Algebraic and Geometric Topology 4 (2004), paper no. 17, pages 311-332.

Shadow world evaluation of the Yang-Mills measure

Charles Frohman, Joanna Kania-Bartoszynska

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

AMS subject classification. Primary: 57M27. Secondary: 57R56, 81T13.

DOI: 10.2140/agt.2004.4.311

E-print: arXiv:math.GT/0205193

Submitted: 17 April 2003. (Revised: 26 March 2004.) Accepted: 28 April 2004. Published: 21 May 2004.

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Charles Frohman, Joanna Kania-Bartoszynska
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
and
Department of Mathematics, Boise State University, Boise, ID 83725, USA

Email: frohman@math.uiowa.edu, kania@math.boisestate.edu
URL: http://www.math.uiowa.edu/~frohman, http://math.boisestate.edu/~kania

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