TFT construction of RCFT correlators II: Unoriented world sheets

TL;DR

This work extends the TFT approach to rational conformal field theories by incorporating unoriented world sheets through a reversion on the symmetric special Frobenius algebra $A$, yielding the enriched structure of Jandl algebras. It provides a complete framework for constructing unoriented correlators, including consistent annulus, Möbius strip, and Klein bottle amplitudes, and analyzes defect lines on non-orientable surfaces; the Ising model serves as a concrete demonstration. A central contribution is a Morita-class–aware classification of reversions via a finite set of representative algebras and a module-category–based algorithm, enabling systematic determination of all inequivalent orientifold projections. The approach maintains locality, modular invariance, and factorization across unoriented world sheets and connects these algebraic reversions to familiar orientifold data, boundary conjugation, and crosscap coefficients, with clear lattice and physical interpretations.

Abstract

A full rational CFT, consistent on all orientable world sheets, can be constructed from the underlying chiral CFT, i.e. a vertex algebra, its representation category C, and the system of chiral blocks, once we select a symmetric special Frobenius algebra A in the category C [I]. Here we show that the construction of [I] can be extended to unoriented world sheets by specifying one additional datum: a reversion on A - an isomorphism from the opposed algebra of A to A that squares to the twist. A given full CFT on oriented surfaces can admit inequivalent reversions, which give rise to different amplitudes on unoriented surfaces, in particular to different Klein bottle amplitudes. We study the classification of reversions, work out the construction of the annulus, Moebius strip and Klein bottle partition functions, and discuss properties of defect lines on non-orientable world sheets. As an illustration, the Ising model is treated in detail.