New Witten Paper: Non-Abelian Localization for Chern-Simons Theory

[marcus]

just out
http://arxiv.org/abs/hep-th/0503126

Non-Abelian Localization For Chern-Simons Theory
Chris Beasley, Edward Witten
131 pages


"We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M."

 

[Chronos]

I don't like the last assertion, but this stuff makes me dizzy. Witten is way out of my league, and it's highly likely I'm confused, but, can you apply Chern-Simons gauge theory and derive any kind of self-consistent 3D topology? Maybe that was the point and I just missed it. I'm retreating to my cave now... and hoping evolution kicks in before it's too late.

 

[selfAdjoint]

can you apply Chern-Simons gauge theory and derive any kind of self-consistent 3D topology?

Witten wrote

We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.

Well let's see what we can do with this. A Riemann surface is two dimensional (locally complex); a circle bundle is a mapping from a space (the "top space" or "total space") to the surface in which the preimage of every point (up to some equivalence) is a circle, and it has an associated group, U(1), the group of rotations of the circle (conceived as the unit circle in the complex plane). Non-trivial means this isn't just a cross-product, so there are at least two neighborhoods on the surface where the section maps from the surface into the top space are different. The top space of such a bundle is DEFINED to be a Seifert space, and its topology (cohomology and such) will follow from the standard machinery associated with bundles; for example I imagine there is a Leray-Serre spectral sequence which converges after two terms to give the cohomology of the top space as the cohomology of the base with coefficients in the cohomology of the fibre; since the fibre is just a circle that coefficient module is nice and simple. This is all ancient algebraic topology and was in place before Witten was out of diapers.

The Seifert space is obviously three dimensional (locally a complex plane cross the circle), but has non trivial topology, as the Riemann surface does too. Now somebody else can tell us about the Chern-Simons path integral. I know who Chern and SImons are, anyway!

 

[Haelfix]

The Chern Simmons path integral is by definition topological, it is the textbook example in 3d of something that doesn't know about metrics or any such construct put in by man, eg no local degrees of freedom. It is the most pure nontrivial geometrical construct that I am aware off in all physics. Basically the observables of the theory are formally obtained as products of wilson loops at each link and found to be simply the jones invariant of the link.

Its an absolutely beautiful theory, and as tends to be the case with tqfts exactly soluble (kinda) given a few reasonable inputs and a tiny bit of mathematical superstition (there are certain axioms defined by Atiyah that we need to abide by).

Now this paper doesn't really give *much* new stuff, other than rederiving existing results using nonabelian localization (there it gets technical and somewhat out of my league at this time), the nice thing is it seems to make some of those arguments more or less precise mathematically and recasting it into nice symplectic geometry and analysis of critical points. I haven't read the full paper, but I imagine Witten has in mind some stringy duality whereby some computational formalism in some sector of that theory might yield a sensible and easy computation in c-s theory and its generalizations.

Incidentally this paper is absolutely wonderful to read, and accessible to both mathematicians and theoretical physicists. He is one of the few people in the field who bridges the gap it seems with every paper he writes