Volume 4, issue 1 (2004)

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Shadow world evaluation of the Yang–Mills measure

Charles Frohman and Joanna Kania-Bartoszynska

Algebraic & Geometric Topology 4 (2004) 311–332
 arXiv: math.GT/0205193
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.

Keywords
Yang–Mills measure, shadows, links, skeins, $SU(2)$–characters of a surface
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57R56, 81T13