340
17 Superstring
central charge c = (15, 26). If spacetime has D non-compact dimensions, then the
matter CFT is made of
• a free theory of D scalars X μ and D left-moving fermions ψ μ (μ = 0, . . . , D −
1) such that c free = 3D/2 and ¯
c free = D;
• an internal theory with c int = 15 − 3D/2 and ¯
c int = 26 − D.
The critical dimension is reached when c int = 0, which corresponds to D = 10.
The diffeomorphisms are generated by the energy–momentum tensor T (z);
correspondingly, supersymmetry is generated by its super-partner G(z) (sometimes
also denoted by T F ). The OPEs of the algebra formed by T (z) and G(z) are
T (z)T (w) ∼
c/2
(z − w) 4 +
2T (w)
(z − w) 2 +
∂T (w)
z − w
,
(17.1a)
G(z)G(w) ∼
2c/3
(z − w) 3 +
2T (w)
(z − w)
,
(17.1b)
T (z)G(w) ∼
3
2
G(w)
(z − w) 2 +
∂G(w)
(z − w)
.
(17.1c)
The superconformal ghosts form a first-order system (see Sect. 7.2) with = −1
and λ = 3/2. Hence, they have conformal weights
h(β) =
3
2
, 0
,
h(γ)=
−
1
2
, 0
(17.2)
and OPEs
γ (z)β(w) ∼
1
z − w
,
β(z)γ (w) ∼ −
1
z − w
.
(17.3)
The expressions of the ghost energy–momentum tensors are
T
gh
= −2b ∂c + c∂b,
T
βγ
=
3
2
β∂γ +
1
2
γ ∂β.
(17.4)
The ghost numbers of the different fields are
N gh (b) = N gh (β) = −1,
N gh (c) = N gh (γ ) = 1.
(17.5)
The worldsheet scalars satisfy periodic boundary conditions. On the other hand,
fermions can satisfy anti-periodic or periodic conditions: this leads to two different
sectors, called Neveu–Schwarz (NS) and Ramond (R), respectively.