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Mathematics > Quantum Algebra
Title: Geometry of Localized Effective Theories, Exact Semi-classical Approximation and the Algebraic Index
(Submitted on 25 Nov 2019 (v1), last revised 6 Nov 2020 (this version, v3))
Abstract: In this paper we propose a general framework to study the quantum geometry of $\sigma$-models when they are effectively localized to small quantum fluctuations around constant maps. Such effective theories have surprising exact descriptions at all loops in terms of target geometry and can be rigorously formulated. We illustrate how to turn the physics idea of exact semi-classical approximation into a geometric set-up in this framework, using Gauss-Manin connection. As an application, we carry out this program in details by the example of topological quantum mechanics, and explain how to implement the idea of exact semi-classical approximation into a proof of the algebraic index theorem. The proof resembles much of the physics derivation of Atiyah-Singer index theorem and clarifies the geometric face of many other mathematical constructions.
Submission history
From: Si Li [view email][v1] Mon, 25 Nov 2019 19:13:41 GMT (64kb)
[v2] Wed, 11 Dec 2019 04:49:54 GMT (70kb)
[v3] Fri, 6 Nov 2020 07:56:53 GMT (71kb)
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