Properties of RG interfaces for 2D boundary flows

TL;DR

This paper investigates the properties of RG interfaces in 2D boundary flows, proposing conjectures about their conformal primary nature and operator product expansions, supported by perturbative, variational, and numerical methods.

Contribution

It introduces conjectures on the conformal primary status of RG operators and their OPEs, supported by multiple computational approaches, advancing understanding of boundary RG flows.

Findings

RG operators are conjectured to be conformal primaries.

OPE of RG operators with their conjugates contain specific operators.

Variational and numerical methods support the conjectures.

Abstract

We consider RG interfaces for boundary RG flows in two-dimensional QFTs. Such interfaces are particular boundary condition changing operators linking the UV and IR conformal boundary conditions. We refer to them as RG operators. In this paper we study their general properties putting forward a number of conjectures. We conjecture that an RG operator is always a conformal primary such that the OPE of this operator with its conjugate must contain the perturbing UV operator when taken in one order and the leading irrelevant operator (when it exists) along which the flow enters the IR fixed point, when taken in the other order. We support our conjectures by perturbative calculations for flows between nearby fixed points, by a non-perturbative variational method inspired by the variational method proposed by J.~Cardy for massive RG flows, and by numerical results obtained using boundary TCSA. The variational method has a merit of its own as it can be used as a first approximation in charting the global structure of the space of boundary RG flows. We also discuss the role of the RG operators in the transport of states and local operators. Some of our considerations can be generalised to two-dimensional bulk flows, clarifying some conceptual issues related to the RG interface put forward by D.~Gaiotto for bulk $\phi_{1,3}$ flows.