Beta ensembles, quantum Painlev\'e equations and isomonodromy systems

Abstract

This is a review of recent developments in the theory of beta ensembles of random matrices and their relations with conformal filed theory (CFT). There are (almost) no new results here. This article can serve as a guide on appearances and studies of quantum Painlev\'e and more general multidimensional linear equations of Belavin-Polyakov-Zamolodchikov (BPZ) type in literature. We demonstrate how BPZ equations of CFT arise from $\beta$-ensemble eigenvalue integrals. Quantum Painlev\'e equations are relatively simple instances of BPZ or confluent BPZ equations, they are PDEs in two independent variables ("time" and "space"). While CFT is known as quantum integrable theory, here we focus on the appearing links of $\beta$-ensembles and CFT with {\it classical} integrable structure and isomonodromy systems. The central point is to show on the example of quantum Painlev\'e II (QPII)~\cite{betaFP1} how classical integrable structure can be extended to general values of $\beta$ (or CFT central charge $c$), beyond the special cases $\beta=2$ ($c=1$) and $c\to\infty$ where its appearance is well-established. We also discuss an \'a priori very different important approach, the ODE/IM correspondence giving information about complex quantum integrable models, e.g.~CFT, from some stationary Schr\"odinger ODEs. Solution of the ODEs depends on (discrete) symmetries leading to functional equations for Stokes multipliers equivalent to discrete integrable Hirota-type equations. The separation of "time" and "space" variables, a consequence of our integrable structure, also leads to Schr\"odinger ODEs and thus may have a connection with ODE/IM methods.