The Thom Isomorphism in Gauge-Equivariant \(K\)-Theory of \(C^*\)-Bundles

IF 1.7 3区物理与天体物理 Q2 PHYSICS, MATHEMATICAL

Russian Journal of Mathematical Physics Pub Date : 2025-05-05 DOI:10.1134/S106192082460168X

D. Fufaev, E. Troitsky

引用次数: 0

Abstract

For a bundle of compact Lie groups \(p\colon {\cal G} \to B\) over a compactum \(B\) (with the structure group of automorphisms of the corresponding group), we introduce the gauge-equivariant \(K\)-theory group \(K_{{\cal G}}^{0}(X; {\mathcal A} )\) of a bundle \(\pi_{X}\colon X \to B\) endowed with a continuous action of \({\cal G}\) constructed using bundles \(E\to X\) with the typical fiber being a projective finitely generated module over a unital \(C^*\)-algebra \( {\mathcal A} \). The index of a family of gauge-invariant (= \({\cal G}\)-equivariant) Fredholm operators naturally takes values in these groups. We introduce and study products and use them to define the Thom homomorphism in gauge-equivariant \(K\)-theory and prove that this homomorphism is an isomorphism.

DOI 10.1134/S106192082460168X

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规范等变\(K\) - \(C^*\) -束理论中的Thom同构

对于一束紧李群 \(p\colon {\cal G} \to B\) 在一个紧凑的 \(B\) （利用相应群的自同构结构群），引入规范等变 \(K\)-理论群 \(K_{{\cal G}}^{0}(X; {\mathcal A} )\) 一捆的 \(\pi_{X}\colon X \to B\) 具有持续行动能力的 \({\cal G}\) 使用捆绑包构造 \(E\to X\) 典型的光纤是一个射影有限生成模块在一个单位 \(C^*\)-代数 \( {\mathcal A} \)．一类规范不变量(= \({\cal G}\)-等变)Fredholm算子自然取这些组中的值。我们引入并研究了积，并用它们来定义标准等变中的Thom同态 \(K\)-理论并证明这个同态是一个同构。Doi 10.1134/ s106192082460168x

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来源期刊

Russian Journal of Mathematical Physics 物理-物理：数学物理

CiteScore

3.10

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.