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arXiv:2508.18357 · Progress of Theoretical and Experimental Physics

New Crosscap States

Wataru Harada, Justin Kaidi, Yuya Kusuki, Yuefeng Liu

Published Jan 30, 2026• 1 citation
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Abstract

Abstract We investigate crosscap states in 2D rational conformal field theories (RCFTs), with an emphasis on the role of noninvertible symmetries. In particular, we argue for the existence of crosscap states labeled by each Verlinde line in the RCFT, extending previous constructions involving simple currents. Evidence for the existence of these new states is obtained by deriving a generalized Cardy condition incorporating both crosscaps and topological defects, which we check in some concrete examples. Finally, we briefly discuss how these crosscap states transform under the action of Verlinde lines, as well as the connection to mixed anomalies between parity and internal symmetries.

DOI
10.1093/ptep/ptag022

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Abstract We investigate crosscap states in 2D rational conformal field theories (RCFTs), with an emphasis on the role of noninvertible symmetries. In particular, we argue for the existence of crosscap states labeled by each Verlinde line in the RCFT, extending previous constructions involving simple currents. Evidence for the existence of these new states is obtained by deriving a generalized Cardy condition incorporating both crosscaps and topological defects, which we check in some concrete examples. Finally, we briefly discuss how these crosscap states transform under the action of Verlinde lines, as well as the connection to mixed anomalies between parity and internal symmetries.
Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory. 1 1. Boundaries and crosscaps In this letter we report progress on the problem of finding boundaries and crosscaps for all conformal field theories that can be obtained as simple current invariants of a given rational conformal field theory (RCFT). This line of research was initiated by Cardy [1], who obtained the (bulk symmetry preserving) boundaries in the case where the torus partition function is the charge conjugation invariant. Later, in [2] the corresponding crosscap coefficients were obtained. In a series of subsequent papers [3,4,5,6,7,8,9,10,11,12,13] various more general situations were considered. In particular one may choose a different Klein bottle projection and a different modular invariant partition function (MIPF) χiZij ¯χ j for the bulk theory. The basic data one would like to determine are the set {m} of Ishibashi labels, the set {a} of boundary labels, a matrix Bm,a of boundary coefficients and a vector Γm of crosscap coefficients.
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 []. 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 nonorientable manifolds.
We explore several aspects of categorical symmetry-resolved entanglement entropy (SREE) directly within two-dimensional rational conformal field theory (RCFT) (without invoking any SymTFT construction arXiv:2409.02806). We derive a general formula applicable whenever the action of the relevant topological defect lines on the annulus Hilbert space is known. This framework accommodates weakly and strongly symmetric boundaries, cloaking states, and fusion rings with multiplicities. We verify the formula in a range of diagonal unitary and non-unitary examples, including theories with generalized Haagerup-Izumi modular data. Furthermore, we extend the analysis to non-diagonal RCFTs. The 12E6\frac{1}{2}E_621​E6​ example demonstrates that closed-channel modular data and NIM-rep multiplicities alone do not suffice to determine the defect action on the complete open-channel Hilbert space.
We construct Cardy states, defect lines and chiral operators for rational coset conformal field theories on the lattice. The bulk theory is obtained by taking the overlap between tensor network representations of different string-nets, while the primary fields emerge from using the topological superselection sectors of the anyons in the original topological theory. This mapping provides an explicit manifestation of the equivalence between conformal field theories in two dimensions and topological field theories in three dimensions: their groundstates and elementary excitations are represented by exactly the same tensors.
Abstract We give a general construction relating Narain rational conformal field theories (RCFTs) and associated 3d Chern-Simons (CS) theories to quantum stabilizer codes. Starting from an abelian CS theory with a fusion group consisting of n even-order factors, we map a boundary RCFT to an n-qubit quantum code. When the relevant ’t Hooft anomalies vanish, we can orbifold our RCFTs and describe this gauging at the level of the code. Along the way, we give CFT interpretations of the code subspace and the Hilbert space of qubits while mapping error operations to CFT defect fields.