Crosscap Defects

Abstract: We introduce a novel class of defects, termed {\it crosscap defects}, in conformal field theory (CFT) in general dimensions. These arise from quotienting the spacetime by a $\mathbb{Z}_2$ automorphism, and provide higher-codimension generalisations of CFT on real projective space ($\mathbb{RP}^{d}$). Crosscap defects extend along a $p$-dimensional fixed locus of the $\mathbb{Z}_2$ 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 {\it 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.