By M. Gaberdiel

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**Extra info for An Introduction to Conformal Field Theory**

**Example text**

A group G acts on the space of states of a conformal field theory H , if each g ∈ G defines a linear map g : H → H (that leaves the dense subspace F invariant, g : F → F ), the composition of maps respects the group structure of G, and the amplitudes satisfy V (gψ1; z1, z¯1 ) · · · V (gψn ; zn , z¯n ) = V (ψ1 ; z1, z¯1) · · · V (ψn ; zn , z¯n ) . } , (176) together with additional twisted sectors, one for each conjugacy class in G. Generically the meromorphic subtheory of the resulting theory consists of those meromorphic fields of the original theory that are invariant under G, but in general it may also happen that some of the twisted sectors contain additional meromorphic fields.

The operator product expansion of J and b is regular, so that the corresponding commutator vanishes. This theory then exhibits superconformal invariance; in the present context this means that the theory has in addition to the Conformal Field Theory 31 stress-energy-tensor L, the superpartner field G of conformal weight 3/2, whose modes satisfy m [Lm , Gr ] = − r Gm+r 2 (163) c 2 1 r − δr,−s . {Gr , Gs } = 2Lr+s + 3 4 Together with the Virasoro commutation relations for L (109), this is called the (Neveu Schwarz [NS] sector of the) N = 1 superconformal algebra.

E. they commute with their adjoint (or transpose) since N†i = Ni∨ . This implies that each Ni can be diagonalised, and since the different Ni commute, there exists a common matrix S that diagonalises all (l) Ni simultaneously. If we denote the different eigenvalues of Ni by λi , we therefore have that (l) Sjl λi δlm S −1 Nijk = k m (l) Sjl λi = lm S −1 k l . e. j = 0, then Ni0 = δik if all representations labelled by i are irreducible. In this case, multiplying all three expressions of (263) by Skn from the right (and summing over k), we find (l) Sin = (n) S0l λi δln = S0n λi .

### An Introduction to Conformal Field Theory by M. Gaberdiel

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