N-Extention of double-graded supersymmetric and ......parafermionic string theory [Zheltukhin,1987] generalized nucler quasispin [Jarvis,Yang,Wybourne, 1987] Physical application was

N -Extention of double-graded supersymmetric

and superconformal quantum mechanics

Kosuke Amakawa

Dept. of Physical Science,Osaka Prefecture University, Japan

Kosuke Amakawa N -Extention of double-graded SQM and SCM

Introduction

1 dimensionN = 1 Z2 × Z2-graded SUSY QM [Bruce, Duplij, 2019]

H =1

2{Q,Q} =

1

2{Q, Q}, [Q, Q] = H, (H central)

H,Q, Q, H · · · Z2 × Z2-graded superalgebra

Graded superalgebra

generalization of Z2-grading of Lie superalgebra [Ree,1960]

(e.g. Z2 → Z22 = Z2 ⊗ Z2, Z

3n, etc.)

mathematical aspects · · · continuously studied till today[Scheunert, 1983], [Silvestov, 1997], [Chen,et al, 2006], [Bruce, 2019] etc.

physical application · · · not widely known


Introduction

Early works

unification of space-time and internal symmetries[Lukierski, Rittenberg, 1979]

symmetry of de Sitter SUGRA [Vasiliev, 1985]

parafermionic string theory [Zheltukhin, 1987]

generalized nucler quasispin [Jarvis, Yang, Wybourne, 1987]

Physical application was very limited compared with Liesuperalgebras and physical meanings were not clear.

Recent works

connection with parastatistics [Tolstoy, 2014]

symmetry of partial differential equations[Aizawa, Kuznetsova, Tanaka, Toppan 2016]

Z22-graded SUSY QM (Bruce-Duplij model) [Bruce, Duplij, 2019]


Introduction

Our main results

N SQM −→ N Z22-graded SQM

↓ +conformal symmetry

Z22-graded superconformal mechanics

[N. Aizawa,K. Amakawa, S. Doi, 2019]

Today I will talk about

a theorem which plays an important roles to realize Z22-graded

version of SQM

how Bruce-Duplij model is extended to higher values of Nanalyze N = 1 Z2

2-graded SCM


Plan of talk

1 Introduction

2 Z22-graded superalgebra

3 From Lie superalgebra to Z22-graded superalgebra

4 N = 2n SUSY QM

5 Analysis of Z22-graded Superconformal mecanics


Z22-graded superalgebra

Z22 = Z2 × Z2 = {(0, 0), (0, 1), (1, 0), (1, 1)}

grading vector : a = (a1, a2), b = (b1, b2)

sum : a+ b = (a1 + b1, a2 + b2) (mod2,mod2)

e.g . (1, 0) + (1, 1) = (0, 1)

inner product : a · b = a1b1 + a2b2 (mod2)

e.g . (1, 0) · (1, 1) = 1 + 0 = 1


g : vector space on C,

g = g(0,0) ⊕ g(0,1) ⊕ g(1,0) ⊕ g(1,1),∀Xa ∈ ga

bilinear operation : [[Xa,Xb]] ∈ ga+b



Def. Z22-graded superalgebra

If g satisfies the identities

[[Xa,Xb]] = −(−1)a·b[[Xb,Xa]][[·, ·]] : a · b = 0 ⇒ [ , ], a · b = 1 ⇒ { , }

Jacobi identity ：[[Xa, [[Xb,Xc]]]] = [[[[Xa,Xb]],Xc]] + (−1)a·b[[Xb, [[Xa,Xc]]]]

then g is called Z22-graded superalgebra.

e.g. [[g(0,0), g(1,1)]] = [g(0,0), g(1,1)] ⊆ g(1,1),

[[g(0,1), g(1,1)]] = {g(0,1), g(1,1)} ⊆ g(1,0)


Z22-graded SUSY algebra [Bruce, 2019]

g(0,0) g(0,1) g(1,0) g(1,1)

H Q Q H

Relations :

(0, 1) · (0, 1) = (1, 0) · (1, 0) = 1, {Q,Q} = {Q, Q} = 2H(0, 1) · (1, 0) = 0, [Q, Q] = 2H(0, 0) · (a1, a2) = 0, [H,X ] = 0, X = Q, Q, H(1, 1) · (0, 1) = (1, 1) · (1, 0) = 1, {H,Q} = {H, Q} = 0

(H,Q), (H, Q) : standard SUSY algebra

The whole system · · · doubling of N = 1 SQM closed in a Z22

-graded superalgebra


Our goal

N SQM −→ N Z22-graded SQM



Extensions of Bruce-Duplij model to

1 N pairs of SUSY algebra

g(0,0) g(0,1) g(1,0) g(1,1)

H Qa Qa H a = 1, 2, . . . ,N

{Qa,Qb} = {Qa, Qb} = 2δabH, [Qa, Qb] = 2δabH2 superconformal mechanics

{Sa,Sb} = {Sa, Sb} ∼ δabK, {Qa,Sb} = {Qa, Sb} ∼ δabD

so(1, 2) = 〈 H, D, K 〉Kosuke Amakawa N -Extention of double-graded SQM and SCM

From Lie superalgebra to Z22-graded superalgebra

Lie superalgebra : T ai , a ∈ Z2 = {0, 1}

[T 0i ,T

0j ] = if kij T

0k , [T

0i ,T

1j ] = ihkijT

0k , {T 1

i ,T1i } = gk

ij T1k

Assumption

matrix representations are given∃Γ s.t. {Γ,T 1

i } = 0, Γ2 = I2m

Theorem

T 0i = 1⊗ T 0

i , T 1i = 1⊗ T 1

i , T 1i = σ1 ⊗ iT 1

i Γ, T 0i = σ1 ⊗ T 0

i Γ(0, 0) (0, 1) (1, 0) (1, 1)

form a Z22-graded superalgebra


Extention of Lie superalgebra to Z22-graded superalgebra

Proof

relations → direct computation

Z22-graded Jacobi → Jacobi for Lie superalgebra

relations：

[T 0i ,T 0

j ] = if kij T 0k , [T 0

i ,T 1j ] = ihkijT 1

k ,

[T 0i , T 1

j ] = ihkij T 1k , [T 0

i , T 0j ] = if kij T 0

k ,

{T 1i ,T 1

j } = {T 1i , T 1

j } = gkij T 0

k , [T 0i , T 0

j ] = if kij T 0k ,

[T 1i , T 1

j ] = igkij T 0

k , {T 0i ,T 1

j } = −hkij T 1k ,

{T 0i , T 1

j } = hkijT 1k

same structure constants as Lie superalgebra∃Γ s.t. {Γ,T 1

i } = 0, Γ2 = I2m ⇒ Z22-graded superalgebra


N = 2n SUSY QM

Akulov-Kudinov construction [V.Akulov, M.Kudinov, 1999]

block-antidiagonal matrices (2n × 2n)

{γI , γJ} = 2δIJ 1, γ†I = γI , I = 1, 2, . . . , 2n

define matrices

γ±a =1

2(γ2a−1 ± iγ2a), Γa = iaγ1γ2 . . . γ2a, a = 1, 2, . . . , n

properties of Γa

{γ±k , Γa} = 0 (k ≤ a), Γ2a = I2n

2n supercharge

Q+a =

1√2γ+a (p+iW (n)

a (x , Γ1, . . . , Γn)), Q−a = (Q+

a )†, a = 1, 2, · · · , n


N = 2n SUSY QM

Akulov-Kudinov construction [V.Akulov, M.Kudinov, 1999]

recursive definition of superpotentials

n = 1 W (1) = w0(x)

n = 2 W(2)1 = w0(x) + Γ2w1(x), w1(x) =

∂xw0(x)

2w0(x)

N = 2n SUSY algebra

{Q+a ,Q

−b } = δabH , [H ,Q±

a ] = 0

{Q±a , Γn} = 0, ∀a ⇒ theorem applicable


Z22-graded version of N = 2n SUSY QM

Z22-graded Super-Poincare algebra

H = I2 ⊗ H, Q±a = I2 ⊗ Q±

a , Q±a = iQ±

a A, H = HA

A = σ1 ⊗ Γn

{Q+a ,Q−

b } = {Q+a , Q−

b } = δabH, [Q±a , Q∓

b ] = iδabH

Example : N = 2 (n = 1)

Q± =

(

Q± 00 Q±

)

, Q± = ±i

(

0 Q±

Q± 0

)

,

H = {Q+,Q−} = {Q+, Q−} = diag(H1,H2,H1,H2),

H =1

i[Q±, Q∓] =

0 0 −H1 00 0 0 H2

−H1 0 0 00 H2 0 0

Z22-graded SQM with higher values of N can be realized


N = 1 Superconformal mechanics

N = 1 Superconformal mechanics [S.Fubini, E.Rabinovici, 1984]

Q =1√2

(

σ1p − σ2β

x

)

, S =x√2σ1

H =1

2

(

p2 +β2

x2

)

I2 +β

2x2σ3, D = −1

4{x , p}I2, K =

x2

2I2

β > 0 : coupling constant

{Q,Q} = 2H , {S , S} = 2K , {Q, S} = −2D,

〈 H ,D,K 〉 = so(1, 2)

spectrum of H · · · continuous

H has no normalizable ground state

Γ = σ3 satisfies {Q, Γ} = {S , Γ} = 0 ⇒ theorem applicable


N = 1 Z22-graded Superconformal mechanics

Z22-graded Superconformal mechanics model

(0, 0) : H =

(

H 00 H

)

, D =

(

D 00 D

)

, K =

(

K 00 K

)

,

(0, 1) : Q =

(

Q 00 Q

)

, S =

(

S 00 S

)

,

(1, 0) : Q = i

(

0 Qσ3Qσ3 0

)

, S = i

(

0 Sσ3Sσ3 0

)

,

(1, 1) : H =

(

0 Hσ3Hσ3 0

)

, D =

(

0 Dσ3Dσ3 0

)

, K =

(

0 Kσ3Kσ3 0

)

H has no normalizable ground state⇒ standerd procedure in conformal mechanics



new operators　R = H +K (Hamiltonian) (0, 0)　 a = S + iQ, a = S + iQ 　 (annihilation)　 (0, 1) (1, 0) 　　　　　　　　

R =1

2{a, a†} =

1

2{a, a†},

[R, a] = −a, [R, a†] = a†, [R, a] = −a, [R, a†] = a†

[a, a†] = [a, a†] = I4 − 2βF , F =

(

σ3 00 σ3

)

⇒ R = a†a+1

2(I4 − 2βF) = a†a+

1

2(I4 − 2βF)



Hilbert space : H = H00 ⊕ H01 ⊕ H11 ⊕ H10

ψ(x) = (φ00, φ01, φ11, φ10)T

ground state of R : aΨ(x) = aΨ(x) = 0

⇒ 2-fold degeneracy H00 ⊕ H11 or H01 ⊕ H10

⇒ spectrum of R is descrete

theorem applicable for N = 2, 4, 8, etc.


Summary

N SUSY QM −→ N Z22-graded SUSY QM



[N. Aizawa,K. Amakawa, S. Doi, 2019]

Extension of Bruce-Duplij model is realized by

the theorem which converts a given Lie superalgebra to a Z22

-graded superalgebra

Akulov-Kudinov construction (for the case of N extension)

⇒ factrizable Hamiltonian may have Z22-graded supersymmetry

our results + examples in Introduction⇓

graded superalgebra would have deep connection with physics



spectrum of R

H00 H01 H11 H10

a† a a

† aa†

a

a† a a

† aa†

a

a†

a

•

a†

a

◦

a†

a•

a†

a

◦

12(1− 2β)

　


hyper Pauli matrices (2n+1 × 2n+1)

A =

(

0 ΓnΓn 0

)

, Σ±a =

(

γ±a 00 γ±a

)

, α±a = iΣ±

a A (1)

Z22-degree : deg(I2n+1) = (0, 0), deg(Σ±

a ) = (0, 1)

deg(α±a ) = (1, 0), deg(A) = (1, 1)

⇒ 〈(1), I2n+1〉 : Z22-graded superalgebra

{Σ+a ,Σ

−b } = {α+

a , α−b } = δabI2n+1 , [Σ±

a , α∓b ] = iδabA

non-zero entries :

(00) (01) (11) (10)(01) (00) (10) (11)(11) (10) (00) (01)(10) (11) (01) (00)


Example: N = 2 (n = 1) double-graded SUSY QM

Properties

Hilbert space

H = L2(R)⊗ C4 = H00 ⊕ H01 ⊕ H11 ⊕ H10

ψ(x) = (ϕ00, ϕ01, ϕ11, ϕ10)T ∈ H

ground state: Q±ψ = Q±ψ = 0

Q−, Q− ⇒ ψ =

C1

0C3

0

e−∫w0dx , Q+, Q+ ⇒ ψ =

0C2

0C4

e∫w0dx

normalizable states ⇒ 2-fold degenerate

(H00 ⊕ H11 or H01 ⊕ H10)


Example: N = 2 (n = 1) double-graded SUSY QM

Properties

excited states · · · 4-fold degeneracy

Hψ00 = Eψ00

Cf. Hψ00 = −Eψ11

0

ψ00

ψ11

0

ψ010 ψ10 0

Q+ Q+

Q+ Q+

Q−

Q−

Q+Q−

Q+

Q−

Q−

Q+

Q+

Q−


Documents

N-Extention of double-graded supersymmetric and ......parafermionic string theory [Zheltukhin,1987] generalized nucler quasispin [Jarvis,Yang,Wybourne, 1987] Physical application was