Non-SUSY physics and the Atiyah-Singer index theorem

TL;DR

This paper presents a non-supersymmetric derivation of the Atiyah-Singer index theorem using quantum statistics, establishing a new mathematical framework that links spectral properties to topological invariants without supersymmetry.

Contribution

It introduces a direct derivation of the index theorem from non-supersymmetric quantum statistics and develops a spectral-sheaf framework for infinite-dimensional cases.

Findings

Grand partition functions correspond to Chern characters.

A generalized index theorem is formulated in infinite dimensions.

Quantum statistics alone can underpin topological invariants.

Abstract

The Atiyah-Singer index theorem, a cornerstone of modern mathematics, has traditionally been derived from supersymmetric (SUSY) physics. This paper demonstrates a direct derivation from non-supersymmetric quantum statistics by establishing a fundamental correspondence: the grand partition functions of non-interacting bosonic and fermionic systems are precisely the Chern characters of certain vector bundles. Furthermore, we generalize this correspondence to infinite dimensions, where we construct a novel mathematical framework of spectral-sheaf pairs. Within this framework, we formulate a generalized index theorem, identifying the topological index with a regularized spectral product. This work not only circumvents the need for supersymmetry but also provides a deeper unifying perspective, revealing quantum statistics as a sufficient foundation for topological invariants.