  In this paper, we present robust evidence that general finite temperature
quantum field theory (QFT) path integrals are invariant under reflecting
temperatures to negative values (T-reflection), up to a possible anomaly phase.
Our main focus is on two-dimensional conformal field theories (2d CFTs) on the
two-torus. Modular invariance for 2d CFT path integrals follows from demanding
invariance under redundant encodings of the two-torus shape in the path
integral. We emphasize that identical logic implies 2d CFTs are invariant under
T-reflection, up to phases. We compute T-reflection anomaly phases for certain
2d CFT path integrals via a continuation, and via an extension of modular forms
from the upper half-plane to the double half-plane. Crucially, they perfectly
agree. Requiring QFT path integrals to be invariant under redundant encodings
of the spacetime geometry implies (i) that 2d CFTs are both modular and
T-reflection invariant and (ii) that general QFT path integrals are invariant
under T-reflection. This quite board argument suggests T-reflection phases may
indicate previously unnoticed anomalies and consistency conditions for general
QFT.
