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By N F Sneddon, I N Mott

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Ii) The OPE of two operators of the form : eiα X (w ) is (: eiα X (z ) :)(: e−iβ X (w ) :) = : eiα X (w ) e−iβ X (w ) : . 9 Path integral quantization So far we have been using canonical quantization. Before proceeding to the general structure of affine current algebra and Virasoro algebra we introduce the quantization of a free massless scalar field using the Euclidean path integral ˆ z¯) can be approximated approach. As usual the functional integration DX(z, by discretizing the two-dimensional space and representing the functional integral by products of ordinary integrals.

49) To get the character of a minimal model one has to determine the irreducible Verma module using the quotient procedure discussed in the previous section. 51) and hr s (p, p ) = (pr − p s)2 − (p − p )2 . 52) Note that these are the non-unitary minimal models, except for the cases p − p = ±1, which coincide with the cases of the previous section with the identification of p = m or p = m. 9 Correlators and the conformal Ward identity Now that the Hilbert space of states has been analyzed we would like to determine the correlation functions of all possible operators of a given CFT.

The scalar products of simple roots define the Cartan matrix as follows: Aij = 2α(i) · α(j ) . 10) The Cartan matrix is a rank(G) × rank(G) matrix with integer components and with diagonal elements which take the value of 2. The off diagonal elements are either negative or vanishing. 42 Theories invariant under affine current algebras Ar 1 2 3 r−2 r−1 su(r+1) r r−2 r−1 r Br Cr 1 2 3 r−2 r−1 r 1 2 3 r−2 r−1 r 1 2 3 r−3 so(2r+1) sp(r) r Dr r−2 so(2r) r−1 E6 E7 E8 F4 G2 Fig. 1. Dynkin diagrams for all the simple Lie algebras.

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