# HET seminar: David McGady – University of Copenhagen

Title: Modularity, Path Integrals, and T-reflection

Abstract: In this talk, I will discuss a (perhaps surprising) symmetry of finite-temperature quantum field theory (QFT) path integrals, Z(T), under reversing the sign of temperature: T-reflection invariance. The basic argument is simple: Finite temperatures, T, set a characteristic timescale, 1/T. When we put a QFT at finite temperature, we identify points in the (Euclidean) time direction that differ by an integer multiple of this timescale: t_E ~ t_E + n/T. Path integrals for these finite-T QFTs depend only on the *periodicity* condition set by this timescale 1/T. Crucially, this periodicity condition does *not* depend on the sign of 1/T. On the strength of this, we may expect path integrals to also formally be invariant under reversing the sign of T: Z(+T) = e^{i \gamma} Z(-T).

Today, I will pursue this logic in QFT and in the mathematics of modular forms, a class of functions that both naturally describe finite-temperature QFT path integrals in low dimensions and are naturally invariant under T-reflections. Time permitting, I will comment on consequences of this new symmetry seemingly buried in the heart of path integrals in general QFT, in condensed matter systems, and for modular forms.