Functorial quantum field theory

String theory



The term topological conformal field theory (TCFT) is used for a linearization or stabilization of something that is like a conformal field theory (CFT) up to homotopy. It is a notion somewhere half-way between a (2-dimensional) TQFT and a CFT.

(Actually, the remnant of conformal structure here should be just an artefact of the way to parameterize the moduli space of surfaces. As the classification result by Lurie discussed below shows, TCFTs are really (,2)(\infty,2)-TFTs.)

This formalizes the physics notion of “the topological string”, a topologically twisted superconformal field theory, such as, notably, the A-model and the B-model. TCFTs are therefore a tool for formalizing homological mirror symmetry.

Recall that an ordinary conformal field theory (CFT) is, in FQFT-language, a symmetric monoidal functor on a category Bord 2 confBord_2^{conf} whose objects are disjoint unions of intervals and circles, and whose morphisms are Riemann surfaces with these 1d manifolds as incoming and outgoing punctures.

Since Riemann surfaces form a well-understood moduli space, one can turn this also into a Top-enriched category, i.e. an (∞,1)-category, Bord 2 conf,topBord_{2}^{conf,top} whose hom-spaces are these moduli spaces of Riemann surfaces with given 1d manifolds as incoming and outgoing punctures.

A “truly topological conformal field theory” would be an (∞,1)-functor of the form

Bord 2 conf,topGrpd Bord_2^{conf,top} \to \infty Grpd

or similar. But what is actually called a “topological conformal field theory” is the linearization or stabilization of this:

in a TCFT, this (∞,1)-category of conformal cobordisms is replaced by a stable (∞,1)-category whose hom-objects (when modeled by a dg-category) are just the homology chain complexs of the original hom-spaces.

Write Bord 2 conf,dgBord_2^{conf,dg} for the resulting symmetric monoidal dg-category of Riemann cobordisms. Then a TCFT is a an homotopy-symmetric monoidal chain complex-enriched functor

F:Bord 2 conf,dgCh F : Bord_2^{conf,dg} \to Ch_\bullet

to the symmetric monoidal dg-category of chain complexes.

This means in particular that when two Riemann surfaces Σ 1\Sigma_1 and Σ 2\Sigma_2 are homologous as chains in the moduli space of Riemann surfaces, then the TCFT will send them to two equivalent morphisms f Σ 1f_{\Sigma_1} and f Σ 2f_{\Sigma_2} of chain complexes between the in- and the output states. The equivalence between f Σ 1f_{\Sigma_1} and f Σ 2f_{\Sigma_2}, however, is not unique neither up to equivalence. Rather, it funtorially depends on the 1-chain realizing the homology equivalence between Σ 1\Sigma_1 and Σ 2\Sigma_2 as 0-chains in the moduli space. In particular, two non-homologous 1-chains between Σ 1\Sigma_1 and Σ 2\Sigma_2 will in general lead to non-equivalent equivalences between f Σ 1f_{\Sigma_1} and f Σ 2f_{\Sigma_2}.


According to ClassTFT the original definition of the domain for TCFTs can be formulated as follows (without reference to any conformal or Riemann structure).

Definition The (,2)(\infty,2)-category Bord 2 ncBord^{nc}_2 of non-compact 2-dimensional cobordism is defined as follows:

Then, the cobordism hypothesis-theorem for Bord 2 ncBord^{nc}_2 becomes


Let CC be a symmetric monoidal (∞,2)-category. Then symmetric monoidal (∞,2)-functors

Z:Bord 2 ncC Z : Bord^{nc}_2 \to C

are equivalent to Calabi-Yau objects AA in CC: the functor ZZ sends the point to AA.

This is ClassTFT, theorem 4.2.11. One can “unfold” Bord 2 ncBord^{nc}_2 and the theorem above, obtaining a statement in terms of symmetric monoidal (∞,1)-categories. Actually it was the unfolded version to be proven first, (Costello 04).

in the particular case C=Ch C=Ch_\bullet. We state it below in the general version given by Jacob Lurie in ClassTFT.


Let 𝒪𝒞\mathcal{OC} be the (infinity,1)-category of open-closed strings, described as follows:

  1. objects are oriented 1-manifolds with boundary;

  2. morphisms are oriented bordisms between 1-manifolds such that each connected component has non-vanishing intersection with the codomain 1-manifold;

  3. the higher morphisms are given by orientation preserving diffeomorphisms, isotopies between these, and so forth.

Write 𝒪\mathcal{O} for the full sub-(∞,1)-category on disjoint unions of intervals (open strings sector).

This is ClassTFT, above theorem 4.2.13.


With coefficients in (algebras in) chain complexes

The original statement of the classification result for TCFTs concerned symmetric homotopy-monoidal functors Bord 2 conf,dgCh Bord_2^{conf,dg} \to Ch_\bullet:


(Costello, following Kontsevich)

  1. The category of open TCFTs with set Λ\Lambda of D-branes is equivalent to that of Calabi-Yau categories with set Λ\Lambda of objects.

  2. The homology of the chain complex of closed states of the universal extension of an open TCFT to an open-closed TCFT is the Hochschild homology of the corresponding Calabi-Yau category.

In (Costello 04) this is proven using information about cell decompositions of the moduli space of punctured Riemann surfaces, thus effectively presenting Bord 2 conf,dgBord_2^{conf,dg} by generators-and-relations, The then theorem amounts to noticing that representations of these generators and relations define the operations in an A A_\infty-category with pairing operation.

General version


Let CC be a symmetric monoidal (∞,1)-category. Then symmetric monoidal (∞,1)-functors

Z:𝒪C Z : \mathcal{O} \to C

are equivalent to Calabi-Yau algebra objects AA in CC: the functor ZZ sends the interval [0,1][0,1] to AA.

This is the result of spring Cos04 reformulated and generalized according to ClassTFT, theorem 4.2.14.

This is a special case of the general cobordism hypothesis-theorem.

The idea of the proof is that a topological open string theory, i.e., a symmetric monoidal (∞,1)-functor Z:𝒪CZ : \mathcal{O} \to C has a Kan extension to an open-closed topological string theory, i.e., to a symmetric monoidal (∞,1)-functor Z:𝒪𝒞CZ : \mathcal{OC} \to C, which is the unfolded version of a symmetric monoidal (∞,2)-functor from Bord 2 ncBord^{nc}_2 to a symmetric monoidal (,2)(\infty,2)-category CC'.

Worldsheet and effective background theories

One imagines generally that one obtains TCFTs, in their formal definition given above, from worldsheet action functionals as familiar from the physics literature (such as on the A-model and the B-model) by performing the path integral and finding from it a collection of differential forms on moduli space of bosonic field configurations.

It seems there is at this point no literature giving a direct construction along these lines, but there is the following:

In Cos06 is constructed from the geometric input datum of a generalized Calabi-Yau space (X,Q)(X,Q) and it is shown that

  1. there is a collection of differential forms K g,h()K_{g,h}(\cdots)on the moduli space g h,n\mathcal{M}_{g}^{h,n} of Riemann surfaces such that these define a 2d TCFT;

    (In the discussion leading up to Lemma 4.5.1 there. The proof that this yields a TCFT is theorem 4.5.4.)

  2. the partition function of the string perturbation series for the above TCFT is

    g,n0,h>02g2+h+n2λ 2g2+hN h1n! g h,nK g,h(a n) \sum_{{g,n \geq 0, h \gt 0} \atop {2g-2+h+\frac{n}{2}}} \lambda^{2g-2+h}N^h \frac{1}{n!} \int_{\mathcal{M}_g^{h,n}} K_{g,h}(a^\otimes n)

    which is shown to be the partition function of a background Chern-Simons theory coming from the action functional

    aS(a)= X12aQa+13a 3. a \mapsto S(a) = \int_X \frac{1}{2} a Q a + \frac{1}{3}a^3 \,.

So this constructs a 2d TCFT and shows that its effective background quantum field theory is a Chern-Simons theory. While the action functionl on the worldsheet itself, whose path integral should give the differential forms on moduli space considered above, is not explicitly considered here, this does formalizes at least some aspects of an observation that was earlier made in (Witten 92) where it was observed that Chern-Simons theory is the effective background string theory of 2d TFTs obtained from action functionals of the A-model and the B-model.

Similarly the effective background QFT of the B-model topological string can be identified. This is known as Kodeira-Spencer gravity or as BCOV theory.

(See also at world sheets for world sheets for a similar mechanism and see at super 1-brane in 3d for related “physical” strings.)

So via the detour over the effective background field theory, this sort of shows that the physicist’s A-model and B-model are indeed captured by the abstract FQFT definition of TCFT as given above.


The concept is essentially a formalization of what used to be called cohomological field theory in

The definition was given independently by


The classification of TCFTs by Calabi-Yau categories was discussed in

following conjectures by Maxim Kontsevich, e.g.

This classification is a precursor of the full cobordism hypothesis-theorem. This, and the reformulation of the original TCFT constructions in full generality is in

Here are notes from a seminar on these definitions and results:

Discussion of the construction of TCFTs from differential forms on moduli space and the way this induces by “second quantization” effective background Chern-Simons theories is in

formalizing at least aspects of the observations in

Discussion of how the second quantization of the B-model yields Kodeira-Spencer gravity/BCOV theory is in