Post Made Community Wiki by Anton Geraschenko♦♦
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I think it might be worth pointing out that there are two kinds of topological quantum field theory, (Albert) Schwarz-type theories and Witten-type theories. In Schwarz type theories (like Chern-Simons theory and BF-theory), you have an action which is explicitly independent of the metric and you expect that the correlation functions computed by the path integral will also be independent of the metric. In Witten-type theories (Donaldson theory, Gromov-Witten theory), metric independence is a little bit more subtle. In these theories, you do have to choose a metric to get started. But you have some extra structure that allows you to compute some quantities which are metric independent.

(Slightly) more precisely: In a Witten-type theory, you have some operator Q which squares to zero, which you think of as a differential. (Witten type theories are also called cohomological field theories.) You also have an operator T, taking values in (2,0)-tensors, which a) is Q-exact ( T = [Q,G] for some G), and b) generates changes in the metric g. The latter means that if we compute the expectation value < epsilon(T)A > as a function of g, we find that it's equal to the expectation value of A computed with respect to g + epsilon. Here epsilon is a "small" (0,2) tensor we pair with T to get a scalar. In these theories, you can show that the correlation functions of operators which are Q-exact must vanish, which implies that small deformations of g don't change the correlation functions of Q-closed operators A. If you choose A so that its expectation value behaves like a function on the space of metrics, this tells you it's constant on the space of metricmetrics. If you choose some fancier A so that the correlation functions behave like differential forms on the space of metrics, cohomological complications can arise.

Most of the references here are for Schwarz-type theories. For a physics treatment of Witten-type theories, it's worth looking at Witten's "Introduction to cohomological field theory". There's also a long set of lecture notes by Cordes, Moore, & Ramgoolam. The mathematical treatments of the idea are less complete. Hopkins, Lurie, & Costello's stuff is about the most comprehensive, but it's pretty far removed from actions and path integrals. For a starter, you might enjoy Teleman's classification of 2d semi-simple "families topological field theories".

2 added 15 characters in body

I think it might be worth pointing out that there are two kinds of topological quantum field theory, (Albert) Schwarz-type theories and Witten-type theories. In Schwarz type theories (like Chern-Simons theory and BF-theory), you have an action which is explicitly independent of the metric and you expect that the correlation functions computed by the path integral will also be independent of the metric. In Witten-type theories (Donaldson theory, Gromov-Witten theory), metric independence is a little bit more subtle. In these theories, you do have to choose a metric to get started. But you have some extra structure that allows you to compute some quantities which are metric independent.

(Slightly) more precisely: In a Witten-type theory, you have some operator Q which squares to zero, which you think of as a differential. (Witten type theories are also called cohomological field theories.) You also have an operator T, taking values in (2,0)-tensors, which a) is Q-exact ( T = [Q,G] for some G), and b) generates changes in the metric g. The latter means that if we compute the expectation value < epsilon(T)A > as a function of g, we find that it's equal to the expectation value of A computed with respect to g + epsilon. Here epsilon is a "small" (0,2) tensor we pair with T to get a scalar. In these theories, you can show that the correlation functions of operators which are Q-exact must vanish, which implies that small deformations of g don't change the metriccorrelation functions. If you choose A so that its expectation value behaves like a function on the space of metrics, this tells you constant on the space of metric. If you choose some fancier A so that the correlation functions behave like differential forms on the space of metrics, cohomological complications can arise.

Most of the references here are for Schwarz-type theories. For a physics treatment of Witten-type theories, it's worth looking at Witten's "Introduction to cohomological field theory". There's also a long set of lecture notes by Cordes, Moore, & Ramgoolam. The mathematical treatments of the idea are less complete. Hopkins, Lurie, & Costello's stuff is about the most comprehensive, but it's pretty far removed from actions and path integrals. For a starter, you might enjoy Teleman's classification of 2d semi-simple "families topological field theories".

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