Quantum harmonic oscillator, index theorem and anomaly

TL;DR

This work investigates whether a purely bosonic quantum system can exhibit topological anomalies by analyzing the quantum harmonic oscillator. The authors show that the oscillator's partition function can be identified with the $Chern\,character$ via the Grothendieck–Riemann–Roch theorem, establishing a bridge between statistical mechanics, quantum anomalies, and topological invariants such as the Atiyah–Singer index theorem and Gromov–Witten theory. They reveal that the internal energy realizes the Hirzebruch $\mathcal{L}$-genus and that the $\beta$-derivative of the partition function encodes a local index density, signaling a bosonic quantum anomaly due to spectral asymmetry and a breaking of time-translation symmetry in the thermal ensemble. Additionally, the partition function is recast as the $Chern\,character$ of a physical sheaf over spacetime, and connections to Gromov–Witten theory are drawn via GRR, offering a unified dictionary that links statistical mechanics, anomalies, and topological geometry with potential broad implications.

Abstract

We report a bosonic anomaly emerging in the quantum harmonic oscillator, whose partition function is rigorously identified as the Chern character via the Grothendieck-Riemann-Roch theorem, establishing a new connection among statistical mechanics, anomaly, Atiyah-Singer index theorem and Gromov-Witten theory. We investigate how its internal energy relates to the Atiyah-Singer index theorem, showing that the partition function can be interpreted as the Chern character of "physical sheaf" over Eucildean spacetime by using Grothentic-Riemann-Roch theorem. This correspondence reveals the internal energy of oscillator as a concrete non-SUSY manifestation of the index theorem. Moreover, we show that this connection naturally leads to the emergence of a quantum anomaly. Furthermore, we arrive at Gromov-Witten theory through a more direct and physically intuitive approach. As a result, the internal energy of the quantum harmonic oscillator serves as a bridge linking two key concepts in physics -- statistical mechanics and anomalies -- with three fundamental mathematical frameworks: the Atiyah-Singer index theorem, the Grothendieck-Riemann-Roch theorem, and Gromov-Witten theory.

Figures (1)

• Figure 1: The integral contour of \ref{['integral after variable substitution']}