Bootstrapping critical Ising model on three-dimensional real projective space

TL;DR

This work uses crosscap conformal bootstrap equations to numerically solve the Lee-Yang model as well as the critical Ising model on a three dimensional real projective space, opening up a novel way to solve conformal field theories on nontrivial geometries.

Abstract

Given a conformal data on a flat Euclidean space, we use crosscap conformal bootstrap equations to numerically solve the Lee-Yang model as well as the critical Ising model on a three-dimensional real projective space. We check the rapid convergence of our bootstrap program in two-dimensions from the exact solutions available. Based on the comparison, we estimate that our systematic error on the numerically solved one-point functions of the critical Ising model on a three-dimensional real projective space is less than one percent. Our method opens up a novel way to solve conformal field theories on non-trivial geometries.

Figures (1)

• Figure 1: Involutions used in two-dimensional real projective spaces: The left panel describes the involution $(t,\vec{\Omega}) \to (-t,-\vec{\Omega})$ on the Lorentzian cylinder while the right panel describes the involution $\vec{x} \to - \frac{\vec{x}}{\vec{x}^2}$ on the Euclidean space. The fundamental domain can be chosen $t\ge0$ or $r^2 = \vec{x}^2 \ge 1$.