Degenerations of flat connections on Riemann surfaces

Mattia Biancotto; Eric D'Hoker; Axel Kleinschmidt; Michele Santagata; Oliver Schlotterer

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arXiv · 2607.05656

Degenerations of flat connections on Riemann surfaces

Mattia Biancotto·Eric D'Hoker·Axel Kleinschmidt·Michele Santagata·Oliver Schlotterer

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Abstract

The integration kernels for polylogarithm functions on a compact Riemann surface of arbitrary genus $h$ are shown to close as the surface undergoes a non-separating degeneration to one of genus $h{-}1$. Explicit formulas are obtained for the non-separating degeneration of the multivariable Enriquez connection for genus $h$ with an arbitrary number of variables to the Enriquez connection for genus $h{-}1$ with two additional punctures whose Lie algebra generators are related to the original ones by the characteristic Bernoulli generating functions known from the degeneration at $h=1$. Analogous degeneration formulas are obtained for the single-valued DHS kernels at the leading order in the real degeneration parameter that is adapted to relating modular tensors at genus $h$ and $h{-}1$.

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Mattia Biancotto, Eric D'Hoker, Axel Kleinschmidt, Michele Santagata, Oliver Schlotterer. 2026-07-06. Degenerations of flat connections on Riemann surfaces. https://arxiv.org/abs/2607.05656

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