The local data for a CFT in dimension $d$ allows to assign to each $d$-dimensional cobordism $\Sigma$ a vector space of “possible correlators”: those functions on the space of conformal structures on $\Sigma$ 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 $Bl(\Sigma)$. 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 $Bl(\Sigma)$ 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 $\Sigma$ is equivalently the space of quantum states of the TQFT on $\Sigma$. See at quantization of 3d Chern-Simons theory for more on this.
For $\Phi$ any finite set (“of lables”) write $\mathcal{S}_{\Phi}$ for the category whose objects are Riemann surfaces with boundary circles labeled by elements of $\Phi$, and whose morphisms are holomorphic maps $X \to X_{sewed}$, where $X_{sewed}$ is obtained from $X$ by sewing along boundary circles carrying the same labels.
(Segal 04, section 4, section 5)
This category, being essentially the total Riemann moduli space is naturally a complex analytic stack.
For $\Sigma$ a topological surface write $\mathcal{S}_\Sigma$ for the component of $\mathcal{S}$ on Riemann surfaces whose underlying topological surface is $\Sigma$.
If $\Sigma$ has at least one hole (boundary component), then the fundamental group $\pi_1(\mathcal{S}_\Sigma)$ is the mapping class group of $\Sigma$.
A modular functor is a holomorphic functor
(i.e. a morphism of complex analytic stacks from the Riemann moduli space to the stack of holomorphic vector bundles, in general with super vector space-fibers)
such that
$E$ is strong monoidal: $E(X \coprod Y) \simeq E(X)\otimes E(Y)$;
$E$ respects seqing: if $X_\phi$ is obtained from $X$ by cutting along a circle and giving the same label $\phi \in \Phi$ to both resulting boundaries, then the natural transformation
is a natural isomorphism.
normalization: for $X = S^2$ the Riemann sphere we have $E(S^1) = 1$ (thetensor unit vector space).
Any modular functor defines a central extension of the semigroup of conformal annuli?.
These correspond precisely to group extensions of $Diff^+(S^1)$ by $\mathbb{C}^\times$.
These in turn are classified by $(c,h) \in \mathbb{C} \times \mathbb{C}/\mathbb{Z}$.
Here in terms of standard 2d CFT terminology
$c$ is the central charge
$h$ is the eigenvalues of $L_0$.
Given a modular functor $E$ as in def. 2 and given a non-closed topological labelled surface $\Sigma$ with $E_\Sigma \to \mathcal{S}_\Sigma$ the resulting vector bundle, then this bundle carries a canonical projectively flat connection $\nabla_\Sigma$ compatible with the sewing operation of def. 1.
When thinking of the modular functor $E$ 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 $E$ 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).
The connection of prop. 1 is a genuine flat connection (not projective) precisely if the central charge, 3, vanishes.
For $\phi,\chi,\psi \in \Phi$ three labels, write $P_{\phi,\chi,\psi}$ for the three-holed sphere (“pair of pants”, “trinion”) with inner circles labeled by $\phi$ and $\chi$ and outer circle labeled by $\psi$.
For $E$ a modular functor as in def. 2, write
for the dimension of the vector space that it assigns to this surface.
Then the free abelian group $\mathbb{Z}[\Phi]$ on the set of labels inherits the structure of an associative algebra via
The Verlinde algebra?.
(Segal 04, section 5, p. 36-37)
By prop. 1 and prop. 4, if $E$ is a modular functor of central charge $c$ then the tensor product
with a possibly fractional power of the determinant line bundle, def. 6, produces a modular functor with vanishing central charge.
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
The category of rigged surfaces $\hat {\mathcal{S}}_\Phi$ is the central extension of that of smooth manifold surfaces such that for genus $\gt 1$ it gives the universal central extension of the diffeomorphism group.
For instance the category of surfaces equipped with a choice of universal covering space of the circle group-principal bundle underlying the determinant line bundle over $\mathcal{S}_\Sigma$.
(Segal 04, def. (5.10) and following, also Bakalov-Kirillov, def. 5.7.5)
Given a modular functor $E$, def. 2 of central charge $c$, def. 3, then the tensor product $\tilde E \coloneqq E \otimes Det^{\otimes c/2}$ is well defined on the category $\hat S_{\phi}$ of rigged surfaces, def. 5.
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.
There is a natural functor from smooth surfaces $\Sigma$ equipped with 3-framing (trivialization of $T \Sigma\oplus \underline{\mathbb{R}}$) to that equipped with Atiyah 2-framing in prop. 3.
thanks to Chris Schommer-Pries for highlighting this point.
For $n \in \mathbb{Z}$ let $E = Det^{\otimes n}$ be the functor which sends a Riemann surface to the $c$th power of its determinant line (i.e. that of its Laplace operator).
The determinant lines of def. 6 constitute precisely the modular functors, def. 2, for which $dim(E(X)) = 1$ for all $X$.
The central charge, def. 3, of the determinant line $E = Det$, def. 6, is
The modular functor for $G$-Chern-Simons theory restricted to genus-1 surfaces (elliptic curves) is essentially what is encoded in the universal $G$-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
Bojko Bakalov, Alexander Kirillov, chapter 5 of Lectures on tensor categories and modular functor (web, pdf)
Krzysztof Gawędzki, section 5.6 of Conformal field theory: a case study (arXiv:hep-th/9904145)
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