Topological and Geometric Universal Thermodynamics in Conformal Field Theory

Abstract

Universal thermal data in conformal field theory (CFT) offer a valuable means for characterizing and classifying criticality. With improved tensor network techniques, we investigate the universal thermodynamics on a nonorientable minimal surface, the crosscapped disk (or real projective plane, RP2). Through a cut-and-sew process, RP2 is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term 12lnk related to a nonorientable genus, with k a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term c4ln$\beta$, with $\beta$ the manifold size and c the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.