FZZT Brane Relations in the Presence of Boundary Magnetic Fields

TL;DR

This paper introduces a matrix model approach to realize and analyze boundary states in minimal string theories with boundary magnetic fields, extending previous work on boundary renormalization flows.

Contribution

It develops a new matrix model method to represent boundary states beyond the (1,1) Cardy state, including boundary magnetic fields, generalizing Ising model results to (m,m+1) minimal models.

Findings

Derived spectral curve for free spin boundary state

Extended saddle-point method for non-identical potentials

Generalized boundary renormalization flow analysis

Abstract

We show how a boundary state different from the (1,1) Cardy state may be realised in the (m,m+1) minimal string by the introduction of an auxiliary matrix into the standard two hermitian matrix model. This boundary is a natural generalisation of the free spin boundary state in the Ising model. The resolvent for the auxiliary matrix is computed using an extension of the saddle-point method of Zinn-Justin to the case of non-identical potentials. The structure of the saddle-point equations result in a Seiberg-Shih like relation between the boundary states which is valid away from the continuum limit, in addition to an expression for the spectral curve of the free spin boundary state. We then show how the technique may be used to analyse boundary states corresponding to a boundary magnetic field, thereby allowing us to generalise the work of Carroll et al. on the boundary renormalisation flow of the Ising model, to any (m,m+1) model.