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Periodic and Quasi-motivic Pencils of Flat Connections by Etingof, Pavel; Varchenko, Alexander is a scholarly article available to read on EtoBox.

What is Periodic and Quasi-motivic Pencils of Flat Connections about?

We introduce a new notion of a periodic pencil of flat connections on a smooth algebraic variety $X$. This is a family $\nabla(s_1,...,s_n)$ of flat connections on a trivial vector bundle on $X$ depending linearly on parameters $s_1,...,s_n$ and generically invariant, up to isomorphism, under the shifts $s_i\mapsto s_i+1$ for all $i$. If in addition $\nabla$ has regular singularities, we call it a quasi-motivic pencil. We use tools from complex analysis to establish various remarkable properties of such pencils over $\mathbb C$. For example, we show that the monodromy of a quasi-motivic pencil is defined over the field of algebraic functions in $e^{2\pi is_j}$, and that its singularities are constrained to an arrangement of hyperplanes with integer normal vectors. Then we show that many important examples of families of flat connections, such as Knizhnik-Zamolodchikov, Dunkl, and Casimir connections, are quasi-motivic and thus periodic pencils. Besides being interesting in its own right, the periodic property of a pencil of flat connections turns out to be very useful in computing the eigenvalues of the $p$-curvature of its reduction to positive characteristic. This will be done in

Author

Etingof, Pavel; Varchenko, Alexander

Published

2024

Language

EN