Stokes factors and multilogarithms

TL;DR

This paper establishes an explicit universal Lie series involving multilogarithms that describes the Stokes factors of certain meromorphic connections on the Riemann sphere, linking complex analysis, algebra, and category theory.

Contribution

It provides an explicit formula for Stokes factors using multilogarithms and connects this to invariants in abelian categories via Lie series inversion.

Findings

Explicit universal Lie series for Stokes factors involving multilogarithms

Inversion of the Stokes map corresponds to a generating function for invariants

Links between complex differential equations and algebraic invariants

Abstract

Let G be a complex, affine algebraic group and D a meromorphic connection on the trivial G-bundle over P^1, with a pole of order 2 at zero and a pole of order 1 at infinity. We show that the map S taking the residue of D at zero to the corresponding Stokes factors is given by an explicit, universal Lie series whose coefficients are multilogarithms. Using a non-commutative analogue of the compositional inversion of formal power series, we show that the same holds for the inverse of S, and that the corresponding Lie series coincides with the generating function for counting invariants in abelian categories constructed by D. Joyce.