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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

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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 $TS^3$.

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  • $\begingroup$ Thank you for the answer! Especially the last part was very helpful. Could you explain a bit more about the metric dependence of the TCFTs, it seems to me that (even though as you state this is only through a certain homology class, I don't really understand how this class is defined) this would make the field theory not topologic? I am asking this because e.g. Lurie gave a description of the closed-string B-model in a completely axiomatic way (as a monoidal functor) in his paper on TFTs, without mentioning anything about a worldsheet metric, is this because we integrate over it? Greetings, $\endgroup$ – Markus Zetto May 5 '20 at 17:27
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    $\begingroup$ No, it's not because of any integration. In his cobordism paper, Lurie only describes the B-model as a TQFT (example 1.4.1). He doesn't say what the category assigned to a point is, and without that info, I don't believe you can construct a chain-level theory. In Lurie's world, the dependence on families of complex structures sneaks in through the cobordism hypothesis. Having a fully dualizable object gives you something that's acted on naturally by families of bordisms of circles, and such bordisms are Riemann surfaces, with all their classical connection to the moduli of curves. $\endgroup$ – user1504 May 5 '20 at 19:01
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    $\begingroup$ You might be better off reading Costello's paper Topological Conformal Field Theories and Calabi-Yau algebras. Lurie's is more generality than needed for this. $\endgroup$ – user1504 May 5 '20 at 19:03

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