Lattice regularisation of a non-compact boundary conformal field theory

TL;DR

This paper develops lattice models with boundary conditions that realize non-compact boundary conformal field theories, advancing understanding of boundary effects in non-compact CFTs relevant to string theory and condensed matter physics.

Contribution

It identifies integrable boundary conditions in a lattice model that produce non-compact boundary CFTs with continuous spectra, extending previous periodic boundary studies.

Findings

Identified boundary conditions leading to continuous boundary spectra.

Connected lattice boundary conditions to non-compact branes in string theory.

Described boundary RG flows from non-compact to compact boundary conditions.

Abstract

Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions. The present paper follows a series of previous works on a $\mathbb{Z}_2$-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable $D_2^2$ vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.