functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
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, Kriz 03, 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 labels”) 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).
For a rational 2d CFT the modular functor is fully determined by the conformal blocks on the Riemann sphere (the genus=0 conformal blocks) – this is proven in Andersen & Ueno 2012.
Closely related is the statement that the braided monoidal structure on the modular representation category of the corresponding vertex operator algebra is fully determined by the genus=0 conformal blocks, a statement that seems to be folklore (highlighted in EGNO 15, p. 266, Runkel, Sec. 4.3).
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. 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. .
(Segal 04, prop. 5.4, see also at Knizhnik-Zamolodchikov equation)
When thinking of the modular functor $E$ as the functor of conformal blocks of a 2d CFT then the projectively flat connection of prop. 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. is a genuine flat connection (not projective) precisely if the central charge, , 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. , 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. and prop. , 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. , 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.
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. of central charge $c$, def. , 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. .
Of course if one has an extension of the diffeomorphism group by a multiple of the universal extension in def. , 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. .
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).
Super-line, see (Kriz-Lai 13)…
The determinant lines of def. constitute precisely the modular functors, def. , for which $dim(E(X)) = 1$ for all $X$.
The central charge, def. , of the determinant line $E = Det$, def. , 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:
Graeme Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol (1989) 22-37.
Graeme Segal, The definition of conformal field theory, in: Ulrike Tillmann (ed.), Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser. 308, Cambridge University Press (2004) 421-577 $[$doi:10.1017/CBO9780511526398.019, pdf, pdf$]$
Igor Kriz, On spin and modularity in conformal field theory, Ann. Sci. ANS (4) 36 (2003), no. 1, 57112 (numdam:ASENS_2003_4_36_1_57_0)
Igor Kriz, Luhang Lai, On the definition and K-theory realization of a modular functor, (arxiv/1310.5174).
Lectures and reviews:
Bojko Bakalov, Alexander Kirillov, chapter 5 of Lectures on tensor categories and modular functor (web, pdf)
Krzysztof Gawedzki, section 5.6 of Conformal field theory: a case study (arXiv:hep-th/9904145)
Eduard Looijenga, From WZW models to Modular Functors (arXiv:1009.2245)
Proof that the values of a modular functor at genus=0 (ie. the conformal blocks on the punctured Riemann sphere) determine the full modular functor:
Discussion in the context of (2,1)-dimensional Euclidean field theories and tmf is in
Discussion in the context of the cobordism hypothesis:
Last revised on June 9, 2022 at 11:16:02. See the history of this page for a list of all contributions to it.