Crosscap states and duality of Ising field theory in two dimensions

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. We further develop a conformal perturbation theory to calculate the Klein bottle entropy as a universal scaling function [Phys. Rev. Lett. 130, 151602 (2023)] in the 2D Ising field theory. The formalism developed in this work is applicable to many other 2D CFTs perturbed by relevant operators.