  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.

Geometric Topology;Differential Geometry