Lectures on String Theory

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Anti-commutators of the modes are
{b µ r , b ν s} = η µν δ r+s .
We again see that we can split oscillators into creation and annihilation operators
according to the sign of their index, namely
• Oscillators with r > 0 are annihilation operators ,
• Oscillators with r < 0 are creation operators .
However, the modes with r = 0 which occur in the Ramond sector only require
special care. Indeed, in the bosonic modes α µ 0 and ᾱ µ 0 correspond to the center of
mass momentum of the string. Analogously, b µ 0 and ¯b µ 0 are distinguished from all the
other modes, in particular, they form the Clifford algebra
and analogously for ¯b µ 0.
{b µ 0, b ν 0} = η µν
The super-Virasoro generators are again defined as normal ordered expressions
L m = 1 ∑
: α −n α m+n : + 1 ∑ (
r + m )
: b −r b m+r : ,
2
2 2
n∈Z
G r = ∑ n∈Z
α −n b r+n .
r
Only the generator L 0 is ambiguous doe to the undetermined normal ordering constant.
Ignoring this constant for the moment we obtain the following answer for the
quantum super-Virasoro algebra
[L m , L n ] = (m − n)L m+n + d 8 m(m2 − 2ω)δ m+n ,
( 1
)
[L m , G r ] =
2 m − n G m+r ,
{G r , G s } = 2L r+s + d (
r 2 − ω )
δ r+s .
2 2
Here ω = 0 for the R-sector and ω = 1 for the NS-sector. Both the R- and NSalgebras
formally agree except the linear terms in anomalies. The linear term can
2
be changed by shifting the L 0 generator. Indeed, one can see that if one shifts
L R 0 → L R 0 + d then both algebras have formally the same structure with ω = 1. Still
16 2
the R- and NS-algebras are very different. For instance, in the NS-sector the five
generators L 1 , L 0 , L −1 , G 1/2 , G −1/2 form a closed superalgebra known as OSp(1|2). In
the R-sector just adding to the generators L 1 , L 0 , L −1 the generator G 0 one generates
the whole infinite-dimensional algebra.