Super Virasoro algebra

In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular importance in superstring theory: the Ramond algebra (named after Pierre Ramond) and the Neveu–Schwarz algebra (named after André Neveu and John Henry Schwarz). Both algebras have N = 1 supersymmetry and an even part given by the Virasoro algebra. They describe the symmetries of a superstring in two different sectors, called the Ramond sector and the Neveu–Schwarz sector.

The N = 1 super Virasoro algebras

There are two minimal extensions of the Virasoro algebra with N = 1 supersymmetry: the Ramond algebra and the Neveu–Schwarz algebra. They are both Lie superalgebras whose even part is the Virasoro algebra: this Lie algebra has a basis consisting of a central element C and generators Lm (for integer m) satisfying

[

L

m

,

L

n

]

=

(

m

−

n

)

L

m

+

n

+

c

12

m

(

m

2

−

1

)

δ

m

+

n

,

0

{\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {c}{12}}m(m^{2}-1)\delta _{m+n,0}}

where

δ

i

,

j

{\displaystyle \delta _{i,j}}

is the Kronecker delta.

The odd part of the algebra has basis

G

r

{\displaystyle G_{r}}

, where

r

{\displaystyle r}

is either an integer (the Ramond case), or half an odd integer (the Neveu–Schwarz case). In both cases,

c

{\displaystyle c}

is central in the superalgebra, and the additional graded brackets are given by

[

L

m

,

G

r

]

=

(

m

2

−

r

)

G

m

+

r

{\displaystyle [L_{m},G_{r}]=\left({\frac {m}{2}}-r\right)G_{m+r}}

{

G

r

,

G

s

}

=

2

L

r

+

s

+

c

3

(

r

2

−

1

4

)

δ

r

+

s

,

0

{\displaystyle \{G_{r},G_{s}\}=2L_{r+s}+{\frac {c}{3}}\left(r^{2}-{\frac {1}{4}}\right)\delta _{r+s,0}}

Note that this last bracket is an anticommutator, not a commutator, because both generators are odd.

The Ramond algebra has a presentation in terms of 2 generators and 5 conditions; and the Neveu—Schwarz algebra has a presentation in terms of 2 generators and 9 conditions.

Representations

The unitary highest weight representations of these algebras have a classification analogous to that for the Virasoro algebra, with a continuum of representations together with an infinite discrete series.