Four-dimensional N=2 Field Theory and Physical Mathematics

TL;DR

This paper reviews d=4, N=2 quantum field theory, highlighting exact results, wall-crossing phenomena, and geometric methods for constructing hyperkahler metrics and computing BPS spectra and operator expectation values.

Contribution

It introduces new geometric constructions that yield exact results for BPS spectra and Wilson line operators in N=2 theories, connecting physical mathematics with geometric analysis.

Findings

Wall-crossing phenomena elucidated

New hyperkahler metric constructions presented

Exact BPS spectrum calculations achieved

Abstract

We give a summary of a talk delivered at the 2012 International Congress on Mathematical Physics. We review d=4, N=2 quantum field theory and some of the exact statements which can be made about it. We discuss the wall-crossing phenomenon. An interesting application is a new construction of hyperkahler metrics on certain manifolds. Then we discuss geometric constructions which lead to exact results on the BPS spectra for some d=4, N=2 field theories and on expectation values of -- for example -- Wilson line operators. These new constructions have interesting relations to a number of other areas of physical mathematics.