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Stokes factors and multilogarithms. (English) Zbl 1288.30037
For \(\mathcal{V}\) a trivial, rank \(n\) complex vector bundle on \(\mathbb{P}^1\) consider a meromorphic connection of the form \(\bigtriangledown =d-\left( \frac{Z}{t^2}+\frac{f}{t}\right) dt\), where \(Z\) and \(f\) are \(n\times n\)-matrices. When \(Z\) is diagonal and with distinct eigenvalues, the computation of the Stokes factors (multipliers) of \(\bigtriangledown\) has been reduced by Balser-Jurkat-Lutz to the analytic continuation of the solutions of a Fuchsian linear system (i.e. a system with logarithmic poles). The monodromy of such a system is expressed by means of multilogarithms. The authors extend these results in the case when \(Z\) is semisimple, but not necessarily with distinct eigenvalues, and when the structure group of the connection \(\bigtriangledown\) is an arbitrary complex affine algebraic group. They show that the map \(\mathcal{S}\) taking the residue of \(\bigtriangledown\) at \(0\) to the corresponding Stokes factors is given by an explicit universal Lie series whose coefficients are multilogarithms. They use a non-commutative analogue of the compositional inversion of formal power series and show that the same holds true for the inverse of \(\mathcal{S}\). The corresponding Lie series is the generating function for counting invariants in abelian categories constructed by D. Joyce.

MSC:
30E99 Miscellaneous topics of analysis in the complex plane
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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