Quantum harmonic oscillator, index theorem and spectral asymmetry

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.