### 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_{G}^{0} (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.

Original language | English (US) |
---|---|

Pages (from-to) | 74-97 |

Number of pages | 24 |

Journal | Russian Journal of Mathematical Physics |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2015 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Russian Journal of Mathematical Physics*,

*22*(1), 74-97. https://doi.org/10.1134/S1061920815010100

}

*Russian Journal of Mathematical Physics*, vol. 22, no. 1, pp. 74-97. https://doi.org/10.1134/S1061920815010100

**Analysis of gauge-equivariant complexes and a topological index theorem for gauge-invariant families.** / Nistor, Victor; Troitsky, E.

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Nistor, Victor

AU - Troitsky, E.

PY - 2015/1/1

Y1 - 2015/1/1

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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UR - http://www.scopus.com/inward/citedby.url?scp=84924359770&partnerID=8YFLogxK

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 -