Difference between TCFTs and 2D TQFTs

Question

I have been reading a lot lately on Topological String Theory and general TQFTs and as I noticed, in most contexts the terms "2-dimensional TQFT" and "Topological Conformal Field Theory" (TCFT) seem to be used quite interchangeably. This does make sense, as conformal invariance should easily follow from the much stronger statement that the theory should only depend on the underlying topology of spacetime, but just to be sure I wanted to ask whether both are indeed identical.

Furthermore, there is also the frequently used term "Topological String Theory", and most introductory papers dwell on a subtle difference between such theories and general TQFTs: To get the transition amplitudes in the first, one must sum over all different topologies of the worldsheet, i.e. over all Riemann surfaces. I wondered how this difference factors into the mathematical description: As far as I have seen (e.g. in Lurie's "On the classification of TFTs"), both are described by monoidal functors from a suitable bordism category to a suitable algebraic category, so I think the space of states should be the same and difference occurs when calculating amplitudes only. Is this true?

I figure these questions might seem a bit trivial for someone already acquainted with the field, but I couldn't find any good references on this (even ncatlab is contradicting itself a bit here...) so I would be very happy to properly understand the relations between those three terms. Greetings,

Markus Zetto

Answer 1

I think these concepts get muddled together a bit. That's language for you.

TQFT is the most general term, I'd say. In a generic TQFT, all one knows is that the observables don't depend on the metric. This could be because the metric doesn't appear in the definition of the theory, like in BF theory.

TCFT is a special sort of 2d TQFT in which the observables are sensitive to the topology of the space of complex surfaces. What makes them special is that the definition of the theory and its observables can involve the metric on the Riemann surface -- or even families of metrics on the Riemann surface -- but the expectation values of observables only depend very loosely on these families. Usually in a TCFT, the observable will only depend on the homology class determined by the family of metrics in the space of complex surfaces. The canonical example here is the A-twisted topological sigma model.

When you have a TCFT, especially if you have a TCFT given by a topological sigma model describing maps from a worldsheet to some ambient spacetime, you can then write down a perturbative string theory (possibly wiht a non-classical target space) by integrating over metrics and summing over topologies. I prefer to use the term topological string theory to describe the resulting physics in the target spacetime. A famous example is Chern-Simons theory, which arises as the target space physics corresponding to a topological A-twisted sigma model to $TS^3$.