nLab modular functor

Idea

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.

Definition

Definition

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 $\Sigma$.

Remark

If $\Sigma$ has at least one hole (boundary component), then the fundamental group $\pi_1(\mathcal{S}_\Sigma)$ is the mapping class group of $\Sigma$.

Definition

A modular functor is a holomorphic functor

$$E \colon \mathcal{S}_\Phi \longrightarrow sVect$$

(i.e. a morphism of complex analytic stacks from the Riemann moduli space to the stack of (finite-rank) holomorphic vector bundles, in general with super vector space-fibers)

such that

1. $E$ is strong monoidal: $E(X \coprod Y) \simeq E(X)\otimes E(Y)$;

2. $E$ respects sewing: 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

   $$\underset{\phi \in \Phi}{\oplus} E(X_\phi) \longrightarrow E(X)$$

   is a natural isomorphism.

3. normalization: for $X = S^2$ the Riemann sphere we have $E(S^1) = 1$ (thetensor unit vector space).

(Segal 04, def. 5.1)

Properties

Relation to genus-zero conformal blocks

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).

Central charge and Central extensions

Any modular functor defines a central extension of the semigroup of conformal annuli?.

These correspond precisely to group extensions of $Diff^+(S^1)$ by $\mathbb{C}^\times$.

(Segal 04, prop. 5.6)

These in turn are classified by $(c,h) \in \mathbb{C} \times \mathbb{C}/\mathbb{Z}$.

(Segal 04, prop. 5.8)

Definition

Here in terms of standard 2d CFT terminology

$c$ is the central charge

$h$ is the eigenvalues of $L_0$.

The Knizhnik–Zamolodchikov-Hitchin connection

Proposition

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)

Remark

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).

Proposition

The connection of prop. is a genuine flat connection (not projective) precisely if the central charge, , vanishes.

(Segal 04, below prop. 5.4)

The Verlinde fusion algebra

Definition

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

$$n_{\phi, \chi,\psi} \coloneqq dim(E(P_{\phi,\chi,\psi}))$$

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

$$(\phi,\chi) \mapsto \underset{\psi}{\sum} n_{\phi,\chi, \psi} \psi \,.$$

The Verlinde algebra.

(Segal 04, section 5, p. 36-37)

Deprojectivization, Cancelling of central charge, topological modular functor

By prop. and prop. , if $E$ is a modular functor of central charge $c$ then the tensor product

$$\tilde E \coloneqq E \otimes Det^{\otimes c/2}$$

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.

Definition

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)

Proposition

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:

Proposition

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.

(Segal 04, p. 46)

Remark

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.

Examples

Powers of the determinant line

Definition

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)…

Proposition

The determinant lines of def. constitute precisely the modular functors, def. , for which $dim(E(X)) = 1$ for all $X$.

(Segal 04, corollary (5.17))

Proposition

The central charge, def. , of the determinant line $E = Det$, def. , is

$$(c,h ) = (-2,0) \,.$$

(Segal 04, p.43)

Related concepts

• quantization of Chern-Simons theory

Properties

Relation to equivariant elliptic cohomology / equivariant $tmf$

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.

References

Original formulations:

• Graeme Segal, Two-dimensional conformal field theories and modular functors, in: IXth International Congress on Mathematical Physics, Swansea 1988, Adam Hilger, Bristol and New York (1989) 22–37 [pdf, pdf]

• 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)

• Jürgen Fuchs, Christoph Schweigert, Simon Wood, Yang Yang, p. 11 of: Algebraic structures in two-dimensional conformal field theory, Encyclopedia of Mathematical Physics [arXiv:2305.02773]

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:

• Jørgen Ellegaard Andersen, Kenji Ueno, Modular functors are determined by their genus zero data, Quantum Topology 3 3/4 (2012) 255-291 $[$doi:10.4171/QT/29$]$

Discussion in the context of (2,1)-dimensional Euclidean field theories and tmf is in

• Stefan Stolz, Peter Teichner, section 5.2 of Supersymmetric field theories and generalized cohomology, in Hisham Sati, Urs Schreiber (eds.), Mathematical foundations of Quantum field theory and String theory, Proceedings of Symposia in Pure Mathematics, Volume 83, AMS (2011)

Discussion in the context of the cobordism hypothesis:

• Domenico Fiorenza, Alessandro Valentino, Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, Commun. Math. Phys. 338 (2015) 1043–1074 $[$arXiv:1409.5723, doi:10.1007/s00220-015-2371-3$]$

Constructing modular functors from pivotal bicategories using string net models:

• Jürgen Fuchs, Christoph Schweigert, Yang Yang, String-net models for pivotal bicategories [arXiv:2302.01468]