On analytic bootstrap for interface and boundary CFT

Abstract

We use analytic bootstrap techniques for a CFT with an interface or a boundary. Exploiting the analytic structure of the bulk and boundary conformal blocks we extract the CFT data. We further constrain the CFT data by applying the equation of motion to the boundary operator expansion. The method presented in this paper is general, and it is illustrated in the context of perturbative Wilson-Fisher theories. In particular, we find constraints on the OPE coefficients for the interface CFT in 4 − ϵ dimensions (upto order $$ \mathcal{O} $$ O (ϵ2)) with ϕ4-interactions in the bulk. We also compute the corresponding coefficients for the non-unitary ϕ3-theory in 6 − ϵ dimensions in the presence of a conformal boundary equipped with either Dirichlet or Neumann boundary conditions upto order $$ \mathcal{O} $$ O (ϵ), or an interface upto order $$ \mathcal{O}\left(\sqrt{\epsilon}\right) $$ O ϵ .

Conformal bootstrap near the edge

“…This means that the CFT data further includes the scaling dimensions of boundary operators and the coefficients of the expansion of bulk degrees of freedom in terms of their boundary counterpart (BOE) [8,9]. Remarkably, 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].…”

“…The most obvious one is the analysis of the analytic structure of the blocks and study of discontinuities of the crossing equation, or more general dispersive techniques, which have the potential to address the infinite sums of blocks that appear in the expansion at first order. The techniques developed by [14,16] have the potential to be transported to this context. The anomalous dimension of the edge operator obtained by Cardy seems like the perfect benchmark to test the full potential of our setup.…”

Spontaneous symmetry breaking in free theories with boundary potentials

“…would acquire a dependence on the normal coordinate and we would need to deal with the renormalisation of φ in the near boundary limit. 5 As a consequence the v.e.v. 's of bulk and boundary fields become unequal, which leads to so called extraordinary phase transitions (see [1] for a comprehensive review of phase transitions with boundaries).…”

“…In particular the machinery of boundary conformal bootstrap [3] allows for high precision evaluation of correlation functions at the Wilson-Fisher (WF) fixed point (f.p.) [4,5], which was recently used in evaluation of layer susceptibility at the extraordinary transition point [6,7]. Alternatively a wealth of information on these models can be obtained by coupling them to a curved background and calculating the resulting partition function [8,9].…”

“…We illustrate how these ideas can be implemented 4 In appendix A we find a general expression for the spacetime and the momentum propagator in a boundary quantum field theory (BQFT) with a mass term in both the bulk and on the boundary. 5 By this we mean that the field enjoys the boundary operator expansion φ = x −∆+∆ ⊥φ +. .…”