Periodic and quasi-motivic pencils of flat connections
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arXiv
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| Format: | Preprint |
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2024
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| author | Etingof, Pavel Varchenko, Alexander |
| author_facet | Etingof, Pavel Varchenko, Alexander |
| contents | 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 $e^{2πis_j}$, 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 our forthcoming paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Periodic and quasi-motivic pencils of flat connections Etingof, Pavel Varchenko, Alexander Algebraic Geometry Complex Variables Quantum Algebra Representation Theory 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 $e^{2πis_j}$, 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 our forthcoming paper. |
| title | Periodic and quasi-motivic pencils of flat connections |
| topic | Algebraic Geometry Complex Variables Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2401.00636 |