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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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