Abstract
We propose a bootstrap program for CFTs near intersecting boundaries which form a co-dimension 2 edge. We describe the kinematical setup and show that bulk 1-pt functions and bulk-edge 2-pt functions depend on a non-trivial cross-ratio and on the angle between the boundaries. Using the boundary OPE (BOE) with respect to each boundary, we derive two independent conformal block expansions for these correlators. The matching of the two BOE expansions leads to a crossing equation. We analytically solve this equation in several simple cases, notably for a free bulk field, where we recover Feynman-diagrammatic results by Cardy.
Highlights
An experimentalist might want to measure his critical sample near its surface
The extension of conformal field theory to this setup is known as boundary conformal field theory (BCFT) [3,4,5,6,7]
The consistency of the bulk operator product expansion with the boundary operator expansion leads to a crossing equation which imposes powerful non-perturbative constraint on the bulk and boundary CFT data, extending the applicability of the conformal bootstrap philosophy [10,11,12,13,14,15,16,17,18]
Summary
We consider a d-dimensional CFT near two intersecting boundaries, which form an edge of co-dimension 2. First recall that a usual bulk CFT possesses SO(d + 1, 1) symmetry, generated by d translations, d special conformal transformations, 1 dilation and d(d − 1)/2 rotations. We can no longer perform rotations that change the normal vector, so we have d − 1 fewer rotations allowed This gives a theory with d − 1 translations d − 1 SCTs, 1 dilation and (d − 1)(d − 2)/2 rotations, which shows that BCFTs have SO(d, 1) symmetry, as is well known. We emphasize that θ is an external parameter of our setup that we can tune as we please This means that the edge CFT data generically depends on θ
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