By Paul Ginsparg
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Extra resources for Applied Conformal Field Theory
Pictorially this generalizes ﬁg. 4 to n-point functions, and is discussed further in the contribution of Dijkgraaf to these proceedings. 4. 1. The Hilbert space of states We now return to consider more carefully the Hilbert space of states of a conformal ﬁeld theory. For the time being it will be suﬃcient to consider only the holomorphic half of the theory. We recall that a highest weight state |h = φ(0)|0 , satisfying L0 |h = h|h , is created by acting with a primary ﬁeld φ of conformal weight h on the SL(2, R) invariant vacuum |0 , which satisﬁes Ln |0 = 0, n ≥ −1.
3) to n-point functions. e. 9)). 21) takes the obvious form 0 = 0| Lk , φ1 (z1 ) . . φn (zn )|0 + · · · + 0|φ1 (z1 ) . . Lk , φn (z1 ) |0 , for k = 0, ±1. 17) we write this equivalently as n ∂i 0|φ1 (z1 ) . . φn (zn )|0 = 0 i=1 n (zi ∂i + hi ) 0|φ1 (z1 ) . . 22) i=1 n (zi2 ∂i + 2zi hi ) 0|φ1 (z1 ) . . φn (zn )|0 = 0 , i=1 implying respectively invariance under translations, dilatations, and special conformal transformations. 15) even though T is not a primary ﬁeld. 9), implying that T is quasi-primary, and that suﬃces to show that its correlation functions transform covariantly under SL(2, C).
If det MN (c, h) vanishes, then there exists a linear combination of states with zero norm for that c, h. e. at least one). The representation of the Virasoro algebra at those values of c and h includes states of negative norm, and is consequently not unitary. 5a) pq≤N is due to Kac and was proven in . 5a) is over all positive integers p, q whose product is less than or equal to N , and αN is a constant independent of c and h. The hp,q (c)’s are most easily expressed by reparametrizing c in terms of the (in general complex) quantity 1 1 m=− ± 2 2 25 − c .
Applied Conformal Field Theory by Paul Ginsparg