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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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
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Publication
Received: 17 April 2003
Revised: 26 March 2004
Accepted: 28 April 2004
Published: 21 May 2004
Authors
Charles Frohman
Department of Mathematics
University of Iowa
Iowa City IA 52242
USA
http://www.math.uiowa.edu/~frohman/
Joanna Kania-Bartoszynska
Department of Mathematics
Boise State University
Boise ID 83725
USA
http://math.boisestate.edu/~kania/