![Page 1: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/1.jpg)
11
An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories.
Tomohisa Takimi (TIFR)Ref) Tomohisa Takimi arXiv:1205.7038 [hep-lat]
8th June 2012 at (NTU)
![Page 2: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/2.jpg)
2
1. Introduction1. Introduction
Supersymmetric gauge theoryOne solution of hierarchy problem Dark Matter, AdS/CFT correspondence
Important issue for particle physics
2
*Dynamical SUSY breaking. *Study of AdS/CFT
Non-perturbative study is important
![Page 3: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/3.jpg)
3
In some cases, we can investigate the non-perturbative quantity in the analytic way,
(For example, by utilizing the duality, holomorphy, so on.)
But if we want to calculate wider class of general dynamical quantities not relying on such structures, direct numerical calculation would be stronger. (For example, non-holomorphic quantities or quantities not restricted by the Chiral properties..)
![Page 4: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/4.jpg)
4
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
![Page 5: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/5.jpg)
5
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
![Page 6: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/6.jpg)
6
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
![Page 7: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/7.jpg)
Lattice formulations free from fine-tuning
7
We call as BRST charge
{ ,Q}=P_
P
Q
A lattice model of Extended SUSY
preserving a partial SUSY
: does not include the translation
![Page 8: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/8.jpg)
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
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
![Page 9: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/9.jpg)
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
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
Irrelevant
![Page 10: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/10.jpg)
Only following operators are 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, requiring no fine-tuning.
![Page 11: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/11.jpg)
11
(2) 4 dimensional case,
If
dimensionless !
All order correction can be relevant or marginal remaining at continuum limit.
Operators with
![Page 12: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/12.jpg)
12
(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.
![Page 13: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/13.jpg)
13
(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
![Page 14: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/14.jpg)
14
Fine-tuning of 4 parameters are required.
The formulation has not been useful..
![Page 15: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/15.jpg)
15
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.
![Page 16: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/16.jpg)
16
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 ?
![Page 17: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/17.jpg)
17
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 ?
Anisotropic treatment !!
![Page 18: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/18.jpg)
18
Anisotropic 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.
![Page 19: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/19.jpg)
19
Anisotropic 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.
![Page 20: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/20.jpg)
20
Anisotropic 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.
![Page 21: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/21.jpg)
21
Anisotropic 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.
![Page 22: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/22.jpg)
22
Anisotropic 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.
![Page 23: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/23.jpg)
23
Anisotropic 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.
![Page 24: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/24.jpg)
24
Anisotropic 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.(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.
![Page 25: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/25.jpg)
25
Anisotropic treatment: (iii) We will take the continuum limit of
the remaining regularized directions. In this steps,
Symmetries restored in the earlier steps help to suppress tough UV divergences in higher dimensions.
![Page 26: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/26.jpg)
26
Anisotropic treatment: (iii) We will take the continuum limit of
the remaining regularized directions. In this steps,
Symmetries restored in the earlier steps help to suppress tough UV divergences in higher dimensions.
The treatment with steps (i) ~ (iii)
will not require fine-tunings.
![Page 27: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/27.jpg)
27
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.
![Page 28: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/28.jpg)
28
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
![Page 29: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/29.jpg)
29
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 .
![Page 30: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/30.jpg)
30
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)
![Page 31: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/31.jpg)
31
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 !!
![Page 32: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/32.jpg)
32
Our work
![Page 33: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/33.jpg)
33
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.
![Page 34: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/34.jpg)
34
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)
![Page 35: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/35.jpg)
35
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)
We introduce a new moduli fixing term with preserving the SUSY on the lattice !!
![Page 36: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/36.jpg)
36
Our Formulation
![Page 37: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/37.jpg)
37
Outline of the way to construct.
![Page 38: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/38.jpg)
38
(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
![Page 39: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/39.jpg)
39
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
![Page 40: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/40.jpg)
40
Detail of how to construct.
![Page 41: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/41.jpg)
41
(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.
![Page 42: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/42.jpg)
42
![Page 43: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/43.jpg)
43
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”
![Page 44: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/44.jpg)
44
The transformation laws are
![Page 45: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/45.jpg)
45
The important property of
Global
generators
:doublets:triplet
If
![Page 46: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/46.jpg)
46
The model has
symmetry with following charge assignment
singlet
Charge is unchanged under the
![Page 47: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/47.jpg)
47
(1) Orbifold lattice gauge theory
![Page 48: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/48.jpg)
(1) Orbifold lattice gauge theoryOrbifold projection operator on fields with
r-charge
![Page 49: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/49.jpg)
49
(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
![Page 50: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/50.jpg)
50
Under the projection, matrix model fields become lattice fields
![Page 51: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/51.jpg)
SUSY on the orbifold lattice theory
SUSY charges commuting with orbifold projection will be the SUSY on the lattice
![Page 52: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/52.jpg)
52
SUSY on the orbifold lattice theory
SUSY charges commuting with orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of fermions with
![Page 53: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/53.jpg)
53
SUSY on the orbifold lattice theory
SUSY charges commuting with orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of fermions with
![Page 54: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/54.jpg)
54
SUSY on the orbifold lattice theory
SUSY charges commuting with orbifold projection will be the SUSY on the lattice
= # of site fermions
# of SUSY on the lattice = # of fermions with
4 fermions
![Page 55: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/55.jpg)
55
SUSY on the orbifold lattice theory
SUSY charges commuting with 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 !!
![Page 56: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/56.jpg)
56
Deconstruction and continuum limit.
*Orbifodling is just picking up the subsector of matrix model. (No space has appeared.)*No kinetic terms.
![Page 57: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/57.jpg)
57
*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.
![Page 58: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/58.jpg)
58
Continuum limit.
*By taking *If fluctuation around is
small, We can obtain the mass deformed 2d SYM with
8SUSY at the continuum limit
![Page 59: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/59.jpg)
59
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.)
![Page 60: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/60.jpg)
60
Proposed new Moduli fixing terms with keeping SUSYWe proposed a new moduli fixing
terms without breaking SUSY !!
![Page 61: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/61.jpg)
61
Proposed new Moduli fixing terms with keeping SUSYWe proposed a new moduli fixing
terms without breaking SUSY !!
We utilized the fact
61
If
![Page 62: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/62.jpg)
62
Orbifold lattice action for 2d mass deformed SYM with moduli fixing terms is
![Page 63: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/63.jpg)
63
![Page 64: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/64.jpg)
64
![Page 65: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/65.jpg)
65
(2) Momentum cut-off on the orbifold lattice theory.
![Page 66: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/66.jpg)
66
To perform the numerical simulation, Remaining one continuum direction also
must be regularized.
We employ the momentum cut-off regularization in Hanada-Nishimura-ShingoTakeuchi
Momentum cut-off is truncating the Fourier expansion in the finite-volume
![Page 67: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/67.jpg)
67
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.
![Page 68: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/68.jpg)
68
Momentum cut-off action on (Hybrid regularized theory) after
gauge fixing.
![Page 69: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/69.jpg)
69
And so on.. (Remaining parts are really boring, so I will omit the parts…)
![Page 70: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/70.jpg)
70
Notes:
(1) About the gauge fixing.
![Page 71: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/71.jpg)
71
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 !!
![Page 72: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/72.jpg)
72
Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
![Page 73: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/73.jpg)
73
Notes:
(2) The cut-off might break the gauge symmetry, is it O.K ?
O.K !
![Page 74: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/74.jpg)
74
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.
![Page 75: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/75.jpg)
75
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
![Page 76: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/76.jpg)
76
(3) Uplifting to 4d by Fuzzy 2-sphere solution
![Page 77: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/77.jpg)
77
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!
![Page 78: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/78.jpg)
78
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
![Page 79: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/79.jpg)
79
We expand the fields in the spherical harmonics:
![Page 80: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/80.jpg)
80
We expand the fields in the spherical harmonics:
field on 2d
spherical harmonics(kind of Fourier basis) on Fuzzy S2
field variable on target 4d space.
![Page 81: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/81.jpg)
81
We expand the fields in the spherical harmonics:
field on 2d
spherical harmonics(kind of Fourier basis) on Fuzzy S2
field variable on target 4d space.
Fuzzy Sphere solution does not break 8 SUSY at all !!
![Page 82: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/82.jpg)
82
By this uplifting, we have completed the construction of non-perturbative formulation for N=2 4d non-commutative SYM theories.
![Page 83: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/83.jpg)
83
How to take the target continuum theory
![Page 84: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/84.jpg)
84
In our formulation, 4-dimensions are divided into 3-parrts.
Regularized by momentum cut-off
sitesparamete
rs
![Page 85: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/85.jpg)
85
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 ?
![Page 86: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/86.jpg)
86
Criteria.
In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries.
![Page 87: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/87.jpg)
87
Criteria.
In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries.
We should undertake the crude regularization first !
![Page 88: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/88.jpg)
88
Regularized by momentum cut-off
sitesparamete
rs
誰是壞人?
![Page 89: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/89.jpg)
89
Regularized by momentum cut-off
sitesparamete
rs
就是你!!
You are so rude !!
![Page 90: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/90.jpg)
90
Regularized by momentum cut-off
sitesparamete
rs
On the other hand,
BPS state, SUSY is well protected.
![Page 91: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/91.jpg)
91
Then order of taking the limit becomes
![Page 92: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/92.jpg)
92
We start from momentum cut-off directions.
![Page 93: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/93.jpg)
93
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.
![Page 94: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/94.jpg)
94
We start from momentum cut-off directions.
![Page 95: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/95.jpg)
Repeating the renormalization discussion in the early stage of this talk….
Renormalization in the 2-dimensional case Let us consider the local operators
Mass dimensions
Quantum corrections of the operators are
Mass dimensions 2!
Super-renormalizable
Relevant or marginal operators show up only at 1-loop level.
Irrelevant
![Page 96: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/96.jpg)
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.
![Page 97: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/97.jpg)
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 !!
![Page 98: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/98.jpg)
98
We start from momentum cut-off directions.
![Page 99: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/99.jpg)
99
In this step, since the full SUSY is preserved, we do not need to mind any quantum correction
![Page 100: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/100.jpg)
100
In this step, since the full SUSY is preserved, we do not need to mind any quantum correction
No fine-tuning !!
![Page 101: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/101.jpg)
101
Notes:
In the case of N=4 theory, we can continuously connect to the commutative theory in
![Page 102: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/102.jpg)
102
Notes:
In the case of N=4 theory, we can continuously connect to the commutative theory in
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.
![Page 103: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/103.jpg)
103
SummarySummary
We provide a simple non-perturbative formulation for N=2 4-dimensional theories, which is easy to put on computer.
![Page 104: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/104.jpg)
104
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)
![Page 105: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/105.jpg)
105
Anisotropic treatment is useful for controlling the UV divergences.
![Page 106: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/106.jpg)
106
End わんたんら
![Page 107: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/107.jpg)
107
Precise discussion
![Page 108: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/108.jpg)
108
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
![Page 109: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/109.jpg)
109
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
![Page 110: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/110.jpg)
110
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
![Page 111: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/111.jpg)
111
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
![Page 112: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/112.jpg)
112
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 !!
![Page 113: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/113.jpg)
113
![Page 114: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/114.jpg)
114
It becomes the theory on
![Page 115: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/115.jpg)
115
![Page 116: 1 1 An anisotropic hybrid non-perturbative formulation for N=2 4d non-commutative supersymmetric Yang-Mills theories. Tomohisa Takimi (TIFR) Ref) Tomohisa](https://reader036.vdocuments.net/reader036/viewer/2022062519/5697c0251a28abf838cd5395/html5/thumbnails/116.jpg)
116