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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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]
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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
Kosuke Amakawa N -Extention of double-graded SQM and 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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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
Z22-graded superalgebra
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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
Z22-graded superalgebra
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)
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
Our goal
N SQM −→ N Z22-graded SQM
↓ +conformal symmetry
Z22-graded superconformal mechanics
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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
N = 1 Z22-graded Superconformal 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)
Kosuke Amakawa N -Extention of double-graded SQM and SCM
N = 1 Z22-graded Superconformal mechanics
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.
Kosuke Amakawa N -Extention of double-graded SQM and SCM
Summary
N SUSY QM −→ N Z22-graded SUSY QM
↓ +conformal symmetry
Z22-graded superconformal mechanics
[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
Kosuke Amakawa N -Extention of double-graded SQM and SCM
N = 1 Z22-graded Superconformal mechanics
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)
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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)
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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)
Kosuke Amakawa N -Extention of double-graded SQM and SCM
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−
Kosuke Amakawa N -Extention of double-graded SQM and SCM