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Shadow world evaluation of the Yang-Mills measure. (English) Zbl 1063.57013

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 invariants 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. Some examples are computed.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)
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