Crosscap states and duality of Ising field theory in two dimensions

TL;DR

This paper introduces two types of crosscap states in 2D Ising field theory, explores their duality, and develops methods to compute correlation functions and Klein bottle entropy, providing insights into non-orientable manifold effects.

Contribution

It constructs and relates two crosscap states in 2D Ising theory, extending bosonization and perturbation techniques to analyze non-orientable geometries.

Findings

Derived Majorana free field representations for crosscap states

Extended bosonization to compute correlation functions with crosscaps

Supported the monotonicity of Klein bottle entropy under perturbations

Abstract

We propose two distinct crosscap states for the two-dimensional (2D) Ising field theory. These two crosscap states, identifying Ising spins or dual spins (domain walls) at antipodal points, are shown to be related via the Kramers-Wannier duality transformation. We derive their Majorana free field representations and extend bosonization techniques to calculate correlation functions of the 2D Ising conformal field theory (CFT) with different crosscap boundaries. Away from criticality, we develop a conformal perturbation theory to calculate the Klein bottle entropy (norm-square of the crosscap overlap) as a universal scaling function [Phys. Rev. Lett. 130, 151602 (2023)]. For the Ising field theory, our analytical results support the conjectured monotonicity of the Klein bottle entropy under relevant perturbations. The formalism provides a general framework for studying perturbed 2D CFTs on non-orientable manifolds.