When can Witten-esque moduli spaces be used to define invariants of geometric structures?

Question

I am trying to understand the big picture around Seiberg-Witten invariants of 4-manifolds. Of course, this points to Taubes work on Gromov-Witten invariants of symplectic manifolds.

It is striking to me that both developments arise, essentially, by studying spaces of functions that satisfy non-linear elliptic equations. For Seiberg-Witten, one studies moduli spaces of sections of bundles over a 4-manifold modulo a gauge group action. In the case of Gromov-Witten, one analyzes spaces of special maps from surfaces to symplectic manifolds.

Perhaps someone has cataloged or systematically studied which elliptic moduli spaces produce "useful" invariants of the underlying geometric structures. Is it a work of genius ala Gromov and Witten to suggest these equations or do we have some methodical process whereby we can define (or guess) such a useful equation and resulting moduli space? For example, might there be a Witten-like equation which helps elucidate contact structures on 7-manifolds, etc?

So my question is, what other Witten-like moduli spaces can be used to define invariants of geometric structures?

Answer 1

One general lesson we can learn from the work of Witten and others is that supersymmetric field theories are a systematic source of deep invariants (in the form of moduli spaces from the classical theories and meaningful linearizations of them from the quantum theories, to oversimplify egregiously). In other words one general principle is to look for representations of super-Lie algebras extending the symmetries of space(time) on spaces of fields (functions, spinors, maps, connections, and other local expressions).

While this might initially seem as 1. an ad hoc principle, and 2. as hopelessly general, it turns out to be neither. It is a unifying theme to many (most?) of the applications of stringy physics to math; and there are remarkable classification results (due to Nahm and others) saying that there are far far fewer such theories than you might originally think. Namely there are strict bounds in terms of the dimension of space to how big a supersymmetry algebra you can find, if you bound the spin of the fields you consider (ie if you want to consider equations involving only scalars, or only scalar and one-forms, or only up to two-forms [gravity]-- physicists tell us to look no further than that, and therefore at no higher than 11 dimensions..)

Since there turn out to be very very few "maximally supersymmetric" equations you can write down, kind of like exceptional Lie groups or other exceptional objects in math, each turns out to be amazingly meaningful and applicable and provides a powerful guiding principle for mathematicians asking the kind of question you're asking. For example you can ask "hmm.. how can I interpret Khovanov homology in terms of elliptic equations or field theory somehow?" Well if you're Witten this summer, you answer: well I need a supersymmetric field theory, I realize it needs to have more than four dimensions, there's basically one super special theory I know that lives in 6 dimensions, so I should look for which geometry to stick in my two extra dimensions to get what I want, and there aren't so many options, I find that taking the plane and modding out by rotations is the most reasonable choice, and bam! I have a new construction of Khovanov and in fact an extension of it to a full-fledged 4d TFT..

I recommend the expository articles of Dan Freed (with Deligne in the IAS QFT volumes, and his Five Lectures of Supersymmetry, and a PCMI proceedings) for intros for mathematicians to supersymmetry, classical and quantum.