Reflection positivity and invertible phases of 2d quantum many-body systems

Reflection positivity is a property that is usually taken as an assumption in the classification of topological phases of matter via continuous quantum field theories. For general quantum many-body systems, this property does not hold. This raises the question of whether it somehow emerges in the effective theory from the microscopic description, thereby justifying the field-theoretic approach.
In this talk, I will discuss reflection positivity in the context of invertible phases of two-dimensional lattice systems. I will explain why every such phase admits a reflection-positive representative, and why inverse phases are represented by complex conjugate states. I will also introduce an index that distinguishes these phases and is conjecturally related to the chiral central charge.