breaking Difficult
* taking continuum limit* numerical study
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Fine-tuning problem
Difficult to perform numerical analysis
Time for computation becomes huge.
To take the desired continuum limit.
SUSY breaking in the UV region
Many SUSY breaking counter terms appear;
is required.
prevents the restoration of the symmetry Fine-Fine-
tuningtuningof the too many parameters.
(To suppress the breaking term effects)
Whole symmetry must be recovered at the limit
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Example). N=1 SUSY with matter fields
gaugino mass, scalar mass
fermion massscalar quartic coupling
Computation time grows as the power of the number of the relevant parameters
By standard lattice action.
(Plaquette gauge action + Wilson fermion action)
too many4 parameters
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Lattice formulations free from fine-tuning
9{ ,Q}=P
_
P
Q
A lattice model of Extended SUSY
preserving a partial SUSY
O.K
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Lattice formulations free from fine-tuning
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We call as BRST charge Q
A lattice model of Extended SUSY
preserving a partial SUSY
: does not include the translation
O.K
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Picking up “BRS” charge from SUSY
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=4There are some scalar supercharges under this diagonal subgroup. If we pick up the charges, they become nilpotent supersymmetry generator which do not include infinitesimal translation in their algebra.
(E.Witten, Commun. Math. Phys. 117 (1988) 353, N.Marcus, Nucl.
Phys. B431 (1994) 3-77
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Does the BRST strategy work to solve the fine-tuning ?
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(1) Let us check the 2-dimensional case
Let us consider the local operators
Mass dimensions
Quantum corrections of the operators are
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
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(1) Let us check the 2-dimensional case
Let us consider the local operators
Mass dimensions
Quantum corrections of the operators are
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
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(1) Let us check the 2-dimensional case
Let us consider the local operators
Mass dimensions
Quantum corrections of the operators are
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
Irrelevant
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(1) Let us check the 2-dimensional case
Let us consider the local operators
Mass dimensions
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
Only these are relevant operators
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Only following operator is relevant:
Relevant
No fermionic partner, prohibited by the SUSY on the
lattice At all order of perturbation, the
absence of the SUSY breaking quantum corrections are guaranteed, no fine-tuning.
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Remaining Task (4 dimensional case)
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(2) 4 dimensional case,
If
dimensionless !
All order correction can be relevant or marginal remaining at continuum limit.
Operators with
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(2) 4 dimensional case,
If
dimensionless !
All order correction can be relevant or marginal remaining at continuum limit.
Prohibited by SUSY and the SU(2)R symmetry on the lattice.
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(2) 4 dimensional case,
If
dimensionless !
All order correction can be relevant or marginal remaining at continuum limit.
Marginal operators are not prohibited only by the SUSY on the lattice
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Fine-tuning of 4 parameters are required.
The formulation has not been useful..
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The reason why the four dimensions have been out of reach.
(1) UV divergences in four dimensions are too tough to be controlled only by little preserved SUSY on the lattice.
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The reason why the four dimensions have been out of reach.
(1) UV divergences in four dimensions are too tough to be controlled only by little preserved SUSY on the lattice.
How should we manage ?
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The reason why the four dimensions have been out of reach.
(1) UV divergences in four dimensions are too tough to be controlled only by little preserved SUSY on the lattice.
How should we manage ?
Can we reduce the 4d system to the 2d system ?
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4d to 2d treatment: (i) We separate the dimensions into
several parts in anisotropic way.(ii) We take the continuum limit of only
a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.
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(i) We separate the dimensions into several parts in anisotropic way.
(ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.
4d to 2d treatment:
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4d to 2d (i) We separate the dimensions into
several parts in anisotropic way.(ii) We take the continuum limit of only
a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.
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4d to 2d treatment (i) We separate the dimensions into
several parts in anisotropic way.(ii) We take the continuum limit of only
a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.
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(i) We separate the dimensions into several parts in anisotropic way.
(ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.
4d to 2d treatment:
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(i) We separate the dimensions into several parts in anisotropic way.
(ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.
4d to 2d treatment:
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(i) We separate the dimensions into several parts in anisotropic way.
(ii) We take the continuum limit of only a part of the four directions. During this step, the theory is regarded as a lower dimensional theory, where the UV divergences are much milder than ones in four -dimensions.(1) Even little SUSY on the
lattice can manage such mild divergences.
(2)A part of broken symmetry can be restored by the first step, to be helpful to suppress the UV divergences in the remaining steps.
4d to 2d treatment:
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(iii) Final step: taking the continuum limit of the remaining directions.
Symmetries restored in the earlier steps help to suppress tough UV divergences in higher dimensions.
4d to 2d treatment:
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The treatment with steps (i) ~ (iii)
will not require fine-tunings.
4d to 2d treatment: (iii) Final step: taking the continuum
limit of the remaining directions. Symmetries restored in the earlier
steps help to suppress tough UV divergences in higher dimensions.
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Non-perturbative formulation using anisotropy.
Hanada-Matsuura-SuginoProg.Theor.Phys. 126 (2012) 597-611
Nucl.Phys. B857 (2012) 335-361
Hanada
JHEP 1011 (2010) 112
Supersymmetric regularized formulation on
Two-dimensional lattice regularized directions.
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Non-perturbative formulation using anisotropy.
Hanada-Matsuura-SuginoProg.Theor.Phys. 126 (2012) 597-611
Nucl.Phys. B857 (2012) 335-361
Hanada
JHEP 1011 (2010) 112
Supersymmetric regularized formulation on
(1) Taking continuum limit of
Full SUSY is recovered in the UV region
Theory on the
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Non-perturbative formulation using anisotropy.
Hanada-Matsuura-SuginoProg.Theor.Phys. 126 (2012) 597-611
Nucl.Phys. B857 (2012) 335-361
Hanada
JHEP 1011 (2010) 112
Supersymmetric regularized formulation on
(1) Taking continuum limit of
Full SUSY is recovered in the UV region
Theory on the
(2) Moyal plane limit or commutative limit of .
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Non-perturbative formulation using anisotropy.
Hanada-Matsuura-SuginoProg.Theor.Phys. 126 (2012) 597-611
Nucl.Phys. B857 (2012) 335-361
Hanada
JHEP 1011 (2010) 112
Supersymmetric regularized formulation on
(1) Taking continuum limit of
Full SUSY is recovered in the UV region
Theory on the
(2) Moyal plane limit or commutative limit of .
Bothering UV divergences are suppressed by fully recovered SUSY in the step (1)
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Non-perturbative formulation using anisotropy.
Hanada-Matsuura-SuginoProg.Theor.Phys. 126 (2012) 597-611
Nucl.Phys. B857 (2012) 335-361
Hanada
JHEP 1011 (2010) 112
Supersymmetric regularized formulation on
(1) Taking continuum limit of
Full SUSY is recovered in the UV region
Theory on the
(2) Moyal plane limit or commutative limit of .
Bothering UV divergences are suppressed by fully recovered SUSY in the step (1)
No fine-tunings !!
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Our work
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We construct the analogous model to
Hanada-Matsuura-Sugino
Advantages of our model: (1) Simpler and easier to put on a computer
(2) It can be embedded to the matrix model easily. (Because we use “deconstruction”)
Easy to utilize the numerical techniques
developed in earlier works.
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Moreover, we resolve the biggest disadvantage of the deconstruction approach of Kaplan et al.
In the conventional approach, it is necessary to introduce SUSY breaking moduli fixing terms, SUSY on the lattice is eventually broken (in IR, still helps to protect from UV divergences)
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Moreover, we resolve the biggest disadvantage of the deconstruction approach of Kaplan et al.
We introduce a new moduli fixing term with preserving the SUSY on the lattice !!
In the conventional approach, it is necessary to introduce SUSY breaking moduli fixing terms, SUSY on the lattice is eventually broken (in IR, still helps to protect from UV divergences)
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Our Formulation
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Schematic explanation
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4 –dimensions are divided into
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4 –dimensions are divided into
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4 –dimensions are divided into
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From this regularized space we want to take the continuum limit without any fine-tuning
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From this regularized space we want to take the continuum limit without any fine-tuning as
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The way to construct(schematic explanation)
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(0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY
(Analogous to BMN matrix model)
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(0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY
Performing Orbifolding
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Performing deconstuction
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Performing deconstuction
Additional 1 dimension emerges
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Performing deconstuction
But this dimension is unstable, fluctuating,
and it can crush
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To stabilize the space, we introduce Moduli fixing term
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To stabilize the space, we introduce Moduli fixing term Then the space would be stabilized
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To stabilize the space, we introduce Moduli fixing term Then the space would be stabilized(I introduce the moduli fixing term
without breaking SUSY on the lattice !)
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(1) Then we obtain the orbifold lattice theory on
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For the numerical study we need to regularize
(1) Then we obtain the orbifold lattice theory on
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We will take momentum cut-off regularization
(1) Then we obtain the orbifold lattice theory on
For the numerical study we need to regularize
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We will take momentum cut-off regularization
(1) Then we obtain the orbifold lattice theory on
For the numerical study we need to regularize
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(2)This is the hybrid regularized theory on
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(2)This is the hybrid regularized theory on
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This is still 2 dimensional theory. Additional 2 dimensions must be emerged.
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(2)This is the hybrid regularized theory on
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Taking Fuzzy Sphere solution.
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(2)This is the hybrid regularized theory on
Taking Fuzzy Sphere solution.
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(3) Finally we obtain the non-perturbative formulation for the 4-d N=2 SYM on
68
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How to take the continuum limit (schematic explanation)
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We manage the momentum cut-off first !
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Continuum limit of the orbifold lattice gauge theory.
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Continuum limit of the orbifold lattice gauge theory.
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Moyal plane limit
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Moyal plane limit
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Moyal plane limit
Until the limit We do not need Fine-tunings !!
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But from
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But from
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But from
This limit is expected not smoothly connected..
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Although our formulation might not be a formulation for the commutative gauge theory,
It can be used for the non-commutative theories.
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Detailed explanation
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(0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY
(Analogous to BMN matrix model)
Orbifolding & deconstruction
(1) Orbifold lattice gauge theory on
4 SUSY is kept on the lattice (UV)And moduli fixing terms will
preserve 2 SUSY
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Momentum cut off
(2) Orbifold lattice gauge theory with momentum cut-off, (Hybrid regularization theory) Theory
on
Uplift to 4D by Fuzzy 2-sphere solution
Actually all of SUSY are broken but “harmless”
(3) Our non-perturbative formulation for 4D N=2 non-commutative SYM theories: Theory
on
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Detail of how to construct.
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(0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY
(Analogous to BMN matrix model)
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(0) The Mass deformed 1 dimensional matrix model
With mN × mN matrices and with 8-SUSY
For later use, we will rewrite the model by complexified fields and decomposed spinor components.
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We also pick up and focus on the specific 2 of 8 SUSY.
By using these 2 supercharges and spnior decomposition and complexified fields, we can rewrite the matrix model action by “the BTFT form”
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The transformation laws are
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The important property of
Global
generators
:doublets:triplet
If
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The model has
symmetry with following charge assignment
singlet
Charge is unchanged under the
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(1) Orbifold lattice gauge theory
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959595
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9696
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(1) Orbifold lattice gauge theoryOrbifold projection operator on fields with
r-charge
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(1) Orbifold lattice gauge theoryOrbifold projection operator on fields with
r-charge
Orbifold projection:Discarding the mN ×mN components except
the ones with
mN ×mN indices
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Example in N=3,
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Example in N=3,
From the gauge transformation law of the above under U(M)3
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Example in N=3,
From the gauge transformation law of the above under U(M)3
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Example in N=3,
From the gauge transformation law of the above under U(M)3
Site
Link
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Under the projection, matrix model fields become lattice fields
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SUSY on the orbifold lattice theory
SUSY charges invariant under orbifold projection will be the SUSY on the lattice
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SUSY on the orbifold lattice theory
SUSY charges invariant under orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of SUSY with
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SUSY on the orbifold lattice theory
SUSY charges invariant under orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of SUSY with
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SUSY on the orbifold lattice theory
SUSY charges invariant under orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of SUSY with
4 fermions
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SUSY on the orbifold lattice theory
SUSY charges invariant under orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of fermions with
4 fermions
4SUSY is preserved on the lattice !!
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I have explained Orbifolding
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Next is Deconstruction
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Next is Deconstruction
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Deconstruction and continuum limit.
*Orbifodling is just picking up the subsector of matrix model. (No space has appeared.)*No kinetic terms.
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*Orbifodling is just picking up the subsector of matrix model. (No space has appeared.)*No kinetic terms.
To provide the kinetic term and continuum limit,
we expand the bosonic link fields around as
Deconstruction and continuum limit.
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Continuum limit.
*By taking *If fluctuation around is
small, We can obtain the mass deformed 2d SYM with
8SUSY at the continuum limit
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Next we need to stabilize the lattice !!
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To provide the proper continuum limit, the fluctuation must be small enough compared with .
But in the SUSY gauge theory, there are flat directions which allows huge fluctuation.
We need to suppress the fluctuation by adding the moduli fixing terms
Moduli fixing terms.
These break the SUSY on the lattice eventually.
(Softly broken, so UV divergence will not be altered.)
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Proposed new Moduli fixing terms with keeping SUSYWe proposed a new moduli fixing
terms without breaking SUSY !!
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Proposed new Moduli fixing terms with keeping SUSYWe proposed a new moduli fixing
terms without breaking SUSY !!
We utilized the fact
118
If
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By a new moduli fixing term, the lattice becomes stabilized !!
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Orbifold lattice action for 2d mass deformed SYM with moduli fixing terms is
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(2) Momentum cut-off on the orbifold lattice theory.
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124124124
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To perform the numerical simulation, Remaining one continuum direction also
must be regularized.
We employ the momentum cut-off regularization in Hanada-Nishimura-Takeuchi
Momentum cut-off is truncating the Fourier expansion in the finite-volume
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Momentum cut-off in gauge theory
To justify the momentum cut-off, we need to fix the gauge symmetry by the gauge fixing condition
These condition fix the large gauge transformation which allows the momentum to go beyond the cut-off.
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Momentum cut-off action on (Hybrid regularized theory) after
gauge fixing.
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And so on.. (Remaining parts are really boring, so I will omit the parts…)
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Notes:
(1) About the gauge fixing.
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Notes:
(1) About the gauge fixing.
Gauge fixing does not spoil the quantum computation based on the gauge symmetry, because it is just putting the BRS exact term to the action, which does not affect the computation of gauge invariant quantity.Rather we should take this fixing as being required to justify the momentum cut-off to be well defined.
Only for this purpose !!
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Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
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Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
O.K !
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Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
O.K ! If the gauge symmetry is recovered only by taking, completely no
problem. I would like to emphasize that what we are
interested in is the theory at , not the theory with finite cut-off.
There is no concern whether the regularized theory break the gauge sym. or not, since it is just a regularization.
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Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
O.K ! If the gauge symmetry is recovered only by taking, completely no
problem. I would like to emphasize that what we are
interested in is the theory at , not the theory with finite cut-off.
There is no concern whether the regularized theory break the gauge sym. or not, since it is just a regularization.
I will explain it later by including the quantum effects
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(3) Uplifting to 4d by Fuzzy 2-sphere solution
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Until here, the theory is still in the 2 dimensions.
We need to uplift the theory to 4 dimensions.
We will use the Fuzzy Sphere solutions!
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Until here, the theory is still in the 2 dimensions.
We need to uplift the theory to 4 dimensions.
We will use the Fuzzy Sphere solutions!
Derivative operators along fuzzy S2
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We expand the fields in the fuzzy sphere basis which is spherical harmonics truncated at spin j:
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We expand the fields in the fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d
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We expand the fields in the Fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d Fuzzy S2 basis
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We expand the fields in the Fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d Fuzzy S2 basis
Fuzzy S2 basis:
(1) If we truncate the spherical harmonic expansion at spin j ⇒ Fuzzy S2 basis
(2) 2j+1 ×2j+1 matrix (Tensor product is altered by
Matrix product of 2j+1 ×2j+1 matrix
⇒Total spin does not exceed j
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We expand the fields in the Fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d Fuzzy S2 basis
field variable on target 4d space.
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We expand the fields in the Fuzzy sphere basis which is spherical harmonics truncated at spin j:
field on 2d Fuzzy S2 basis
field variable on target 4d space.
Fuzzy Sphere solution does not break 8 SUSY at all !!
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By this uplifting, we have completed the construction of non-perturbative formulation for N=2 4d non-commutative SYM theories.
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(ii) How to take the target continuum theory
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In our formulation, 4-dimensions are divided into 3-parrts.
Regularized by momentum cut-off
sitesparamete
rs
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In our formulation, 4-dimensions are divided into 3-parrts.
Regularized by momentum cut-off
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Task.
Which direction should we deal with first ?
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Criteria.
In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries.
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Criteria.
In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries.
We should undertake the crude regularization first !
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Regularized by momentum cut-off
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Regularized by momentum cut-off
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This one !!
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Regularized by momentum cut-off
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On the other hand,
BPS state, SUSY is well protected.
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Regularized by momentum cut-off
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Then the order of taking the continuum limit is
(1)
(2)
(3)
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Then order of taking the limit becomes
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(1) Momentum cut-off directions.
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We manage the momentum cut-off first !
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In finite the theory is one-dimensional theory.
There is no UV divergences.
There is no quantum correction breaking
2 SUSY and gauge symmetry.only by taking, orbifold lattice
theory is recovered.
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(2) Managing the orbifold lattice directions
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Repeating the renormalization discussion in the early stage of this talk….
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(1) Let us check the 2-dimensional case
Let us consider the local operators
Mass dimensions
:bosonic fields :fermionic fields
:derivatives :Some mass
parameters
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
Only these are relevant operators
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Only following operator is relevant:
Relevant
No fermionic partner, prohibited by the SUSY on the
lattice At all order of perturbation, the
absence of the SUSY breaking quantum corrections are guaranteed, no fine-tuning.
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Only following operator is relevant:
Relevant
No fermionic partner, prohibited by the SUSY on the
lattice At all order of perturbation, the
absence of the SUSY breaking quantum corrections are guaranteed, no fine-tuning.
In this step, the full 8 SUSY is restored !!
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(3) Fuzzy S2 directions.
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Moyal plane limit
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Moyal plane limit
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In this step, since the full SUSY is preserved, we do not need to mind any quantum correction
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In this step, since the full SUSY is preserved, we do not need to mind any quantum correction
No fine-tuning !!
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Notes:
In the case of N=4 theory, we can continuously connect to the commutative theory in
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Notes:
On the other hand, N=2 theory, it is expected not to be continuously connectted to the commutative theory in
Our theory is a non-perturbative formulation for the non-commutative gauge theory, but it is useful enough to investigate the non-perturbative aspects of gauge theories.
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But from
This limit is expected not smoothly connected..
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SummarySummary
We provide a simple non-perturbative formulation for N=2 4-dimensional theories, which is easy to put on computer.
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Moreover, we resolve the biggest disadvantage of the deconstruction approach of Kaplan et al.
In the approach, to make the well defined lattice theory from the matrix model, we need to introduce SUSY breaking moduli fixing terms, SUSY on the lattice is eventually broken (in IR, still helps to protect from UV divergences)
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Anisotropic treatment is useful for controlling the UV divergences.
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End
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Precise discussion
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Only following diagrams can provide quantum corrections
Bosonic tadpole with fermionic loop
Bosonic 2-point function with fermionic loop
Bosonic 2-point function with bosonic loop and derivative coupling
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Only following diagrams can provide quantum corrections
Bosonic tadpole with fermionic loop
Bosonic 2-point function with fermionic loop
Bosonic 2-point function with bosonic loop and derivative coupling
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Momentum integration of the odd function
Bosonic tadpole with fermionic loop
Bosonic 2-point function with fermionic loop
Bosonic 2-point function with bosonic loop and derivative coupling
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Momentum integration of the odd function
Bosonic tadpole with fermionic loop
Bosonic 2-point function with fermionic loop
Bosonic 2-point function with bosonic loop and derivative coupling
= 0
= 0
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Momentum integration of the odd function
Bosonic tadpole with fermionic loop
Bosonic 2-point function with fermionic loop
Bosonic 2-point function with bosonic loop and derivative coupling
= 0
= 0
No quantum correction !!
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It becomes the theory on
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