1 1 A lattice formulation of 4 dimensional N=2 supersymmetric Yang-Mills theories Tomohisa Takimi (TIFR) Ref) Tomohisa Takimi arXiv: [hep-lat] 19 th July 2012 Free Meson Seminar

2 1. Introduction Supersymmetric gauge theory One solution of hierarchy problem of SM. Dark Matter, AdS/CFT correspondence Important issue for particle physics 2 *Dynamical SUSY breaking. *Study of AdS/CFT Non-perturbative study is important

3 Lattice: Lattice: A non-perturbative method lattice construction of SUSY field theory is difficult. Fine-tuning problemSUSY breaking Difficult * taking continuum limit * numerical study

4 Lattice: Lattice: A non-perturbative method lattice construction of SUSY field theory is difficult. Fine-tuning problemSUSY breaking Difficult * taking continuum limit * numerical study

5 Lattice: Lattice: A non-perturbative method lattice construction of SUSY field theory is difficult. Fine-tuning problemSUSY breaking Difficult * taking continuum limit * numerical study

6 Lattice: Lattice: A non-perturbative method lattice construction of SUSY field theory is difficult. Fine-tuning problemSUSY breaking Difficult * taking continuum limit * numerical study

7 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-tuning of the too many parameters. (To suppress the breaking term effects) Whole symmetry must be recovered at the limit

8 Example). N=1 SUSY with matter fields gaugino mass,scalar massfermion mass scalar 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 10 We call as BRST charge Q A lattice model of Extended SUSY preserving a partial SUSY : does not include the translation O.K

11 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=4 There 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

12 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.

18 Remaining Task (4 dimensional case)

19 (2) 4 dimensional case, If dimensionless ! All order correction can be relevant or marginal remaining at continuum limit. Operators with

20 (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.

21 (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

22 Fine-tuning of 4 parameters are required. The formulation has not been useful..

23 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.

24 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 ?

25 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 ?

26 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.

27 (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:

28 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.

29 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.

30 (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:

31 (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:

32 (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:

33 (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:

34 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.

35 Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) Nucl.Phys. B857 (2012) Hanada JHEP 1011 (2010) 112 Supersymmetric regularized formulation on Two-dimensional lattice regularized directions.

36 Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) Nucl.Phys. B857 (2012) 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

37 Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) Nucl.Phys. B857 (2012) 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.

38 Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) Nucl.Phys. B857 (2012) 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)

39 Non-perturbative formulation using anisotropy. Hanada-Matsuura-Sugino Prog.Theor.Phys. 126 (2012) Nucl.Phys. B857 (2012) 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 !!

40 Our work

41 We construct the analogous model toHanada-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.

42 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)

43 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)

44 Our Formulation

45 Schematic explanation

46 4 –dimensions are divided into

47 4 –dimensions are divided into

48 4 –dimensions are divided into

49 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

51 The way to construct(schematic explanation)

52 (0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY (Analogous to BMN matrix model)

53 (0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY Performing Orbifolding

54 Performing deconstuction

55 Performing deconstuction Additional 1 dimension emerges

56 Performing deconstuction But this dimension is unstable, fluctuating, and it can crush

57 To stabilize the space, we introduce Moduli fixing term

58 To stabilize the space, we introduce Moduli fixing term Then the space would be stabilized

59 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 !)

60 (1) Then we obtain the orbifold lattice theory on

61 For the numerical study we need to regularize (1) Then we obtain the orbifold lattice theory on

62 We will take momentum cut-off regularization (1) Then we obtain the orbifold lattice theory on For the numerical study we need to regularize

63 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 64

(2)This is the hybrid regularized theory on 65 This is still 2 dimensional theory. Additional 2 dimensions must be emerged.

(2)This is the hybrid regularized theory on 66 Taking Fuzzy Sphere solution.

(2)This is the hybrid regularized theory on Taking Fuzzy Sphere solution. 67

(3) Finally we obtain the non-perturbative formulation for the 4-d N=2 SYM on 68

69 How to take the continuum limit (schematic explanation)

70

71 We manage the momentum cut-off first !

72

73

74 Continuum limit of the orbifold lattice gauge theory.

Moyal plane limit

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

But from

This limit is expected not smoothly connected..

83 Although our formulation might not be a formulation for the commutative gauge theory, It can be used for the non-commutative theories.

84 Detailed explanation

85 (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

86 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

87 Detail of how to construct.

88 (0) Starting from the Mass deformed 1 dimensional matrix model with 8SUSY (Analogous to BMN matrix model) 88

89 (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.

90 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”

91 The transformation laws are

92 The important property of Globalgenerators :doublets :triplet If

93 The model hassymmetry with following charge assignment singlet Charge is unchanged under the

94 (1) Orbifold lattice gauge theory

95

96

(1) Orbifold lattice gauge theory Orbifold projection operator on fields with r-charge

98 (1) Orbifold lattice gauge theory Orbifold projection operator on fields with r-charge Orbifold projection: Discarding the mN ×mN components except the ones with mN ×mN indices

99 Example in N=3,

100 Example in N=3, From the gauge transformation law of the above under U(M) 3

101 Example in N=3, From the gauge transformation law of the above under U(M) 3

102 Example in N=3, From the gauge transformation law of the above under U(M) 3 Site Link

103 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

105 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

106 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

107 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

108 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 !!

109 I have explained Orbifolding 109

110 Next is Deconstruction

111 Next is Deconstruction 111

112 Deconstruction and continuum limit. *Orbifodling is just picking up the subsector of matrix model. (No space has appeared.) *No kinetic terms.

113 *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.

114 Continuum limit. *By taking *If fluctuation around is small, We can obtain the mass deformed 2d SYM with 8SUSY at the continuum limit

115 Next we need to stabilize the lattice !! 115

116 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.)

117 Proposed new Moduli fixing terms with keeping SUSY We proposed a new moduli fixing terms without breaking SUSY !!

118 Proposed new Moduli fixing terms with keeping SUSY We proposed a new moduli fixing terms without breaking SUSY !! We utilized the fact 118 If

119 By a new moduli fixing term, the lattice becomes stabilized !!

120 Orbifold lattice action for 2d mass deformed SYM with moduli fixing terms is

121

122

123 (2) Momentum cut-off on the orbifold lattice theory.

124

125

126 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

127 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.

128 Momentum cut-off action on (Hybrid regularized theory) after gauge fixing.

129 And so on.. (Remaining parts are really boring, so I will omit the parts…)

130 Notes: (1) About the gauge fixing.

131 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 !!

132 Notes: (2) The cut-off might break the gauge symmetry, is it O.K ?

133 Notes: (2) The cut-off might break the gauge symmetry, is it O.K ? O.K !

134 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.

135 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

136 (3) Uplifting to 4d by Fuzzy 2-sphere solution

137

138

139 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!

140 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

141 We expand the fields in the fuzzy sphere basis which is spherical harmonics truncated at spin j:

142 We expand the fields in the fuzzy sphere basis which is spherical harmonics truncated at spin j: field on 2d

143 We expand the fields in the Fuzzy sphere basis which is spherical harmonics truncated at spin j: field on 2d Fuzzy S2 basis

144 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 !!

147 By this uplifting, we have completed the construction of non-perturbative formulation for N=2 4d non- commutative SYM theories.

148 (ii) How to take the target continuum theory

149 In our formulation, 4-dimensions are divided into 3-parrts. Regularized by momentum cut-off sites parameters

150 In our formulation, 4-dimensions are divided into 3-parrts. Regularized by momentum cut-off sites parameters Task. Which direction should we deal with first ?

151 Criteria. In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries.

152 Criteria. In early lower dimensional stage, it is easier to handle the crude regularization breaking much symmetries. We should undertake the crude regularization first !

153 Regularized by momentum cut-off sites parameters

154 Regularized by momentum cut-off sites parameters This one !!

155 Regularized by momentum cut-off sites parameters On the other hand, BPS state, SUSY is well protected.

156 Regularized by momentum cut-off sites parameters Then the order of taking the continuum limit is (1) (2) (3)

157 Then order of taking the limit becomes

158 (1) Momentum cut-off directions.

159 We manage the momentum cut-off first !

160

161 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.

162 (2) Managing the orbifold lattice directions

163

165 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 !!

169 (3) Fuzzy S2 directions.

Moyal plane limit

172 In this step, since the full SUSY is preserved, we do not need to mind any quantum correction

173 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 Summary 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.

180 End

181 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

183 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

186 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 No quantum correction !!

187

188 It becomes the theory on

189

190