📝 SummarXiv

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.