TY - JOUR

T1 - Analysis of gauge-equivariant complexes and a topological index theorem for gauge-invariant families

AU - Nistor, V.

AU - Troitsky, E.

N1 - Publisher Copyright:
© 2015, Pleiades Publishing, Ltd.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - 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 = (Db)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) ∈ KG0 (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 inda(D) is defined using analytic properties of the family D and is essentially the difference of the kernel and cokernel KG-classes of D. The topological index is defined purely in terms of the principal symbol of D.

AB - 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 = (Db)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) ∈ KG0 (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 inda(D) is defined using analytic properties of the family D and is essentially the difference of the kernel and cokernel KG-classes of D. The topological index is defined purely in terms of the principal symbol of D.

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U2 - 10.1134/S1061920815010100

DO - 10.1134/S1061920815010100

M3 - Article

AN - SCOPUS:84924359770

VL - 22

SP - 74

EP - 97

JO - Russian Journal of Mathematical Physics

JF - Russian Journal of Mathematical Physics

SN - 1061-9208

IS - 1

ER -