Infinite-dimensional Dubrovin-Frobenius manifolds and the Stokes phenomenon

TL;DR

This paper investigates the infinite-dimensional Dubrovin-Frobenius manifold related to the 2D Toda hierarchy, revealing complex solution structures and computing the Stokes matrices associated with irregular singularities.

Contribution

It introduces a novel analysis of the Dubrovin equation in infinite dimensions, including the emergence of solutions with infinite parameters and the computation of Stokes matrices.

Findings

Formal solutions depend on infinitely many parameters.

A family of solutions is constructed via integrals along the unit circle.

The infinite-dimensional Stokes matrices are explicitly computed.

Abstract

We study the Dubrovin equation of the infinite-dimensional 2D Toda Dubrovin-Frobenius manifold at its irregular singularity. We first revisit the definition of the canonical coordinates, proving that they emerge naturally as generalized eigenvalues of the operator of multiplication by the Euler vector field. We then show that the formal solutions to the Dubrovin equation with exponential type behaviour at the irregular singular point are not uniquely determined by their leading order, but instead depend on an infinite number of parameters, contrary to what happens in the finite-dimensional case. Next, we obtain a large family of solutions to the Dubrovin equation given by integrals along the unit circle of certain combinations of the superpotentials. Observing that such a family is not complete and has trivial monodromy, we study a larger family of weak solutions obtained via Borel resummation of some distinguished formal solutions. These resummed solutions naturally appear in monodromy-related pairs, finally allowing us to compute the infinite-dimensional analogue of the Stokes matrices.