Existence for prescribed Stokes data in the (3,4) string-equation Riemann–Hilbert problem

Establish solvability of the 3×3 Riemann–Hilbert problem associated with the (3,4) string equation for arbitrary fixed Stokes matrices satisfying the Stokes constraint S_{-7}⋯S_{-1}S_{1}⋯S_{7}=𝒮^T; specifically, prove that for any given admissible Stokes data the Riemann–Hilbert problem has a solution (equivalently, a corresponding solution (U,V) of the (3,4) string equation exists) beyond the asymptotic parameter regimes treated in this paper.

Background

The paper recalls that the (3,4) string equation is equivalent to an isomonodromic 3×3 Riemann–Hilbert problem with rational asymptotics and specified Stokes matrices satisfying a constraint. Prior work established an equivalence between solvability of the Riemann–Hilbert problem and the existence of a solution to the string equation away from singularities, but did not resolve existence for particular Stokes data.

The present work proves existence asymptotically for a special Stokes data set relevant to the multicritical quartic 2-matrix model and in certain double-scaling regimes, leaving the general existence for arbitrary prescribed Stokes data unresolved.