Non-SUSY physics and the Atiyah-Singer index theorem
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.
Cite
@article{arxiv.2512.22887,
title = {Non-SUSY physics and the Atiyah-Singer index theorem},
author = {Shunrui Li and Yang Liu},
journal= {arXiv preprint arXiv:2512.22887},
year = {2025}
}