Abelianisation of Logarithmic sl_2-Connections

Nikita Nikolaev

doi:10.1007/s00029-021-00688-5

Abelianisation of Logarithmic sl_2-Connections

Nikita Nikolaev^*

^*Corresponding author for this work

Mathematics

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Abstract

We prove a functorial correspondence between a category of logarithmic $\frak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \sf{\Sigma} \to X$. The proof is by constructing a pair of inverse functors $\pi^\ab, \pi_\ab$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $\pi_\ast$.

Original language	English
Article number	78
Number of pages	35
Journal	Selecta Mathematica, New Series
Volume	27
Issue number	5
DOIs	https://doi.org/10.1007/s00029-021-00688-5
Publication status	Published - 3 Aug 2021

Keywords

meromorphic connections
spectral curves
spectral networks
Stokes graph
exact WKB
abelianisation
Levelt filtrations
singular differential equations
Higgs bundles
local systems
algebraic geometry
differential geometry
holomorphic geometry

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10.1007/s00029-021-00688-5Licence: Creative Commons: Attribution (CC BY)

NikolaevN2021AbelianisationFinal published version, 1.16 MBLicence: Creative Commons: Attribution (CC BY)

https://arxiv.org/abs/1902.03384

Cite this

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abstract = "We prove a functorial correspondence between a category of logarithmic $\frak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \sf{\Sigma} \to X$. The proof is by constructing a pair of inverse functors $\pi^\ab, \pi_\ab$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $\pi_\ast$.",

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N2 - We prove a functorial correspondence between a category of logarithmic $\frak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \sf{\Sigma} \to X$. The proof is by constructing a pair of inverse functors $\pi^\ab, \pi_\ab$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $\pi_\ast$.

AB - We prove a functorial correspondence between a category of logarithmic $\frak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \sf{\Sigma} \to X$. The proof is by constructing a pair of inverse functors $\pi^\ab, \pi_\ab$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $\pi_\ast$.

KW - meromorphic connections

KW - spectral curves

KW - spectral networks

KW - Stokes graph

KW - exact WKB

KW - abelianisation

KW - Levelt filtrations

KW - singular differential equations

KW - Higgs bundles

KW - local systems

KW - algebraic geometry

KW - differential geometry

KW - holomorphic geometry

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DO - 10.1007/s00029-021-00688-5

M3 - Article

SN - 1022-1824

VL - 27

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

IS - 5

M1 - 78

ER -

Abelianisation of Logarithmic sl_2-Connections

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Keywords

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