precise definition of "moduli space" I'm curious what the precise definition of the moduli space of a QFT is.  One often talks about the classical moduli space, which then can get quantum corrections.  Does this mean the quantum moduli space is something like the set of minima of the effective potential in the 1PI action (or does the 1PI action already presume a choice of vacuum?)?  Is there a non-perturbative definition, without referring to field configurations or effective actions (eg, how do we define it in string theory, where there are no fields)?  Maybe it can be identified with some submanifold of the (projective) Hilbert space?  Can the metric or other structure be defined in such an invariant way?
 A: The most accurate and most quantum definition is that it is the set of all maximally symmetric states (preserving the spacetime symmetries of the flat space or de Sitter space or anti de Sitter space if one allows gravity) in the extended Hilbert space – or one connected component of such a space. By the extension, I mean the formal union of all the superselection sectors.
Equivalently, one may consider the moduli space in QFT to be the set of all superselection sectors that include the maximally supersymmetric ground state.
It just happens that one may associate each such maximally symmetric state with a stationary point (minimum) of the effective action and it is indeed the 1PI effective action. We also talk about the Wilsonian, low-energy effective actions which are conceptually important but they're not an accurate tool to describe the moduli spaces; see some uncontroversial comments about the "right effective action to be used" written by my ex-adviser in his otherwise controversial paper

http://arxiv.org/abs/hep-th/0412129

It's often the case that we know exactly what decides about the fate of the moduli space. In most cases, the theory admits a classical limit and the true moduli space has to be a "deformation" of the classical moduli space – the set of solutions to the classical equations. However, the potential may be modified by quantum corrections that may lift the degeneracy and lower the dimension of the Hilbert space (or set it to zero).
Also, as the ${\mathcal N}=2$ supersymmetric gauge theories show, the quantum effects may introduce monodromy and change the topology of the moduli space etc.
