nLab
modular functor

Contents

Idea

The local data for a CFT in dimension dd allows to assign to each dd-dimensional cobordism Σ\Sigma a vector space of “possible correlators”: those functions on the space of conformal structures on Σ\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(Σ)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(Σ)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 Σ\Sigma is equivalently the space of quantum states of the TQFT on Σ\Sigma. See at quantization of 3d Chern-Simons theory for more on this.

Definition

Definition

For Φ\Phi any finite set (“of lables”) write 𝒮 Φ\mathcal{S}_{\Phi} for the category whose objects are Riemann surfaces with boundary circles labeled by elements of Φ\Phi, and whose morphisms are holomorphic maps XX sewedX \to X_{sewed}, where X sewedX_{sewed} is obtained from XX 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 Σ\Sigma a topological surface write 𝒮 Σ\mathcal{S}_\Sigma for the component of 𝒮\mathcal{S} on Riemann surfaces whose underlying topological surface is Σ\Sigma.

Remark

If Σ\Sigma has at least one hole (boundary component), then the fundamental group π 1(𝒮 Σ)\pi_1(\mathcal{S}_\Sigma) is the mapping class group of Σ\Sigma.

Definition

A modular functor is a holomorphic functor

E:𝒮 ΦsFinVect E \colon \mathcal{S}_\Phi \longrightarrow sFinVect

(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

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

  2. EE respects seqing: if X ϕX_\phi is obtained from XX by cutting along a circle and giving the same label ϕΦ\phi \in \Phi to both resulting boundaries, then the natural transformation

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

    is a natural isomorphism.

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

(Segal 04, def. 5.1)

Properties

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)Diff^+(S^1) by ×\mathbb{C}^\times.

(Segal 04, prop. 5.6)

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

(Segal 04, prop. 5.8)

Definition

Here in terms of standard 2d CFT terminology

cc is the central charge

hh is the eigenvalues of L 0L_0.

The Knizhnik–Zamolodchikov-Hitchin connection

Proposition

Given a modular functor EE as in def. 2 and given a non-closed topological labelled surface Σ\Sigma with E Σ𝒮 ΣE_\Sigma \to \mathcal{S}_\Sigma the resulting vector bundle, then this bundle carries a canonical projectively flat connection Σ\nabla_\Sigma compatible with the sewing operation of def. 1.

(Segal 04, prop. 5.4)

Remark

When thinking of the modular functor EE 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 EE 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. 1 is a genuine flat connection (not projective) precisely if the central charge, 3, vanishes.

(Segal 04, below prop. 5.4)

The Verlinde fusion alegbra

Definition

For ϕ,χ,ψΦ\phi,\chi,\psi \in \Phi three labels, write P ϕ,χ,ψP_{\phi,\chi,\psi} for the three-holed sphere (“pair of pants”, “trinion”) with inner circles labeled by ϕ\phi and χ\chi and outer circle labeled by ψ\psi.

For EE a modular functor as in def. 2, write

n ϕ,χ,ψdim(E(P ϕ,χ,ψ)) 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 [Φ]\mathbb{Z}[\Phi] on the set of labels inherits the structure of an associative algebra via

(ϕ,χ)ψn ϕ,χ,ψψ. (\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. 1 and prop. 4, if EE is a modular functor of central charge cc then the tensor product

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

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.

Definition

The category of rigged surfaces 𝒮^ Φ\hat {\mathcal{S}}_\Phi is the central extension of that of smooth manifold surfaces such that for genus >1\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 𝒮 Σ\mathcal{S}_\Sigma.

(Segal 04, def. (5.10) and following, also Bakalov-Kirillov, def. 5.7.5)

Proposition

Given a modular functor EE, def. 2 of central charge cc, def. 3, then the tensor product E˜EDet c/2\tilde E \coloneqq E \otimes Det^{\otimes c/2} is well defined on the category S^ ϕ\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:

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 Σ\Sigma equipped with 3-framing (trivialization of TΣ̲T \Sigma\oplus \underline{\mathbb{R}}) to that equipped with Atiyah 2-framing in prop. 3.

thanks to Chris Schommer-Pries for highlighting this point.

Examples

Powers of the determinant line

Definition

For nn \in \mathbb{Z} let E=Det nE = Det^{\otimes n} be the functor which sends a Riemann surface to the ccth power of its determinant line (i.e. that of its Laplace operator).

Super-line, see (Kriz-Lai 13)…

Proposition

The determinant lines of def. 6 constitute precisely the modular functors, def. 2, for which dim(E(X))=1dim(E(X)) = 1 for all XX.

(Segal 04, corollary (5.17))

Proposition

The central charge, def. 3, of the determinant line E=DetE = Det, def. 6, is

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

(Segal 04, p.43)

Properties

Relation to equivariant elliptic cohomology / equivariant tmftmf

The modular functor for GG-Chern-Simons theory restricted to genus-1 surfaces (elliptic curves) is essentially what is encoded in the universal GG-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 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 (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 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

Discussion in the context of the cobordism hypothesis is in

Revised on October 27, 2014 12:53:28 by Urs Schreiber (89.204.155.181)