String Theory Demystified

CHAPTER 7 RNS Superstrings 145
The Virasoro operators generate what is known as a super-Virasoro algebra. There
are some differences for the R sector and the NS sector, so we consider each
independently.
NS SECTOR ALGEBRA
For the NS sector, the following relations are satisfi ed:
c 3
[ L , L ] = ( n− m) L + ( n −n)
δ n m n+ m n+ m,
0
12
(7.42)
1
[ L , G ] = ( n− 2 r) G
n r n+ r
2
(7.43)
c 2
{ G , G } = 2L
+ ( 4r−1) δ r s r+ s r+ s,
0
12
(7.44)
The central charge is related to the space-time dimension by c= D+ D/2.
Let ψ
be a physical state in the NS sector. The NS sector super-Virasoro constraints are
( L − a ) ψ = 0
(7.45)
0 NS
L ψ = n>
0 0 (7.46)
n
G ψ = r > 0 0 (7.47)
r
Here, following the quantization of the bosonic string, aNS is a normal-ordering
constant. The open string mass formula is taken by setting L = a , which gives
0 NS
Where the number operator is
The Super-Virasoro Algebra
2 1
m = ( N − a ) (7.48)
NS
α ′
∞
∑ −n
n
∞
∑
n=
1 r=
1/ 2
N = α ⋅ α + rb ⋅b
−r
r
(7.49)