Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-SUSY physics and the Atiyah-Singer index theorem

Shunrui Li, Yang Liu

TL;DR

This work shows that the Atiyah-Singer index theorem can be derived from non-supersymmetric quantum statistics by linking the grand partition functions of bosonic and fermionic systems to Chern characters of vector bundles. It builds a bridge via a thermal spacetime $M\times S^1_\beta$ and a hierarchy of physical and geometric objects (Fock sheaves, spectral-sheaf pairs), then extends the construction to infinite dimensions with a formal spectral framework. Four pairings (Fermi–Bose, Bose–Bose, Fermi–Fermi, Bose–Fermi) reproduce index formulas for Dirac and de Rham operators and introduce two new characteristic classes, the $\hat{B}$-genus and the $\mathrm{Td}^*$-class, showing that quantum statistics encode topological invariants. The infinite-dimensional generalization defines a spectrum-based, regularized determinant approach within an abstract category of spectral sheaves, yielding a generalized index theorem that reduces to the classical Euler class in the nondegenerate limit. Overall, the paper posits quantum statistics as a universal language for topological invariants, offering a unifying perspective on geometry, topology, and physics with potential applications to condensed matter and beyond.

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.

Non-SUSY physics and the Atiyah-Singer index theorem

TL;DR

This work shows that the Atiyah-Singer index theorem can be derived from non-supersymmetric quantum statistics by linking the grand partition functions of bosonic and fermionic systems to Chern characters of vector bundles. It builds a bridge via a thermal spacetime and a hierarchy of physical and geometric objects (Fock sheaves, spectral-sheaf pairs), then extends the construction to infinite dimensions with a formal spectral framework. Four pairings (Fermi–Bose, Bose–Bose, Fermi–Fermi, Bose–Fermi) reproduce index formulas for Dirac and de Rham operators and introduce two new characteristic classes, the -genus and the -class, showing that quantum statistics encode topological invariants. The infinite-dimensional generalization defines a spectrum-based, regularized determinant approach within an abstract category of spectral sheaves, yielding a generalized index theorem that reduces to the classical Euler class in the nondegenerate limit. Overall, the paper posits quantum statistics as a universal language for topological invariants, offering a unifying perspective on geometry, topology, and physics with potential applications to condensed matter and beyond.

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.
Paper Structure (24 sections, 76 equations, 4 tables)

This paper contains 24 sections, 76 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Basics
  3. The grand canonical ensemble and non-interacting system distribution
  4. The grand canonical ensemble
  5. Bose-Einstein distribution and Fermi-Dirac distribution
  6. Todd class and the Atiyah-Singer index theorem
  7. Todd class
  8. The Atiyah-Singer index theorem
  9. From single-particle to grand partition functions
  10. The physical motivation: incorporating finite temperature
  11. The mathematical framework: sheaves and characteristic classes
  12. 1. Single particle
  13. 2. Bosonic system: symmetric algebra (local Fock Bosonic sheaf)
  14. 3. Fermionic system: exterior algebra (local Fock sheaf)
  15. The “physical” derivation of the Atiyah-Singer index theorem from non-interacting system
  16. ...and 9 more sections