Shadow world evaluation of the Yang–Mills measure

Frohman, Charles; Kania-Bartoszynska, Joanna

doi:10.2140/agt.2004.4.311

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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

DOI: 10.2140/agt.2004.4.311

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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <mo class="MathClass-bin">−</mo> <mn>1</mn></math>$ 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/

Volume 4, issue 1 (2004)