Topological Conformal Field Theories.

Question

Hi there!

My question is simple, and I hope you don't misunderstand it. Why should one care about Topological Conformal Field Theories? What are the motivations to study it? I mean, for example: Topological Quantum Field Theories are important because they yield invariants of manifolds and non-trivial representations of the Mapping Class Group of a surface.

Thank you!

Answer 1

The basic problem here is the naming rather than the object itself. The name TCFT as far as I understand indicates that it's a topological field theory, whose origin is in conformal field theory. Namely there is a construction (a "topological twist") starting from an $N=2$ supersymmetric conformal field theory in two dimensions, that produces a topological field theory (in fact two, the A- and B-twists). This is a special case of a general theory of topological twists of SUSY quantum field theories, which is by far the predominant source of topological quantum field theories as far as I know, including many of the most interesting ones coming from SUSY gauge theories in four dimensions (these are sometimes called "TFTs of Witten type" as opposed to the very rare Schwarz type, like Chern-Simons theory, which come with a manifestly topological formulation).

Now when we say "TFT" here it is at a more refined chain level than the classical Atiyah-Segal axiomatic definition --- a synonym for TCFT in the sense of say Costello's beautiful paper on the subject is differential graded TFT. This means roughly that the theory is topological on a derived level -- its outputs are topologically invariant up to coherent homotopies. (This is the kind of refined topological invariance one always gets out of twisting SUSY field theory.)

What makes this very confusing initially is it seems conformal structures on a Riemann surface are playing an essential role: TCFT is defined in terms of chains on moduli spaces of complex structures. However this is a red herring (unless you are interested in the CFT origin of the construction) -- the moduli of complex structures is just playing the role of a nice model for the classifying space $BDiff(\Sigma)$ of topological surfaces, and everything can be said purely topologically (as it is in Hopkins-Lurie's work on the Cobordism Hypothesis). So really we are just defining a TFT on the chain level, in families (ie universally over moduli of topological surfaces). (This is a perspective I learned from Segal and Teleman and Freed and Costello, see Lurie's manuscript on TFTs for the contemporary perspective.)

Answer 2

Let me give you one reason. As you may know moduli spaces of curves are important objects. The homology of moduli spaces stabilizes when the genus increases (by Harer's stability theorem) and is known thanks to the solution to Mumford's conjecture. In the unstable range, computations become very difficult C.-F. Bodigheimer has done some computations (see also V. Godin's paper to Math. Ann.). The solution of Mumford's conjecture (by Galatius-Madsen-Tillmann-Weiss) relies on Segal's foundationnal work on field theories because of the heavy use of topological categories of cobordism. This is a beautiful example of how Segal's ideas together with homotopy theoretic techniques can be applied to a problem in algebraic geometry.

Studying topological conformal field theories is studying the space of representations of the prop of singular chains of these moduli spaces. One hope is to be able to say something about the homology of this prop through the eyes of its representations.

Let me give you an example. If you look at string topology, we are able to build operations by considering the action of spaces of diagrams (which are combinatorial models of moduli spaces they are related to Kontsevich's graph homology) on the singular chains of the free loop space of a closed, oriented manifold $M$. Singular chains of loop spaces or at least Hochschild cochains of the cochains of $M$ give an example of a TCFT (with some boundary conditions), you can view it as a consequence of Costello's work (or of the cobordism hypothesis).

Imagine that you have a geometric cycle in your space of diagrams and that you are able to prove that its action is not homologicaly trivial on a free loop space then you have created a homology class in the homology of the corresponding moduli space. References here are : Papers of Chas and Sullivan, Cohen and Godin, Godin, Kupers, Poirier, Wahl.....(I am sure I have forgotten plenty of people). Nathalie Wahl has a preprint in preparation available on her homepage toward this direction.

Edit : Nathalie Wahl's preprint is now published: "Universal operations in Hochschild homology", J. Reine Angew. Math. 720 (2016), 81-127. arXiv:1212.6498, doi:10.1515/crelle-2014-0037

In section 4 of this paper she exhibits a family of non-trivial string topology operations $t_g$. These operations are unstable homology classes of moduli spaces of genus $g$ riemann surfaces with $1$ incoming and 1 outgoing boundary components.