FQFT and cohomology
the correlators need to glue correctly underly composition of cobordisms.
The spaces of functionals that satisfy the first of these conditions are called conformal blocks . The second condition is called the sewing constraint on conformal blocks.
So conformal blocks are something like “precorrelators” or “potential correlators” of a CFT.
The assignment of spaces of conformal blocks to surfaces and their isomorphisms under diffeomorphisms of these surfaces together constitutes the modular functor. Under CS/WZW holography this is essentially the data also given by the Hitchin connection, see at quantization of 3d Chern-Simons theory for more on this.
From a point of view closer to number theory and geometric Langlands correspondence elements of conformal blocks are naturally thought of (Beauville-Laszlo 93) as generalized theta functions (see there for more).
For the -WZW model the assignment of spaces of conformal blocks, hence by the above equivalently modular functor for -Chern-Simons theory restricted to genus-1 surfaces (elliptic curves) is essentially what is encoded in the universal -equivariant elliptic cohomology (equivariant tmf). In fact equivariant elliptic cohomology remembers also the pre-quantum incarnation of the modular functor as a systems of prequantum line bundles over Chern-Simons phase spaces (which are moduli stacks of flat connections) and remembers the quantization-process from there to the actual space of quantum states by forming holomorphic sections. See at equivariant elliptic cohomology – Idea – Interpretation in Quantum field theory for more on this.
|bulk field theory||boundary field theory|
|wave function||correlation function|
|space of quantum states||conformal blocks|
A review is around p. 30 of
Relation to theta functions:
A.A. Beilinson, Yu.I. Manin, V.V. Schechtman, Sheaves of Virasoro and Neveu-Schwarz algebras, Lecture Notes in Math. 1289, Springer 1987, 52–66
A.Mironov, A.Morozov, Sh.Shakirov, Conformal blocks as Dotsenko-Fateev integral discriminants, arxiv/1001.0563
Conformal blocks for self-dual higher gauge theory are discussed in