FQFT and cohomology
The local data for a CFT in dimension allows to assign to each -dimensional cobordism a vector space of “possible correlators”: those functions on the space of conformal structures on that have the correct behaviour (satisfy the conformal Ward identities) to qualify as the (chiral) correlator of a CFT. This is called a space of conformal blocks . This assignment is functorial under diffeomorphism. The corresponding functor is called a modular functor. (Segal 89, Segal 04, def. 5.1)
To get an actual collection of correlators one has to choose from each space of conformal blocks an element such that these choices glue under composition of cobordism: such that they solve the sewing constraints, see for instance at FRS-theorem on rational 2d CFT.
Dually, under a holographic principle such as CS3/WZW2 the space of conformal blocks on is equivalently the space of quantum states of the TQFT on . See at quantization of 3d Chern-Simons theory for more on this.
For any finite set (“of lables”) write for the category whose objects are Riemann surfaces with boundary circles labeled by elements of , and whose morphisms are holomorphic maps , where is obtained from by sewing along boundary circles carrying the same labels.
A modular functor is a holomorphic functor
These correspond precisely to group extensions of by .
These in turn are classified by .
Here in terms of standard 2d CFT terminology
is the central charge
is the eigenvalues of .
Given a modular functor as in def. 2 and given a non-closed topological labelled surface with the resulting vector bundle, then this bundle carries a canonical projectively flat connection compatible with the sewing operation of def. 1.
When thinking of the modular functor as the functor of conformal blocks of a 2d CFT then the projectively flat connection of prop. 1 would often be called the Knizhnik-Zamolodchikov connection. Thining of dually as the functor assigning spaces of quantum states of Chern-Simons theory then it would typically be called the Hitchin connection. (see also Segal 04, p. 44, p. 84).
For three labels, write for the three-holed sphere (“pair of pants”, “trinion”) with inner circles labeled by and and outer circle labeled by .
For a modular functor as in def. 2, write
for the dimension of the vector space that it assigns to this surface.
The Verlinde algebra?.
To make sense of this however one needs to consistently define the fractional power. For that one needs to pass to surfaces equipped with a bit more structure, sich that the
Of course if one has an extension of the diffeomorphism group by a multiple of the universal extension in def. 5, then this still trivializes the conformal anomaly for all modular functors whose central charge is a corrsponding multiple. In particular:
The category of smooth surfaces equipped with “Atiyah 2-framing” (hence with a trivialization of the spin lift of the double of their tangent bundle) provides an extension of the diffeomorphic group of level 12.
thanks to Chris Schommer-Pries for highlighting this point.
The 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.
Original formulations include
Graeme Segal, Two-dimensional conformal field theories and modular functors, in: IXth International Congress on Mathematical Physics (Swansee 1988), Hilger, Bristol 1989, pp. 22-37
Graeme Segal, section 5 of The definition of conformal field theory, Topology, geometry and quantum field theory London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, (2004), 421-577 (pdf)
Igor Kriz, On spin and modularity in conformal field theory, Ann. Sci. ANS (4) 36 (2003), no. 1, 57112
Lectures and reviews include
A nice review with a new concise construction is in
Discussion in the context of (2,1)-dimensional Euclidean field theories and tmf is in
A generalization of the modular functors is in