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44) k where ckij (x − y) are the coeﬃcient functions which are singular in the limit of x → y. Such expansions were proven to hold in renormalizable ﬁeld theories. The OPEs are an essential tool in exploring quantum ﬁeld theories. On (xn ) >. In particular, one is interested in the behavior of these correlation functions when two or more points approach each other, which is encapsulated in the OPEs. For all applications discussed here the OPEs are treated as asymptotic expansions and only their singular terms will be speciﬁed.

So far we have implemented the global Ward identities. To get the local Ward identity one performs a conformal transformation of an n-point function of primary ﬁelds Gn . This is achieved by integrating (z)T (z) along a contour C which bounds a region that includes all the operators (see Fig. 2) Now using analyticity one can deform the contour into a sum of countours each of which encircles one operator. The result of the integral is therefore, dz (z)T (z)φ(w1 , w ¯1 ) . . φ(wn , w ¯n ) 2πi n φ(w1 , w ¯1 ) .

68) l (n ) which means that λi = S˜in S˜0n and therefore, S˜jl S˜il (S˜−1 )kl Nijk = . 8), we recall that under the S-transformation τ → − τ1 the characters of a given CFT transform as, 1 χj (− ) = Sjk χk (τ ). 70) τ k Verlinde’s formula6 states that the matrix S˜ above is identical to the Stransformation matrix, ˜ S = S. This is a remarkable relation. 6 The Verlinde formula was introduced in the seminal paper [210]. 12 Free Majorana fermions – an example of a CFT The theory of free massless fermions in two dimensions is an example of a 2d conformal theory of the utmost importance.