Quantum harmonic oscillator, index theorem and spectral asymmetry

TL;DR

This paper reveals a topological structure in the quantum harmonic oscillator by linking its partition function to the Chern character and the Atiyah-Singer index theorem, connecting statistical mechanics with topology.

Contribution

It establishes a novel connection between spectral asymmetry in quantum oscillators and topological invariants, introducing a topological interpretation of the partition function.

Findings

Partition function identified as the Chern character.

Spectral asymmetry linked to the Atiyah-Singer index theorem.

Internal energy shown as a non-SUSY manifestation of the index theorem.

Abstract

We report a spectral asymmetry effect in the quantum harmonic oscillator, where its partition function is identified as the Chern character. This establishes a direct link between statistical mechanics, and topological invariants (Atiyah-Singer index theorem), revealing the internal energy as a non-SUSY manifestation of the index theorem. We show that the partition function can be interpreted as the Chern character of "virtual physical sheaf", namely, a Hermitian vector bundle encoding quantum states over spacetime. This work uncovers an underlying topological structure in bosonic quantum systems.