- Source: N = 2 superconformal algebra
- N = 2 superconformal algebra
- Superconformal algebra
- Super Virasoro algebra
- Supersymmetry algebra
- Vertex operator algebra
- Virasoro algebra
- Primary field
- Kac–Moody algebra
- Adrian Kent
- Haag–Łopuszański–Sohnius theorem
Artikel: N = 2 superconformal algebra GudangMovies21 Rebahinxxi
In mathematical physics, the 2D N = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and two-dimensional conformal field theory. It has important applications in mirror symmetry. It was introduced by M. Ademollo, L. Brink, and A. D'Adda et al. (1976) as a gauge algebra of the U(1) fermionic string.
Definition
There are two slightly different ways to describe the N = 2 superconformal algebra, called the N = 2 Ramond algebra and the N = 2 Neveu–Schwarz algebra, which are isomorphic (see below) but differ in the choice of standard basis.
The N = 2 superconformal algebra is the Lie superalgebra with basis of even elements c, Ln, Jn, for n an integer, and odd elements G+r, G−r, where
r
∈
Z
{\displaystyle r\in {\mathbb {Z} }}
(for the Ramond basis) or
r
∈
1
2
+
Z
{\textstyle r\in {1 \over 2}+{\mathbb {Z} }}
(for the Neveu–Schwarz basis) defined by the following relations:
c is in the center
[
L
m
,
L
n
]
=
(
m
−
n
)
L
m
+
n
+
c
12
(
m
3
−
m
)
δ
m
+
n
,
0
{\displaystyle [L_{m},L_{n}]=\left(m-n\right)L_{m+n}+{c \over 12}\left(m^{3}-m\right)\delta _{m+n,0}}
[
L
m
,
J
n
]
=
−
n
J
m
+
n
{\displaystyle [L_{m},\,J_{n}]=-nJ_{m+n}}
[
J
m
,
J
n
]
=
c
3
m
δ
m
+
n
,
0
{\displaystyle [J_{m},J_{n}]={c \over 3}m\delta _{m+n,0}}
{
G
r
+
,
G
s
−
}
=
L
r
+
s
+
1
2
(
r
−
s
)
J
r
+
s
+
c
6
(
r
2
−
1
4
)
δ
r
+
s
,
0
{\displaystyle \{G_{r}^{+},G_{s}^{-}\}=L_{r+s}+{1 \over 2}\left(r-s\right)J_{r+s}+{c \over 6}\left(r^{2}-{1 \over 4}\right)\delta _{r+s,0}}
{
G
r
+
,
G
s
+
}
=
0
=
{
G
r
−
,
G
s
−
}
{\displaystyle \{G_{r}^{+},G_{s}^{+}\}=0=\{G_{r}^{-},G_{s}^{-}\}}
[
L
m
,
G
r
±
]
=
(
m
2
−
r
)
G
r
+
m
±
{\displaystyle [L_{m},G_{r}^{\pm }]=\left({m \over 2}-r\right)G_{r+m}^{\pm }}
[
J
m
,
G
r
±
]
=
±
G
m
+
r
±
{\displaystyle [J_{m},G_{r}^{\pm }]=\pm G_{m+r}^{\pm }}
If
r
,
s
∈
Z
{\displaystyle r,s\in {\mathbb {Z} }}
in these relations, this yields the
N = 2 Ramond algebra; while if
r
,
s
∈
1
2
+
Z
{\textstyle r,s\in {1 \over 2}+{\mathbb {Z} }}
are half-integers, it gives the N = 2 Neveu–Schwarz algebra. The operators
L
n
{\displaystyle L_{n}}
generate a Lie subalgebra isomorphic to the Virasoro algebra. Together with the operators
G
r
=
G
r
+
+
G
r
−
{\displaystyle G_{r}=G_{r}^{+}+G_{r}^{-}}
, they generate a Lie superalgebra isomorphic to the super Virasoro algebra,
giving the Ramond algebra if
r
,
s
{\displaystyle r,s}
are integers and the Neveu–Schwarz algebra otherwise. When represented as operators on a complex inner product space,
c
{\displaystyle c}
is taken to act as multiplication by a real scalar, denoted by the same letter and called the central charge, and the adjoint structure is as follows:
L
n
∗
=
L
−
n
,
J
m
∗
=
J
−
m
,
(
G
r
±
)
∗
=
G
−
r
∓
,
c
∗
=
c
{\displaystyle {L_{n}^{*}=L_{-n},\,\,J_{m}^{*}=J_{-m},\,\,(G_{r}^{\pm })^{*}=G_{-r}^{\mp },\,\,c^{*}=c}}
Properties
The N = 2 Ramond and Neveu–Schwarz algebras are isomorphic by the spectral shift isomorphism
α
{\displaystyle \alpha }
of Schwimmer & Seiberg (1987):
α
(
L
n
)
=
L
n
+
1
2
J
n
+
c
24
δ
n
,
0
{\displaystyle \alpha (L_{n})=L_{n}+{1 \over 2}J_{n}+{c \over 24}\delta _{n,0}}
α
(
J
n
)
=
J
n
+
c
6
δ
n
,
0
{\displaystyle \alpha (J_{n})=J_{n}+{c \over 6}\delta _{n,0}}
α
(
G
r
±
)
=
G
r
±
1
2
±
{\displaystyle \alpha (G_{r}^{\pm })=G_{r\pm {1 \over 2}}^{\pm }}
with inverse:
α
−
1
(
L
n
)
=
L
n
−
1
2
J
n
+
c
24
δ
n
,
0
{\displaystyle \alpha ^{-1}(L_{n})=L_{n}-{1 \over 2}J_{n}+{c \over 24}\delta _{n,0}}
α
−
1
(
J
n
)
=
J
n
−
c
6
δ
n
,
0
{\displaystyle \alpha ^{-1}(J_{n})=J_{n}-{c \over 6}\delta _{n,0}}
α
−
1
(
G
r
±
)
=
G
r
∓
1
2
±
{\displaystyle \alpha ^{-1}(G_{r}^{\pm })=G_{r\mp {1 \over 2}}^{\pm }}
In the N = 2 Ramond algebra, the zero mode operators
L
0
{\displaystyle L_{0}}
,
J
0
{\displaystyle J_{0}}
,
G
0
±
{\displaystyle G_{0}^{\pm }}
and the constants form a five-dimensional Lie superalgebra. They satisfy the same relations as the fundamental operators in Kähler geometry, with
L
0
{\displaystyle L_{0}}
corresponding to the Laplacian,
J
0
{\displaystyle J_{0}}
the degree operator, and
G
0
±
{\displaystyle G_{0}^{\pm }}
the
∂
{\displaystyle \partial }
and
∂
¯
{\displaystyle {\overline {\partial }}}
operators.
Even integer powers of the spectral shift give automorphisms of the N = 2 superconformal algebras, called spectral shift automorphisms. Another automorphism
β
{\displaystyle \beta }
, of period two, is given by
β
(
L
m
)
=
L
m
,
{\displaystyle \beta (L_{m})=L_{m},}
β
(
J
m
)
=
−
J
m
−
c
3
δ
m
,
0
,
{\displaystyle \beta (J_{m})=-J_{m}-{c \over 3}\delta _{m,0},}
β
(
G
r
±
)
=
G
r
∓
{\displaystyle \beta (G_{r}^{\pm })=G_{r}^{\mp }}
In terms of Kähler operators,
β
{\displaystyle \beta }
corresponds to conjugating the complex structure. Since
β
α
β
−
1
=
α
−
1
{\displaystyle \beta \alpha \beta ^{-1}=\alpha ^{-1}}
, the automorphisms
α
2
{\displaystyle \alpha ^{2}}
and
β
{\displaystyle \beta }
generate a group of automorphisms of the N = 2 superconformal algebra isomorphic to the infinite dihedral group
Z
⋊
Z
2
{\displaystyle {\mathbb {Z} }\rtimes {\mathbb {Z} }_{2}}
.
Twisted operators
L
n
=
L
n
+
1
2
(
n
+
1
)
J
n
{\textstyle {\mathcal {L}}_{n}=L_{n}+{1 \over 2}(n+1)J_{n}}
were introduced by Eguchi & Yang (1990) and satisfy:
[
L
m
,
L
n
]
=
(
m
−
n
)
L
m
+
n
{\displaystyle [{\mathcal {L}}_{m},{\mathcal {L}}_{n}]=(m-n){\mathcal {L}}_{m+n}}
so that these operators satisfy the Virasoro relation with central charge 0. The constant
c
{\displaystyle c}
still appears in the relations for
J
m
{\displaystyle J_{m}}
and the modified relations
[
L
m
,
J
n
]
=
−
n
J
m
+
n
+
c
6
(
m
2
+
m
)
δ
m
+
n
,
0
{\displaystyle [{\mathcal {L}}_{m},J_{n}]=-nJ_{m+n}+{c \over 6}\left(m^{2}+m\right)\delta _{m+n,0}}
{
G
r
+
,
G
s
−
}
=
2
L
r
+
s
−
2
s
J
r
+
s
+
c
3
(
m
2
+
m
)
δ
m
+
n
,
0
{\displaystyle \{G_{r}^{+},G_{s}^{-}\}=2{\mathcal {L}}_{r+s}-2sJ_{r+s}+{c \over 3}\left(m^{2}+m\right)\delta _{m+n,0}}
Constructions
= Free field construction
=Green, Schwarz, and Witten (1988a, 1988b) give a construction using two commuting real bosonic fields
(
a
n
)
{\displaystyle (a_{n})}
,
(
b
n
)
{\displaystyle (b_{n})}
[
a
m
,
a
n
]
=
m
2
δ
m
+
n
,
0
,
[
b
m
,
b
n
]
=
m
2
δ
m
+
n
,
0
,
a
n
∗
=
a
−
n
,
b
n
∗
=
b
−
n
{\displaystyle {[a_{m},a_{n}]={m \over 2}\delta _{m+n,0},\,\,\,\,[b_{m},b_{n}]={m \over 2}\delta _{m+n,0}},\,\,\,\,a_{n}^{*}=a_{-n},\,\,\,\,b_{n}^{*}=b_{-n}}
and a complex fermionic field
(
e
r
)
{\displaystyle (e_{r})}
{
e
r
,
e
s
∗
}
=
δ
r
,
s
,
{
e
r
,
e
s
}
=
0.
{\displaystyle \{e_{r},e_{s}^{*}\}=\delta _{r,s},\,\,\,\,\{e_{r},e_{s}\}=0.}
L
n
{\displaystyle L_{n}}
is defined to the sum of the Virasoro operators naturally associated with each of the three systems
L
n
=
∑
m
:
a
−
m
+
n
a
m
:
+
∑
m
:
b
−
m
+
n
b
m
:
+
∑
r
(
r
+
n
2
)
:
e
r
∗
e
n
+
r
:
{\displaystyle L_{n}=\sum _{m}:a_{-m+n}a_{m}:+\sum _{m}:b_{-m+n}b_{m}:+\sum _{r}\left(r+{n \over 2}\right):e_{r}^{*}e_{n+r}:}
where normal ordering has been used for bosons and fermions.
The current operator
J
n
{\displaystyle J_{n}}
is defined by the standard construction from fermions
J
n
=
∑
r
:
e
r
∗
e
n
+
r
:
{\displaystyle J_{n}=\sum _{r}:e_{r}^{*}e_{n+r}:}
and the two supersymmetric operators
G
r
±
{\displaystyle G_{r}^{\pm }}
by
G
r
+
=
∑
(
a
−
m
+
i
b
−
m
)
⋅
e
r
+
m
,
G
r
−
=
∑
(
a
r
+
m
−
i
b
r
+
m
)
⋅
e
m
∗
{\displaystyle G_{r}^{+}=\sum (a_{-m}+ib_{-m})\cdot e_{r+m},\,\,\,\,G_{r}^{-}=\sum (a_{r+m}-ib_{r+m})\cdot e_{m}^{*}}
This yields an N = 2 Neveu–Schwarz algebra with c = 3.
= SU(2) supersymmetric coset construction
=Di Vecchia et al. (1986) gave a coset construction of the N = 2 superconformal algebras, generalizing the coset constructions of Goddard, Kent & Olive (1986) for the discrete series representations of the Virasoro and super Virasoro algebra. Given a representation of the affine Kac–Moody algebra of SU(2) at level
ℓ
{\displaystyle \ell }
with basis
E
n
,
F
n
,
H
n
{\displaystyle E_{n},F_{n},H_{n}}
satisfying
[
H
m
,
H
n
]
=
2
m
ℓ
δ
n
+
m
,
0
,
{\displaystyle [H_{m},H_{n}]=2m\ell \delta _{n+m,0},}
[
E
m
,
F
n
]
=
H
m
+
n
+
m
ℓ
δ
m
+
n
,
0
,
{\displaystyle [E_{m},F_{n}]=H_{m+n}+m\ell \delta _{m+n,0},}
[
H
m
,
E
n
]
=
2
E
m
+
n
,
{\displaystyle [H_{m},E_{n}]=2E_{m+n},}
[
H
m
,
F
n
]
=
−
2
F
m
+
n
,
{\displaystyle [H_{m},F_{n}]=-2F_{m+n},}
the supersymmetric generators are defined by
G
r
+
=
(
ℓ
/
2
+
1
)
−
1
/
2
∑
E
−
m
⋅
e
m
+
r
,
G
r
−
=
(
ℓ
/
2
+
1
)
−
1
/
2
∑
F
r
+
m
⋅
e
m
∗
.
{\displaystyle G_{r}^{+}=(\ell /2+1)^{-1/2}\sum E_{-m}\cdot e_{m+r},\,\,\,G_{r}^{-}=(\ell /2+1)^{-1/2}\sum F_{r+m}\cdot e_{m}^{*}.}
This yields the N=2 superconformal algebra with
c
=
3
ℓ
/
(
ℓ
+
2
)
.
{\displaystyle c=3\ell /(\ell +2).}
The algebra commutes with the bosonic operators
X
n
=
H
n
−
2
∑
r
:
e
r
∗
e
n
+
r
:
.
{\displaystyle X_{n}=H_{n}-2\sum _{r}:e_{r}^{*}e_{n+r}:.}
The space of physical states consists of eigenvectors of
X
0
{\displaystyle X_{0}}
simultaneously annihilated by the
X
n
{\displaystyle X_{n}}
's for positive
n
{\displaystyle n}
and the supercharge operator
Q
=
G
1
/
2
+
+
G
−
1
/
2
−
{\displaystyle Q=G_{1/2}^{+}+G_{-1/2}^{-}}
(Neveu–Schwarz)
Q
=
G
0
+
+
G
0
−
.
{\displaystyle Q=G_{0}^{+}+G_{0}^{-}.}
(Ramond)
The supercharge operator commutes with the action of the affine Weyl group and the physical states lie in a single orbit of this group, a fact which implies the Weyl-Kac character formula.
= Kazama–Suzuki supersymmetric coset construction
=Kazama & Suzuki (1989) generalized the SU(2) coset construction to any pair consisting of a simple compact Lie group
G
{\displaystyle G}
and a closed subgroup
H
{\displaystyle H}
of maximal rank, i.e. containing a maximal torus
T
{\displaystyle T}
of
G
{\displaystyle G}
, with the additional condition that the dimension of the centre of
H
{\displaystyle H}
is non-zero. In this case the compact Hermitian symmetric space
G
/
H
{\displaystyle G/H}
is a Kähler manifold, for example when
H
=
T
{\displaystyle H=T}
. The physical states lie in a single orbit of the affine Weyl group, which again implies the Weyl–Kac character formula for the affine Kac–Moody algebra of
G
{\displaystyle G}
.
See also
Virasoro algebra
Super Virasoro algebra
Coset construction
Type IIB string theory
Notes
References
Ademollo, M.; Brink, L.; D'Adda, A.; D'Auria, R.; Napolitano, E.; Sciuto, S.; Giudice, E. Del; Vecchia, P. Di; Ferrara, S.; Gliozzi, F.; Musto, R.; Pettorino, R. (1976), "Supersymmetric strings and colour confinement", Physics Letters B, 62 (1): 105–110, Bibcode:1976PhLB...62..105A, doi:10.1016/0370-2693(76)90061-7
Boucher, W.; Friedan, D; Kent, A. (1986), "Determinant formulae and unitarity for the N = 2 superconformal algebras in two dimensions or exact results on string compactification", Phys. Lett. B, 172 (3–4): 316–322, Bibcode:1986PhLB..172..316B, doi:10.1016/0370-2693(86)90260-1
Di Vecchia, P.; Petersen, J. L.; Yu, M.; Zheng, H. B. (1986), "Explicit construction of unitary representations of the N = 2 superconformal algebra", Phys. Lett. B, 174 (3): 280–284, Bibcode:1986PhLB..174..280D, doi:10.1016/0370-2693(86)91099-3
Eguchi, Tohru; Yang, Sung-Kil (1990), "N = 2 superconformal models as topological field theories", Mod. Phys. Lett. A, 5 (21): 1693–1701, Bibcode:1990MPLA....5.1693E, doi:10.1142/S0217732390001943
Goddard, P.; Kent, A.; Olive, D. (1986), "Unitary representations of the Virasoro and super-Virasoro algebras", Comm. Math. Phys., 103 (1): 105–119, Bibcode:1986CMaPh.103..105G, doi:10.1007/bf01464283, S2CID 91181508
Green, Michael B.; Schwarz, John H.; Witten, Edward (1988a), Superstring theory, Volume 1: Introduction, Cambridge University Press, ISBN 0-521-35752-7
Green, Michael B.; Schwarz, John H.; Witten, Edward (1988b), Superstring theory, Volume 2: Loop amplitudes, anomalies and phenomenology, Cambridge University Press, Bibcode:1987cup..bookR....G, ISBN 0-521-35753-5
Kazama, Yoichi; Suzuki, Hisao (1989), "New N = 2 superconformal field theories and superstring compactification", Nuclear Physics B, 321 (1): 232–268, Bibcode:1989NuPhB.321..232K, doi:10.1016/0550-3213(89)90250-2
Schwimmer, A.; Seiberg, N. (1987), "Comments on the N = 2, 3, 4 superconformal algebras in two dimensions", Phys. Lett. B, 184 (2–3): 191–196, Bibcode:1987PhLB..184..191S, doi:10.1016/0370-2693(87)90566-1
Voisin, Claire (1999), Mirror symmetry, SMF/AMS texts and monographs, vol. 1, American Mathematical Society, ISBN 0-8218-1947-X
Wassermann, A. J. (2010) [1998]. "Lecture notes on Kac-Moody and Virasoro algebras". arXiv:1004.1287.
West, Peter C. (1990), Introduction to supersymmetry and supergravity (2nd ed.), World Scientific, pp. 337–8, ISBN 981-02-0099-4