Rectangular W-algebras of types so and sp and dual coset CFTs - - PowerPoint PPT Presentation

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Rectangular W-algebras of types so and sp and dual coset CFTs - - PowerPoint PPT Presentation

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Rectangular W-algebras of types so and sp and dual coset CFTs Takahiro Uetoko (Ritsumeikan Univ.) Based on: [arXiv:1906.05872] w/ Thomas Creutzig (Alberta Univ.), Yasuaki Hikida (YITP, Kyoto Univ.) Aug. 22 (2019) @YITP Strings and Fields

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SLIDE 1

Rectangular W-algebras

  • f types so and sp

and dual coset CFTs

Takahiro Uetoko (Ritsumeikan Univ.)

Based on: [arXiv:1906.05872] w/ Thomas Creutzig (Alberta Univ.), Yasuaki Hikida (YITP, Kyoto Univ.)

  • Aug. 22 (2019) @YITP “Strings and Fields 2019”

1

slide-2
SLIDE 2

Introduction

  • Strings and Higher spins

2

String theory

First Regge trajectory Vasiliev theory

Higher spin gravity Tensionless limit

[Gross ’88]

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SLIDE 3

Introduction

  • Strings and Higher spins

3

String theory

First Regge trajectory Vasiliev theory

Higher spin gravity

  • String spectrum

(mass)2

(spin)

(mass)2

(spin)

First Regge trajectory Vasiliev theory

Tensionless limit

[Gross ’88]

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SLIDE 4

[Gross ’88]

Introduction

  • Strings and Higher spins

4

String theory

First Regge trajectory Vasiliev theory

Higher spin gravity Tensionless limit

  • String spectrum

(mass)2

(spin)

(mass)2

(spin)

First Regge trajectory Vasiliev theory

How to explain the higher Regge trajectories?

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SLIDE 5

Introduction

  • Strings and Extended higher spins

5

String theory

All Regge trajectory Vasiliev theory with matrix valued fields

M × M

Higher spin gravity

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SLIDE 6

Introduction

  • Strings and Extended higher spins

6

String theory

All Regge trajectory Vasiliev theory with matrix valued fields

M × M

Higher spin gravity

  • Matrix extension of 3d Prokushkin-Vasiliev theory may be analyzed

with the infinite dimensional symmetry of 2d CFT

[Creutzig-Hikida ’13]

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SLIDE 7

Introduction

  • Strings and Extended higher spins

7

String theory

All Regge trajectory Vasiliev theory with matrix valued fields

M × M

Higher spin gravity

  • Matrix extension of 3d Prokushkin-Vasiliev theory may be analyzed

with the infinite dimensional symmetry of 2d CFT

  • Dual model is 2d Grassmannian-like coset

su(N + M)k su(N)k ⊕ u(1)kNM(N+M)

With , this reduce to the Gaberdiel-Gopakumar duality

M = 1

  • Evidence: spectrum, asymptotic symmetry, …

[Creutzig-Hikida ’13] [Gaberdiel-Gopakumar ’10] [Creutzig-Hikida-Rønne ’13, Creutzig-Hikida ’18]

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SLIDE 8

Introduction

  • Strings and Extended higher spins

8

String theory

All Regge trajectory Vasiliev theory with matrix valued fields

M × M

Higher spin gravity

  • Matrix extension of 3d Prokushkin-Vasiliev theory may be analyzed

with the infinite dimensional symmetry of 2d CFT

  • Dual model is 2d Grassmannian-like coset

su(N + M)k su(N)k ⊕ u(1)kNM(N+M)

With , this reduce to the Gaberdiel-Gopakumar duality

M = 1

  • Evidence: spectrum, asymptotic symmetry, …

Can we generalize this analysis to other models?

[Creutzig-Hikida ’13] [Gaberdiel-Gopakumar ’10] [Creutzig-Hikida-Rønne ’13, Creutzig-Hikida ’18]

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SLIDE 9

Introduction

  • Our question and summary

9

Can we generalize this analysis to other models?

  • We consider 2 ways to truncate the DOF
  • Restricted matrix extensions;
  • Even spin truncation of
  • We propose the dual coset model

and examine the asymptotic symmetry

so(M), sp(M) hs[λ]

An answer (my talk)

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SLIDE 10
  • 1. Introduction
  • 2. HS gravity with

gauge sector

  • 3. Some generalization for extended HS gravity
  • 4. Summary

sl(M)

10

Plan of talk

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SLIDE 11
  • 1. Introduction
  • 2. HS gravity with

gauge sector

  • 3. Some generalization for extended HS gravity
  • 4. Summary

sl(M)

11

Plan of talk

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SLIDE 12
  • Chern-Simons description of HS gravity

12

HS gravity with gauge sector

sl(M)

  • 3d gravity:

CS theory

sl(2)

[Witten ’88]

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SLIDE 13
  • Chern-Simons description of HS gravity

13

HS gravity with gauge sector

sl(M)

  • 3d gravity:

CS theory

sl(2)

Include higher spin

  • 3d HS gravity:
  • hs[λ]

hs[λ] = B[λ] ⊖ 1

B[λ] = U(sl(2)) ⟨C2 − 1

4 (λ2 − 1)1⟩ [Prokushkin-Vasiliev ’98] [Witten ’88]

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SLIDE 14
  • Chern-Simons description of HS gravity

14

HS gravity with gauge sector

sl(M)

  • 3d gravity:

CS theory

sl(2)

[

  • ]

λ = n (n = 2,3,…)

Include higher spin

  • 3d HS gravity:
  • hs[λ]
  • CS theory with gravitational

sl(n) sl(2)

Ex) principal embedding

sl(n) = sl(2) ⊕ (

n

⨁

s=3

g(s) )

hs[λ] = B[λ] ⊖ 1

B[λ] = U(sl(2)) ⟨C2 − 1

4 (λ2 − 1)1⟩ [Prokushkin-Vasiliev ’98] [Witten ’88]

spin-(s-1) representation

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SLIDE 15
  • Chern-Simons description of HS gravity

15

HS gravity with gauge sector

sl(M)

  • 3d gravity:

CS theory

sl(2)

[

  • ]

λ = n (n = 2,3,…)

  • 3d HS gravity with

fields:

M × M

Include higher spin

  • 3d HS gravity:
  • hs[λ]
  • CS theory with gravitational

sl(n) sl(2)

Matrix extension Ex) principal embedding

sl(n) = sl(2) ⊕ (

n

⨁

s=3

g(s) )

hs[λ] = B[λ] ⊖ 1

B[λ] = U(sl(2)) ⟨C2 − 1

4 (λ2 − 1)1⟩

hsM[λ] ≃ gl(M) ⊗ B[λ] ⊖ 1M ⊗ 1 ≃ sl(M) ⊗ 1 ⊕ 1M ⊗ hs[λ] ⊕ sl(M) ⊗ hs[λ]

[Prokushkin-Vasiliev ’98] [Witten ’88]

spin-(s-1) representation

[Gaberdiel-Gopakumar ’13, Creutzig-Hikida-Rønne ’13]

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SLIDE 16
  • Chern-Simons description of HS gravity

16

HS gravity with gauge sector

sl(M)

  • 3d gravity:

CS theory

sl(2)

[

  • ]

λ = n (n = 2,3,…)

  • 3d HS gravity with

fields:

M × M

sl(M) ⊗ 1n ⊕ 1M ⊗ sl(n) ⊕ sl(M) ⊗ sl(n) ≃ sl(Mn)

Include higher spin

  • 3d HS gravity:
  • hs[λ]
  • CS theory with gravitational

sl(n) sl(2)

Matrix extension

[ ]

λ = n

Ex) principal embedding

sl(n) = sl(2) ⊕ (

n

⨁

s=3

g(s) )

hs[λ] = B[λ] ⊖ 1

B[λ] = U(sl(2)) ⟨C2 − 1

4 (λ2 − 1)1⟩

hsM[λ] ≃ gl(M) ⊗ B[λ] ⊖ 1M ⊗ 1 ≃ sl(M) ⊗ 1 ⊕ 1M ⊗ hs[λ] ⊕ sl(M) ⊗ hs[λ]

Include gravitational sl(2)

[Prokushkin-Vasiliev ’98] [Witten ’88]

spin-(s-1) representation

[Gaberdiel-Gopakumar ’13, Creutzig-Hikida-Rønne ’13]

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SLIDE 17

[Gaberdiel-Gopakumar ’13, Creutzig-Hikida-Rønne ’13]

  • Chern-Simons description of HS gravity

17

HS gravity with gauge sector

sl(M)

  • 3d gravity:

CS theory

sl(2)

[

  • ]

λ = n (n = 2,3,…)

  • 3d HS gravity with

fields:

M × M

sl(M) ⊗ 1n ⊕ 1M ⊗ sl(n) ⊕ sl(M) ⊗ sl(n) ≃ sl(Mn)

Include higher spin

  • 3d HS gravity:
  • hs[λ]
  • CS theory with gravitational

sl(n) sl(2)

Matrix extension

[ ]

λ = n

Ex) principal embedding

sl(n) = sl(2) ⊕ (

n

⨁

s=3

g(s) )

hs[λ] = B[λ] ⊖ 1

B[λ] = U(sl(2)) ⟨C2 − 1

4 (λ2 − 1)1⟩

hsM[λ] ≃ gl(M) ⊗ B[λ] ⊖ 1M ⊗ 1 ≃ sl(M) ⊗ 1 ⊕ 1M ⊗ hs[λ] ⊕ sl(M) ⊗ hs[λ]

Include gravitational sl(2)

[Prokushkin-Vasiliev ’98]

We examine CS theory decomposed as

  • sl(Mn)

sl(Mn) ≃ sl(M) ⊗ 1n ⊕ 1M ⊗ sl(n) ⊕ sl(M) ⊗ sl(n)

[Witten ’88]

spin-(s-1) representation

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SLIDE 18
  • Gauge field (Solution of EOM)

18

HS gravity with gauge sector

sl(M)

  • 3d gravity: A = e−ρL0a(z)eρL0dz + L0dρ

ρ → ∞ z, ¯ z

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SLIDE 19
  • Gauge field (Solution of EOM)

19

HS gravity with gauge sector

sl(M)

  • 3d gravity:

Include higher spin

  • 3d HS gravity:

A = e−ρL0a(z)eρL0dz + L0dρ

ρ → ∞ z, ¯ z

A = e−ρV 2

0a(z)eρV 2 0dz + V2

0dρ

Ex) case of s = 2

Vs

−s+1,⋯,s−1

V 2

0,±1 ≡ L0,±1

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SLIDE 20
  • Gauge field (Solution of EOM)

20

HS gravity with gauge sector

sl(M)

  • 3d gravity:

Include higher spin

  • 3d HS gravity:

Matrix extension

  • 3d HS gravity with

fields:

M × M

A = e−ρL0a(z)eρL0dz + L0dρ

ρ → ∞ z, ¯ z

A = e−ρV 2

0a(z)eρV 2 0dz + V2

0dρ

A = e−ρ(1M⊗V 2

0)a(z)eρ(1M⊗V 2 0)dz + (1M ⊗ V2

0)dρ

Ex) case of s = 2

Vs

−s+1,⋯,s−1

V 2

0,±1 ≡ L0,±1

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SLIDE 21

21

HS gravity with gauge sector

sl(M)

  • 3d gravity:
  • 3d HS gravity:
  • 3d HS gravity with

fields:

M × M

A = e−ρL0a(z)eρL0dz + L0dρ

A = e−ρV 2

0a(z)eρV 2 0dz + V2

0dρ

A = e−ρ(1M⊗V 2

0)a(z)eρ(1M⊗V 2 0)dz + (1M ⊗ V2

0)dρ

(A − AAdS)|ρ→∞ = 𝒫((eρ)0)

Asymptotically AdS

  • Asymptotically AdS condition
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SLIDE 22

22

HS gravity with gauge sector

sl(M)

  • 3d gravity:
  • 3d HS gravity:
  • 3d HS gravity with

fields:

M × M

A = e−ρL0a(z)eρL0dz + L0dρ

A = e−ρV 2

0a(z)eρV 2 0dz + V2

0dρ

A = e−ρ(1M⊗V 2

0)a(z)eρ(1M⊗V 2 0)dz + (1M ⊗ V2

0)dρ

a(z) = L1 + 1 kCS T(z)L−1

(A − AAdS)|ρ→∞ = 𝒫((eρ)0)

Asymptotically AdS Virasoro generator

  • Asymptotically AdS condition
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SLIDE 23

23

HS gravity with gauge sector

sl(M)

  • 3d gravity:
  • 3d HS gravity:
  • 3d HS gravity with

fields:

M × M

A = e−ρL0a(z)eρL0dz + L0dρ

A = e−ρV 2

0a(z)eρV 2 0dz + V2

0dρ

A = e−ρ(1M⊗V 2

0)a(z)eρ(1M⊗V 2 0)dz + (1M ⊗ V2

0)dρ

a(z) = L1 + 1 kCS T(z)L−1 a(z) = V2

1 + n

∑

s=2

W(s)(z)Vs

−s+1

(A − AAdS)|ρ→∞ = 𝒫((eρ)0)

Asymptotically AdS Virasoro generator

  • generators

WN

  • Asymptotically AdS condition
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SLIDE 24

24

HS gravity with gauge sector

sl(M)

  • 3d gravity:
  • 3d HS gravity:
  • 3d HS gravity with

fields:

M × M

A = e−ρL0a(z)eρL0dz + L0dρ

A = e−ρV 2

0a(z)eρV 2 0dz + V2

0dρ

A = e−ρ(1M⊗V 2

0)a(z)eρ(1M⊗V 2 0)dz + (1M ⊗ V2

0)dρ

a(z) = L1 + 1 kCS T(z)L−1 a(z) = V2

1 + n

∑

s=2

W(s)(z)Vs

−s+1

a(z) = Ja(z)(ta ⊗ 1n) + 1M ⊗ V2

1

+

n

∑

s=2

W(s)(z)(1M ⊗ Vs

−s+1) + n

∑

s=2

Q(s)

a (z)(ta ⊗ Vs −s+1)

(A − AAdS)|ρ→∞ = 𝒫((eρ)0)

Asymptotically AdS Virasoro generator

  • generators

WN

  • Asymptotically AdS condition

sl(Mn) ≃ sl(M ) ⊗ 1n ⊕ 1M ⊗ sl(n) ⊕ sl(M ) ⊗ sl(n)

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SLIDE 25

25

HS gravity with gauge sector

sl(M)

  • 3d gravity:
  • 3d HS gravity:
  • 3d HS gravity with

fields:

M × M

A = e−ρL0a(z)eρL0dz + L0dρ

A = e−ρV 2

0a(z)eρV 2 0dz + V2

0dρ

A = e−ρ(1M⊗V 2

0)a(z)eρ(1M⊗V 2 0)dz + (1M ⊗ V2

0)dρ

a(z) = L1 + 1 kCS T(z)L−1 a(z) = V2

1 + n

∑

s=2

W(s)(z)Vs

−s+1

a(z) = Ja(z)(ta ⊗ 1n) + 1M ⊗ V2

1

+

n

∑

s=2

W(s)(z)(1M ⊗ Vs

−s+1) + n

∑

s=2

Q(s)

a (z)(ta ⊗ Vs −s+1)

(A − AAdS)|ρ→∞ = 𝒫((eρ)0)

Asymptotically AdS Virasoro generator

  • generators

WN

  • Asymptotically AdS condition
  • generators

WN generators of sl(M) Charged higher spin generators

sl(Mn) ≃ sl(M ) ⊗ 1n ⊕ 1M ⊗ sl(n) ⊕ sl(M ) ⊗ sl(n)

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SLIDE 26

26

HS gravity with gauge sector

sl(M)

  • 3d gravity:
  • 3d HS gravity:
  • 3d HS gravity with

fields:

M × M

A = e−ρL0a(z)eρL0dz + L0dρ

A = e−ρV 2

0a(z)eρV 2 0dz + V2

0dρ

A = e−ρ(1M⊗V 2

0)a(z)eρ(1M⊗V 2 0)dz + (1M ⊗ V2

0)dρ

a(z) = L1 + 1 kCS T(z)L−1 a(z) = V2

1 + n

∑

s=2

W(s)(z)Vs

−s+1

a(z) = Ja(z)(ta ⊗ 1n) + 1M ⊗ V2

1

+

n

∑

s=2

W(s)(z)(1M ⊗ Vs

−s+1) + n

∑

s=2

Q(s)

a (z)(ta ⊗ Vs −s+1)

(A − AAdS)|ρ→∞ = 𝒫((eρ)0)

Asymptotically AdS Virasoro generator

  • generators

WN

  • generators

WN generators of sl(M) Charged higher spin generators

  • Asymptotically AdS condition

They are the generators of asymptotic symmetries

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SLIDE 27

27

HS gravity with gauge sector

sl(M)

  • Hamiltonian reduction of

with the embedding

sl(Mn) sl(2)

  • Asymptotic symmetry

We obtain the rectangular W-algebra

Principal embedding correspond to the partition Young diagram

  • f rectangular type

⋮ ⋮ ⋮ ⋯ ⋯ ⋯

⋱

n M

sl(Mn) ≃ sl(M) ⊗ 1n ⊕ 1M ⊗ sl(n) ⊕ sl(M) ⊗ sl(n)

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SLIDE 28

28

HS gravity with gauge sector

sl(M)

  • Hamiltonian reduction of

with the embedding

sl(Mn) sl(2)

  • Asymptotic symmetry

We obtain the rectangular W-algebra

  • The central charge and level of

are evaluated

sl(M)

  • These are consistent with the dual coset model with finite N
  • We need the proper t’ Hooft parameter

λ = k k + N

Principal embedding correspond to the partition Young diagram

  • f rectangular type

⋮ ⋮ ⋮ ⋯ ⋯ ⋯

⋱

n M

(e.g. [Kac-Wakimoto ’03] ) [Creutzig-Hikida ’18]

sl(Mn) ≃ sl(M) ⊗ 1n ⊕ 1M ⊗ sl(n) ⊕ sl(M) ⊗ sl(n)

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SLIDE 29

29

HS gravity with gauge sector

sl(M)

  • Hamiltonian reduction of

with the embedding

sl(Mn) sl(2)

  • Asymptotic symmetry

We obtain the rectangular W-algebra

  • embedding correspond to the partition

sl(2) Mn =

M

n + n + ⋯ + n

Rectangular Young diagram

  • The central charge and level of

are evaluated

sl(M)

  • We need the proper t’ Hooft parameter

λ = k k + N or λ′ = − k k + N + M

Principal embedding correspond to the partition Young diagram

  • f Rectangular type

⋮ ⋮ ⋮ ⋯ ⋯ ⋯

⋱

n M

Let’s turn to the other matrix extension!

(e.g. [Kac-Wakimoto ’03] )

  • These are consistent with the dual coset model with finite N

[Creutzig-Hikida ’18]

sl(Mn) ≃ sl(M) ⊗ 1n ⊕ 1M ⊗ sl(n) ⊕ sl(M) ⊗ sl(n)

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SLIDE 30
  • 1. Introduction
  • 2. HS gravity with

gauge sector

  • 3. Some generalization for extended HS gravity
  • 4. Summary

sl(M)

30

Plan of talk

slide-31
SLIDE 31
  • Restricted

matrix extension

M × M

31

Some generalization

  • 2 restrictions on the extra matrix DOF

sl(M)

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SLIDE 32
  • Restricted

matrix extension

M × M

32

Some generalization

  • 2 restrictions on the extra matrix DOF

sl(M) so(M) sp(2m) for M = 2m

Ex) AT A = AAT = 1M

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SLIDE 33
  • Restricted

matrix extension

M × M

33

Some generalization

  • 2 restrictions on the extra matrix DOF

sl(M) so(M) sp(2m) for M = 2m

  • Even spin truncation

can be truncated to

hs[λ] hse[λ] hs[λ] ≃ hse[λ] ⊕ hso[λ]

Ex) AT A = AAT = 1M

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SLIDE 34
  • Restricted

matrix extension

M × M

34

Some generalization

  • 2 restrictions on the extra matrix DOF

sl(M) so(M) sp(2m) for M = 2m

  • Even spin truncation

can be truncated to

hs[λ] hse[λ] hs[λ] ≃ hse[λ] ⊕ hso[λ]

λ = 2n + 1 λ = 2n

so(2n + 1) sp(2n)

Ex) AT A = AAT = 1M

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SLIDE 35
  • Restricted

matrix extension

M × M

35

Some generalization

  • 2 restrictions on the extra matrix DOF

sl(M) so(M) sp(2m) for M = 2m

  • Even spin truncation

can be truncated to

hs[λ] hse[λ] hs[λ] ≃ hse[λ] ⊕ hso[λ]

λ = 2n + 1 λ = 2n

so(2n + 1) sp(2n)

Ex) (Asymm)T = Asymm

(Aanti−symm)T = − Aanti−symm

We consider 4 CS gravities

so(M) sp(2m) so(2n + 1) sp(2n) so(M(2n + 1)) sp(2Mn) sp(2m(2n + 1)) so(4mn)

M × M hs[λ]

sl(Mn) ≃ sl(M) ⊗ 1n ⊕ 1M ⊗ sl(n) ⊕ sl(M) ⊗ sl(n)

Ex) Decomposition of previous model

slide-36
SLIDE 36
  • Asymptotic symmetry

36

Some generalization

  • We obtain 4 types of rectangular W-algebras

by Hamiltonian reduction Their central charge and level are evaluated

so(M(2n + 1)) sp(2Mn) sp(2m(2n + 1)) so(4mn)

hsso(M)[λ] hssp(2m)[λ]

Matrix type

slide-37
SLIDE 37
  • Asymptotic symmetry

37

Some generalization

  • We obtain 4 types of rectangular W-algebras

by Hamiltonian reduction Their central charge and level are evaluated

  • Dual coset model
  • We propose 2 dual coset models

Their central charge and level are consistent with above algebras with finite N

so(N + M)k so(N)k sp(2N + 2m)k sp(2N)k

with proper t’ Hooft parameters

so(M(2n + 1)) sp(2Mn) sp(2m(2n + 1)) so(4mn)

hsso(M)[λ] hssp(2m)[λ]

Matrix type

λ = k + 1 k + N + 1

slide-38
SLIDE 38

so(M(2n + 1)) sp(2Mn) sp(2m(2n + 1)) so(4mn)

so(N + M)k so(N)k sp(2N + 2m)k sp(2N)k

hsso(M)[λ] hssp(2m)[λ]

Matrix type

Dual with λ =

k − 2 k + N − 2

  • Asymptotic symmetry

38

Some generalization

  • We obtain 4 types of rectangular W-algebras

by Hamiltonian reduction Their central charge and level are evaluated

  • Dual coset model
  • We propose 4 types of dual coset models,

respectively

so(N + M)k so(N)k

  • sp(M|2N)k

sp(2N)k

Other checks of the duality We focus on the OPEs

Their central charge and level are consistent with above algebras with finite N

slide-39
SLIDE 39
  • OPEs (for

)

λ = 2

39

Some generalization

  • with

matrix

sl(2) M × M

slide-40
SLIDE 40
  • We compute the OPEs among generators of each algebras

by requiring associatively

  • OPEs (for

)

λ = 2

40

Some generalization

  • with

matrix

sl(2) M × M

slide-41
SLIDE 41
  • We compute the OPEs among generators of each algebras

by requiring associatively spin1: spin2:

Ja T, Qa

41

Some generalization

  • with

matrix

sl(2) M × M

  • r

currents

so(2M) sp(2m)

EM tensor Spin 2 charged currents

  • OPEs (for

)

λ = 2

slide-42
SLIDE 42
  • We compute the OPEs among generators of each algebras

by requiring associatively spin1: spin2:

Ja T, Qa

42

Some generalization

  • with

matrix

sl(2) M × M

  • r

currents

so(2M) sp(2m)

EM tensor Spin 2 charged currents

  • We compute OPEs among generators of each coset algebras

Above OPEs are reproduced !!

  • OPEs (for

)

λ = 2

slide-43
SLIDE 43
  • 1. Introduction
  • 2. HS gravity with

gauge sector

  • 3. Some generalization for extended HS gravity
  • 4. Summary

sl(M)

43

Plan of talk

slide-44
SLIDE 44

Summary

  • Our question and summary

44

Can we generalize this analysis to other models?

  • We consider 2 ways to truncate the DOF
  • Restricted matrix extensions;
  • Even spin truncation of
  • We propose the dual coset model

and examine the asymptotic symmetry

so(M), sp(M) hs[λ]

An answer (my talk)

That’s all for my presentation

slide-45
SLIDE 45

Back up slides

45

slide-46
SLIDE 46

46

Some generalization

super symmetric models

𝒪 = 1

  • sp(M(2n + 1)|2Mn)

so(2N + 1 + M)k ⊕ so((2N + 1)M)1 so(2N + 1)k+M sp(2N + 2m)k ⊕ so(4Nm)1 sp(2N)k+m

shsso(M)[λ] shssp(2m)[λ]

Matrix type

  • sp(M(2n − 1)|2Mn)
  • sp(4mn|2m(2n + 1)) osp(4mn|2m(2n − 1))
  • We analyze 4 types algebras and propose 2 coset models

with proper t’ Hooft parameters