Wronskians, dualities and FZZT-Cardy branes

TL;DR

This paper explores generalized Wronskians in matrix models, revealing their role as non-perturbative degrees of freedom linked to FZZT-Cardy branes, extending spectral dualities and aligning with Liouville theory.

Contribution

It introduces generalized Wronskians of Baker-Akhiezer systems as new non-perturbative operators, extending spectral dualities, and identifies them with FZZT-Cardy branes in Liouville theory.

Findings

Derived isomonodromy systems for Wronskian operators

Extended spectral dualities to these systems

Proposed FZZT-Cardy branes as bound states of elemental branes

Abstract

The resolvent operator plays a central role in matrix models. For instance, with utilizing the loop equation, all of the perturbative amplitudes including correlators, the free-energy and those of instanton corrections can be obtained from the spectral curve of the resolvent operator. However, at the level of non-perturbative completion, the resolvent operator is generally not sufficient to recover all the information from the loop equations. Therefore it is necessary to find a sufficient set of operators which provide the missing non-perturbative information. In this paper, we study generalized Wronskians of the Baker-Akhiezer systems as a manifestation of these new degrees of freedom. In particular, we derive their isomonodromy systems and then extend several spectral dualities to these systems. In addition, we discuss how these Wronskian operators are naturally aligned on the Kac table. Since they are consistent with the Seiberg-Shih relation, we propose that these new degrees of freedom can be identified as FZZT-Cardy branes in Liouville theory. This means that FZZT-Cardy branes are the bound states of elemental FZZT branes (i.e. the twisted fermions) rather than the bound states of principal FZZT-brane (i.e. the resolvent operator).