- Source: Superconformal algebra
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group (in two Euclidean dimensions, the Lie superalgebra does not generate any Lie supergroup).
Superconformal algebra in dimension greater than 2
The conformal group of the
(
p
+
q
)
{\displaystyle (p+q)}
-dimensional space
R
p
,
q
{\displaystyle \mathbb {R} ^{p,q}}
is
S
O
(
p
+
1
,
q
+
1
)
{\displaystyle SO(p+1,q+1)}
and its Lie algebra is
s
o
(
p
+
1
,
q
+
1
)
{\displaystyle {\mathfrak {so}}(p+1,q+1)}
. The superconformal algebra is a Lie superalgebra containing the bosonic factor
s
o
(
p
+
1
,
q
+
1
)
{\displaystyle {\mathfrak {so}}(p+1,q+1)}
and whose odd generators transform in spinor representations of
s
o
(
p
+
1
,
q
+
1
)
{\displaystyle {\mathfrak {so}}(p+1,q+1)}
. Given Kac's classification of finite-dimensional simple Lie superalgebras, this can only happen for small values of
p
{\displaystyle p}
and
q
{\displaystyle q}
. A (possibly incomplete) list is
o
s
p
∗
(
2
N
|
2
,
2
)
{\displaystyle {\mathfrak {osp}}^{*}(2N|2,2)}
in 3+0D thanks to
u
s
p
(
2
,
2
)
≃
s
o
(
4
,
1
)
{\displaystyle {\mathfrak {usp}}(2,2)\simeq {\mathfrak {so}}(4,1)}
;
o
s
p
(
N
|
4
)
{\displaystyle {\mathfrak {osp}}(N|4)}
in 2+1D thanks to
s
p
(
4
,
R
)
≃
s
o
(
3
,
2
)
{\displaystyle {\mathfrak {sp}}(4,\mathbb {R} )\simeq {\mathfrak {so}}(3,2)}
;
s
u
∗
(
2
N
|
4
)
{\displaystyle {\mathfrak {su}}^{*}(2N|4)}
in 4+0D thanks to
s
u
∗
(
4
)
≃
s
o
(
5
,
1
)
{\displaystyle {\mathfrak {su}}^{*}(4)\simeq {\mathfrak {so}}(5,1)}
;
s
u
(
2
,
2
|
N
)
{\displaystyle {\mathfrak {su}}(2,2|N)}
in 3+1D thanks to
s
u
(
2
,
2
)
≃
s
o
(
4
,
2
)
{\displaystyle {\mathfrak {su}}(2,2)\simeq {\mathfrak {so}}(4,2)}
;
s
l
(
4
|
N
)
{\displaystyle {\mathfrak {sl}}(4|N)}
in 2+2D thanks to
s
l
(
4
,
R
)
≃
s
o
(
3
,
3
)
{\displaystyle {\mathfrak {sl}}(4,\mathbb {R} )\simeq {\mathfrak {so}}(3,3)}
;
real forms of
F
(
4
)
{\displaystyle F(4)}
in five dimensions
o
s
p
(
8
∗
|
2
N
)
{\displaystyle {\mathfrak {osp}}(8^{*}|2N)}
in 5+1D, thanks to the fact that spinor and fundamental representations of
s
o
(
8
,
C
)
{\displaystyle {\mathfrak {so}}(8,\mathbb {C} )}
are mapped to each other by outer automorphisms.
Superconformal algebra in 3+1D
According to the superconformal algebra with
N
{\displaystyle {\mathcal {N}}}
supersymmetries in 3+1 dimensions is given by the bosonic generators
P
μ
{\displaystyle P_{\mu }}
,
D
{\displaystyle D}
,
M
μ
ν
{\displaystyle M_{\mu \nu }}
,
K
μ
{\displaystyle K_{\mu }}
, the U(1) R-symmetry
A
{\displaystyle A}
, the SU(N) R-symmetry
T
j
i
{\displaystyle T_{j}^{i}}
and the fermionic generators
Q
α
i
{\displaystyle Q^{\alpha i}}
,
Q
¯
i
α
˙
{\displaystyle {\overline {Q}}_{i}^{\dot {\alpha }}}
,
S
i
α
{\displaystyle S_{i}^{\alpha }}
and
S
¯
α
˙
i
{\displaystyle {\overline {S}}^{{\dot {\alpha }}i}}
. Here,
μ
,
ν
,
ρ
,
…
{\displaystyle \mu ,\nu ,\rho ,\dots }
denote spacetime indices;
α
,
β
,
…
{\displaystyle \alpha ,\beta ,\dots }
left-handed Weyl spinor indices;
α
˙
,
β
˙
,
…
{\displaystyle {\dot {\alpha }},{\dot {\beta }},\dots }
right-handed Weyl spinor indices; and
i
,
j
,
…
{\displaystyle i,j,\dots }
the internal R-symmetry indices.
The Lie superbrackets of the bosonic conformal algebra are given by
[
M
μ
ν
,
M
ρ
σ
]
=
η
ν
ρ
M
μ
σ
−
η
μ
ρ
M
ν
σ
+
η
ν
σ
M
ρ
μ
−
η
μ
σ
M
ρ
ν
{\displaystyle [M_{\mu \nu },M_{\rho \sigma }]=\eta _{\nu \rho }M_{\mu \sigma }-\eta _{\mu \rho }M_{\nu \sigma }+\eta _{\nu \sigma }M_{\rho \mu }-\eta _{\mu \sigma }M_{\rho \nu }}
[
M
μ
ν
,
P
ρ
]
=
η
ν
ρ
P
μ
−
η
μ
ρ
P
ν
{\displaystyle [M_{\mu \nu },P_{\rho }]=\eta _{\nu \rho }P_{\mu }-\eta _{\mu \rho }P_{\nu }}
[
M
μ
ν
,
K
ρ
]
=
η
ν
ρ
K
μ
−
η
μ
ρ
K
ν
{\displaystyle [M_{\mu \nu },K_{\rho }]=\eta _{\nu \rho }K_{\mu }-\eta _{\mu \rho }K_{\nu }}
[
M
μ
ν
,
D
]
=
0
{\displaystyle [M_{\mu \nu },D]=0}
[
D
,
P
ρ
]
=
−
P
ρ
{\displaystyle [D,P_{\rho }]=-P_{\rho }}
[
D
,
K
ρ
]
=
+
K
ρ
{\displaystyle [D,K_{\rho }]=+K_{\rho }}
[
P
μ
,
K
ν
]
=
−
2
M
μ
ν
+
2
η
μ
ν
D
{\displaystyle [P_{\mu },K_{\nu }]=-2M_{\mu \nu }+2\eta _{\mu \nu }D}
[
K
n
,
K
m
]
=
0
{\displaystyle [K_{n},K_{m}]=0}
[
P
n
,
P
m
]
=
0
{\displaystyle [P_{n},P_{m}]=0}
where η is the Minkowski metric; while the ones for the fermionic generators are:
{
Q
α
i
,
Q
¯
β
˙
j
}
=
2
δ
i
j
σ
α
β
˙
μ
P
μ
{\displaystyle \left\{Q_{\alpha i},{\overline {Q}}_{\dot {\beta }}^{j}\right\}=2\delta _{i}^{j}\sigma _{\alpha {\dot {\beta }}}^{\mu }P_{\mu }}
{
Q
,
Q
}
=
{
Q
¯
,
Q
¯
}
=
0
{\displaystyle \left\{Q,Q\right\}=\left\{{\overline {Q}},{\overline {Q}}\right\}=0}
{
S
α
i
,
S
¯
β
˙
j
}
=
2
δ
j
i
σ
α
β
˙
μ
K
μ
{\displaystyle \left\{S_{\alpha }^{i},{\overline {S}}_{{\dot {\beta }}j}\right\}=2\delta _{j}^{i}\sigma _{\alpha {\dot {\beta }}}^{\mu }K_{\mu }}
{
S
,
S
}
=
{
S
¯
,
S
¯
}
=
0
{\displaystyle \left\{S,S\right\}=\left\{{\overline {S}},{\overline {S}}\right\}=0}
{
Q
,
S
}
=
{\displaystyle \left\{Q,S\right\}=}
{
Q
,
S
¯
}
=
{
Q
¯
,
S
}
=
0
{\displaystyle \left\{Q,{\overline {S}}\right\}=\left\{{\overline {Q}},S\right\}=0}
The bosonic conformal generators do not carry any R-charges, as they commute with the R-symmetry generators:
[
A
,
M
]
=
[
A
,
D
]
=
[
A
,
P
]
=
[
A
,
K
]
=
0
{\displaystyle [A,M]=[A,D]=[A,P]=[A,K]=0}
[
T
,
M
]
=
[
T
,
D
]
=
[
T
,
P
]
=
[
T
,
K
]
=
0
{\displaystyle [T,M]=[T,D]=[T,P]=[T,K]=0}
But the fermionic generators do carry R-charge:
[
A
,
Q
]
=
−
1
2
Q
{\displaystyle [A,Q]=-{\frac {1}{2}}Q}
[
A
,
Q
¯
]
=
1
2
Q
¯
{\displaystyle [A,{\overline {Q}}]={\frac {1}{2}}{\overline {Q}}}
[
A
,
S
]
=
1
2
S
{\displaystyle [A,S]={\frac {1}{2}}S}
[
A
,
S
¯
]
=
−
1
2
S
¯
{\displaystyle [A,{\overline {S}}]=-{\frac {1}{2}}{\overline {S}}}
[
T
j
i
,
Q
k
]
=
−
δ
k
i
Q
j
{\displaystyle [T_{j}^{i},Q_{k}]=-\delta _{k}^{i}Q_{j}}
[
T
j
i
,
Q
¯
k
]
=
δ
j
k
Q
¯
i
{\displaystyle [T_{j}^{i},{\overline {Q}}^{k}]=\delta _{j}^{k}{\overline {Q}}^{i}}
[
T
j
i
,
S
k
]
=
δ
j
k
S
i
{\displaystyle [T_{j}^{i},S^{k}]=\delta _{j}^{k}S^{i}}
[
T
j
i
,
S
¯
k
]
=
−
δ
k
i
S
¯
j
{\displaystyle [T_{j}^{i},{\overline {S}}_{k}]=-\delta _{k}^{i}{\overline {S}}_{j}}
Under bosonic conformal transformations, the fermionic generators transform as:
[
D
,
Q
]
=
−
1
2
Q
{\displaystyle [D,Q]=-{\frac {1}{2}}Q}
[
D
,
Q
¯
]
=
−
1
2
Q
¯
{\displaystyle [D,{\overline {Q}}]=-{\frac {1}{2}}{\overline {Q}}}
[
D
,
S
]
=
1
2
S
{\displaystyle [D,S]={\frac {1}{2}}S}
[
D
,
S
¯
]
=
1
2
S
¯
{\displaystyle [D,{\overline {S}}]={\frac {1}{2}}{\overline {S}}}
[
P
,
Q
]
=
[
P
,
Q
¯
]
=
0
{\displaystyle [P,Q]=[P,{\overline {Q}}]=0}
[
K
,
S
]
=
[
K
,
S
¯
]
=
0
{\displaystyle [K,S]=[K,{\overline {S}}]=0}
Superconformal algebra in 2D
There are two possible algebras with minimal supersymmetry in two dimensions; a Neveu–Schwarz algebra and a Ramond algebra. Additional supersymmetry is possible, for instance the N = 2 superconformal algebra.
See also
Conformal symmetry
Super Virasoro algebra
Supersymmetry algebra
References
Kata Kunci Pencarian:
- Superconformal algebra
- N = 2 superconformal algebra
- Virasoro algebra
- Supersymmetry algebra
- M-theory
- Kac–Moody algebra
- Vertex operator algebra
- Superstring theory
- Supersymmetry
- Loop algebra
