We recall that in section II.4 we replaced the integration over the embeddings X by the determinant of the Laplace operator, and that in section II.5 we implemented the Faddeev–Popov procedure to discard the integration over D0 (the group of smooth diffeomorphisms homotopic to the identity) by introducing the Faddeev–Popov determinant, which essentially coincides with the determinant of the operator Pg given by (1.4.10). In section II.6, we interpreted the integration over the conformal deformations in terms of a family of Liouville measures indexed by Teichmüller parameters. In this section we shall investigate the integration over Teichmüller space; the measure on the latter is given by the Weil–Petersson metric. There are many articles concerning a “direct” description of the Polyakov measure in critical dimension, e.g. [BJ], [BK], [BM], but these do not discuss, as we did, its derivation from functional integrals considered for all d.

As before, we let Tp,k be the Teichmüller space of surfaces of genus p with k boundaries. As discussed in section 1.7, it can be considered as a totally real subspace of Tq, with q = 2p + k − 1, the genus of the Schottky double. We recall that the cotangent space of Tp,k is given by the holomorphic quadratic differentials which are real on the boundary, and that on this space we have the Weil–Petersson metric. Complexifying, we obtain the space of holomorphic quadratic differentials on closed surfaces of genus q, and these differentials vary holomorphically with the conformal structure of the surface.

The aim of this section is to give a description of the Faddeev–Popov procedure for gauge field theories first introduced by Faddeev and Popov in the case of Yang–Mills theories [FP]. It yields a device to compute formally functional integrals over configurations of fields, the integrand being invariant under the action of the gauge group. This invariance leads to an infinite expression and the Faddeev–Popov procedure gives a prescription for “factoring out” the infinite volume of the gauge group. More precisely, one considers the integration of a G-invariant function on a principal fiber bundle with structure group G, picking out a well chosen set of representatives (a local section of the bundle) of the orbits on which the integration will actually be done. The Faddeev–Popov procedure then tells us how to relate the integral along this section with the integral on the whole orbit space. The presentation that follows is close to [Pa2], [Pa3]. Further discussions concerning the interpretation of the Faddeev–Popov procedure were presented in [AP2], [AP3].

Going from an integral on the configuration space to an integral on a section gives rise to a Jacobian determinant, the Faddeev–Popov determinant; it is, up to a finite-dimensional determinant which depends on the choice of the slice, entirely determined by the geometric data, namely, the fiber bundle P → P/G equipped with a Riemannian structure.