Quantum Stokes matrices and quantum Riemann-Hilbert-Birkhoff maps

Abstract

In this paper, we introduce quantum Stokes matrices for a noncommutative version of meromorphic linear systems of ordinary differential equations with a pole of order $p+1$. We prove that these quantum Stokes matrices satisfy natural quantum exchange relations. These relations allow us to interpret the quantum Stokes matrices as an associative algebra homomorphism, which may be viewed as a deformation quantization of the Riemann-Hilbert-Birkhoff map, regarded as a Poisson map, for meromorphic connections with a pole of order $p+1$.