# precise definition of "moduli space"

+ 3 like - 0 dislike
79 views

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?

This post has been migrated from (A51.SE)

+ 1 like - 0 dislike

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.

This post has been migrated from (A51.SE)
answered Apr 24, 2012 by (10,278 points)
Thanks for the answer. So let me see if I understand. A point in moduli space is associated to certain superselection sectors, which in turn are classically associated to the asymptotic values of the fields. In some of these sectors there is a vacuum state (annihilated by the poincare generators), with excitations above it described by the effective theory about that point in moduli space. Perturbatively, one finds a saddle point with the given asymptotic behavior and quantizes perturbations around it. Still a little confused about what the superselection sectors are quantum mechanically.

This post has been migrated from (A51.SE)
Right! And concerning the last question, superselection *only* make sense quantum mechanically. http://en.wikipedia.org/wiki/Superselection_sector

This post has been migrated from (A51.SE)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.