Crosscap Defects

TL;DR

This paper introduces crosscap defects in conformal field theories, analyzing their properties, crossing equations, and examples in the $O(N)$ model, revealing novel features like the absence of certain operators.

Contribution

It presents the concept of crosscap defects in CFTs, derives crossing equations, and studies their properties and examples, expanding the understanding of defect structures in higher dimensions.

Findings

Crosscap defects generalize real projective space in CFTs.

Crosscap crossing equations are derived and analyzed.

Displacement and tilt operators are absent for generic p, indicating novel defect features.

Abstract

We introduce a novel class of defects, termed crosscap defects, in conformal field theory (CFT) in general dimensions. These arise from quotienting the spacetime by a $Z_2$ automorphism, and provide higher-codimension generalisations of CFT on real projective space ($RP^{d}$). Crosscap defects extend along a $p$-dimensional fixed locus of the $Z_2$ action and preserve an $SO(p+1,1)\times PO(d-p)$ subgroup of the conformal group. The two-point functions of operators in this setup exhibit three operator product expansion channels: bulk, image, and defect. These lead to several crosscap crossing equations, which we present. We analyse conformal block decompositions and show that the blocks are identical to defect CFT blocks up to a redefinition of cross ratios. As concrete examples, we study crosscap defects in the $O(N)$ model at the Gaussian and Wilson--Fisher fixed points in the $\varepsilon$-expansion. We compute explicitly the associated CFT data as a function of $p$ and find that, unlike standard defects, displacement and tilt operators are absent for generic $p$. They provide examples of defect conformal manifolds without exactly marginal operators.