Conformal constraints on defects

TL;DR

Conformal invariance imposes strong constraints on defects in CFTs. The authors develop an embedding-space framework to describe defects, establish a defect expansion that expresses defects in terms of bulk local operators, and derive conformal blocks and Casimir equations for correlators of two defects. They solve several key special cases (notably codimension-1 and codimension-2 defects) and outline a general, covariant approach to defect correlation functions with potential applications to Casimir forces and entanglement measures. This work provides a systematic foundation for extracting defect data and pursuing defect bootstrap in conformal field theories.

Abstract

In this paper we study the constraints imposed by conformal invariance on extended objects a.k.a defects in a conformal field theory. We identify a particularly nice class of defects that is closed under conformal transformations. Correlation function of the defect with a bulk local operator is fixed by conformal invariance up to an overall constant. This gives rise to the notion of defect expansion, where the defect itself is expanded in terms of local operators. This expansion generalizes the idea of the boundary state. We will show how one can fix the correlation function of two defects from the knowledge of the defect expansion. The defect correlator admits a number of conformal cross-ratios depending on their dimensionality. We find the differential equation obeyed by the conformal block and solve them in certain special cases.

Figures (6)

• Figure 1: Null cone in the $3$-dimensional embedding space and its intersection with a $2$-hyperplane resulting in a spherical $0$-defect (i.e. pair of points, denoted by solid dots). The orthogonal vectors $P_\alpha$ parametrize the hyperplane and hence the defect.
• Figure 2: A generic configuration of a circular defect and bulk-local operator.
• Figure 3: A spherical slice enclosing the bulk operator $\Phi(X)$ and cutting a $2$-dimensional defect ${\mathcal{D}}(P_\alpha)$. The state induces on the slice can be expanded in terms of defect local operators $o(Y)$.
• Figure 4: The quantization slice encloses the defect. The state induced on the sphere is expanded in terms of bulk local operators.
• Figure 5: A generic configuration of co-dimension $m$ and co-dimension $1$ defect.