An index theorem for loop spaces

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2024-08-10 DOI:10.1016/j.geomphys.2024.105291

Doman Takata

引用次数: 0

Abstract

We formulate and prove an index theorem for loop spaces of compact manifolds in the framework of KK-theory. It is a strong candidate for the noncommutative geometrical definition (or the analytic counterpart) of the Witten genus. In order to specify an “appropriate form” of the index theorem to formulate a loop space version, we formulate and prove an equivariant index theorem for non-compact $S^{1}$ -manifolds with a compact fixed-point set. In order to formulate it, we use a ring of formal power series.

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循环空间的索引定理

我们在 KK 理论框架内提出并证明了紧凑流形环空间的索引定理。它是维滕属的非交换几何定义（或解析对应物）的有力候选者。为了指定索引定理的 "适当形式"，以提出环空间版本，我们提出并证明了具有紧凑定点集的非紧凑 S1 流形的等变索引定理。为了提出这个定理，我们使用了形式幂级数环。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Geometry and Physics 物理-物理：数学物理

CiteScore

2.90

自引率

6.70%

发文量

205

审稿时长

64 days

期刊介绍： The Journal of Geometry and Physics is an International Journal in Mathematical Physics. The Journal stimulates the interaction between geometry and physics by publishing primary research, feature and review articles which are of common interest to practitioners in both fields. The Journal of Geometry and Physics now also accepts Letters, allowing for rapid dissemination of outstanding results in the field of geometry and physics. Letters should not exceed a maximum of five printed journal pages (or contain a maximum of 5000 words) and should contain novel, cutting edge results that are of broad interest to the mathematical physics community. Only Letters which are expected to make a significant addition to the literature in the field will be considered. The Journal covers the following areas of research: Methods of: • Algebraic and Differential Topology • Algebraic Geometry • Real and Complex Differential Geometry • Riemannian Manifolds • Symplectic Geometry • Global Analysis, Analysis on Manifolds • Geometric Theory of Differential Equations • Geometric Control Theory • Lie Groups and Lie Algebras • Supermanifolds and Supergroups • Discrete Geometry • Spinors and Twistors Applications to: • Strings and Superstrings • Noncommutative Topology and Geometry • Quantum Groups • Geometric Methods in Statistics and Probability • Geometry Approaches to Thermodynamics • Classical and Quantum Dynamical Systems • Classical and Quantum Integrable Systems • Classical and Quantum Mechanics • Classical and Quantum Field Theory • General Relativity • Quantum Information • Quantum Gravity