Reflection and transmission for conformal defects

Abstract

We consider conformal defects joining two conformal field theories along a line. We define two new quantities associated to such defects in terms of expectation values of the stress tensors and we propose them as measures of the reflectivity and transmissivity of the defect. Their properties are investigated and they are computed in a number of examples. We obtain a complete answer for all defects in the Ising model and between certain pairs of minimal models. In the case of two conformal field theories with an enhanced symmetry we restrict ourselves to non-trivial defects that can be obtained by a coset construction.

The reflection coefficient for minimal model conformal defects from perturbation theory

“…There has also been a great deal of study of defects between different conformal field theories, with exact classifications in a few cases [2], exact proposals [9] for defects related to renormalisation group flows, and perturbative calculations [10].…”

“…One characteristic of a conformal defect is its transmission coefficient T , or equivalently its reflection coefficient R = 1 − T , which was defined in [2]. These take the values R = 0 for a topological defect and R = 1 for a factorised defect, and 0 < R < 1 for a general conformal defect in a unitary theory [11].…”

“…These sewing constraints have been solved in [14] for the (1, 2) defect in the non-unitary Lee-Yang model, the (A 1 , A 4 ) theory, in which D (1,2) is the only non-trivial defect and {1, φ,φ, ϕ α } are the only non-trivial primary defect fields. In this paper we extend this analysis to the fields {1, φ,φ, ϕ α } on defects of type D (r,2) in all the (A p , A q ) models.…”

“…Defects in two-dimensional systems have been studied for a long time, see eg [1,2] and references therein. In conformal field theory, attention has been focused primarily on defects which preserve some or all of the conformal symmetry.…”

“…The situation of more general conformal defects is much less clear. The conformal defects in the Ising model were classified in [5] (and in the much simpler Lee-Yang model in [2]), but in general the only results found are either perturbative or numerical [6]. More recently, we have also found exact expressions for conformal defects in the tricritical Ising model [7] (based on ideas in [8]).…”