In this article correlation functions in rational conformal field theory on a possibly nonorientable surface with boundary are studied from the point of view of three-dimensional topological field theory. The starting point is the set-up of modular categories in the sense of V. G. Turaev [Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, 18, Walter de Gruyter, Berlin (1994; Zbl 0812.57003)]. In particular, there exists a set \(I\) of distinguished simple objects. A conformal field theory is an assignment of a correlation function \(C(X)\) to each labeled surface \(X\). A labeled surface \(X\) consists of a (not necessarily orientable) compact two dimensional manifold with (possibly empty) oriented boundary, a set of marked points on it, and “boundary conditions”. The boundary conditions are a coloring by \(I\) of the boundary arcs between marked boundary points. The correlation function is a vector in the space of states of the topological field theory (i.e. in the space of conformal blocks).
The main result of this article is the construction of such a correlation function which obeys the required modular and factorization properties. In this construction the vector is the vector \(Z(M_X)\) associated to the connecting 3-manifold (with ribbon graph) \(M_X\) of \(X\) whose boundary is the double \(\widehat X\) of \(X\). The vector \(Z(M_X)\) depends linearly on the colorings of the vertices of the ribbon graph by morphisms of the modular category. The correlation functions for the elementary building blocks: the disk with three boundary points, the disk with one interior and one boundary point, and the projective plane with one point are determined. The authors call these correlation functions “structure constants”. Also formulae for the annulus, the Klein bottle and the Möbius strip without marked points are given. In these cases the double is a torus, and it is shown that the coefficients of the correlation functions with respect to a natural basis of the space of conformal blocks are integers.