Abstract
We continue our study of gauge equivariant K-theory. We thus study the analysis of complexes endowed with the action of a family of compact Lie groups and their index in gauge equivariant K-theory. We introduce various index functions, including an axiomatic one, and show that all index functions coincide. As an application, we prove a topological index theorem for a family D = (D b ) b∈B of gauge-invariant elliptic operators on a G-bundle X → B, where G → B is a locally trivial bundle of compact groups, with typical fiber G. More precisely, one of our main results states that a-ind(D) = t-ind(D) ∈ K 0 G (X), that is, the equality of the analytic index and of the topological index of the family D in the gauge-equivariant K-theory groups of X. The analytic index ind a (D) is defined using analytic properties of the family D and is essentially the difference of the kernel and cokernel K G -classes of D. The topological index is defined purely in terms of the principal symbol of D.
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V. N. was partially supported by ANR-14-CE25-0012-01. E. T. was partially supported by RFBR Grant 13-01-00232 and by the grant of the Government of the Russian Federation N2010-220-01-077 contract 11.G34.31.0005. The present joint research was started under the hospitality of MPIM (Bonn).
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Nistor, V., Troitsky, E. Analysis of gauge-equivariant complexes and a topological index theorem for gauge-invariant families. Russ. J. Math. Phys. 22, 74–97 (2015). https://doi.org/10.1134/S1061920815010100
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DOI: https://doi.org/10.1134/S1061920815010100