Topological and Geometric Universal Thermodynamics in Conformal Field Theory

TL;DR

This paper explores universal thermodynamic properties of conformal field theories on nonorientable surfaces, revealing topological and geometric contributions to thermal data, with analytical and numerical methods highlighting their significance.

Contribution

It introduces a novel analysis of CFT thermodynamics on nonorientable surfaces, identifying fractional topological and geometric terms and connecting them to fundamental CFT constants.

Findings

Discovery of a fractional topological term $rac{1}{2} ln{k}$ in the thermodynamics.

Identification of a geometric term $rac{c}{4} ln{eta}$ related to the central charge.

Analytical derivation of the rainbow boundary term and its link to Cardy-Peschel singularity.

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, $\mathbb{RP}^2$). Through a cut-and-sew process, $\mathbb{RP}^2$ is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term $\frac{1}{2} \ln{k}$ related to nonorientable genus, with $k$ a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{c}{4} \ln{\beta}$, 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.