Path integrals, finite temperature, and lattices

Surprisingly, partition functions for some model systems in statistical mechanics are invariant under formally reflecting the sign of temperature, T: +T -> -T. We call this T-reflection invariance. Clearly, partition functions for generic statistical systems cannot be invariant under T-reflection. However, in this talk we focus on finite-temperature path integrals and give a general picture for why finite-temperature path integrals in quantum field theory *should* behave well under T-reflection. We probe this general picture in the context of the harmonic oscillators (in one-dimension) and in conformal field theories on the two-torus (in two-dimensions) and in the mathematics of modular forms. We find that the relevant path integrals are often invariant only up to overall T-independent phases, which could be naturally interpreted as new anomalies under large coordinate transforms.