A lattice formulation of 4 dimensional N=2 supersymmetric Yang-Mills theories

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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

Lattice formulations free from fine-tuning

9{ ,Q}=P

_

P

Q

A lattice model of Extended SUSY

preserving a partial SUSY

O.K

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 ?

(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

(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.

(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

(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

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

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

(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.

(2)This is the hybrid regularized theory on

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Taking Fuzzy Sphere solution.

(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

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How to take the continuum limit (schematic explanation)

7070

7171

We manage the momentum cut-off first !

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Continuum limit of the orbifold lattice gauge theory.

Continuum limit of the orbifold lattice gauge theory.

Moyal plane limit

Moyal plane limit

Moyal plane limit

Until the limit We do not need Fine-tunings !!

But from

But from

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

959595

9696

(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

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

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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.

124124124

125125

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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

137137

138138

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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

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.

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

sitesparamete

rs

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

sitesparamete

rs

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Regularized by momentum cut-off

sitesparamete

rs

This one !!

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Regularized by momentum cut-off

sitesparamete

rs

On the other hand,

BPS state, SUSY is well protected.

156

Regularized by momentum cut-off

sitesparamete

rs

Then the order of taking the continuum limit is

(1)

(2)

(3)

157

Then order of taking the limit becomes

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(1) Momentum cut-off directions.

159159

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….

(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

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.

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.

Moyal plane limit

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 !!

174

Notes:

In the case of N=4 theory, we can continuously connect to the commutative theory in

175

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.

But from

This limit is expected not smoothly connected..

177

SummarySummary

We provide a simple non-perturbative formulation for N=2 4-dimensional theories, which is easy to put on computer.

178

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)

179

Anisotropic treatment is useful for controlling the UV divergences.

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End

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Precise discussion

182

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

184

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

185

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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