Shadow world evaluation of the Yang–Mills measure

Abstract

A new state-sum formula for the evaluation of the Yang–Mills measure in the Kauffman bracket skein algebra of a closed surface is derived. The formula extends the Kauffman bracket to diagrams that lie in surfaces other than the plane. It also extends Turaev’s shadow world invariant of links in a circle bundle over a surface away from roots of unity. The limiting behavior of the Yang–Mills measure when the complex parameter approaches $-1$ is studied. The formula is applied to compute integrals of simple closed curves over the character variety of the surface against Goldman’s symplectic measure.