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

Research output: Contribution to journal › Article › peer-review

32 Downloads (Pure)

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

Access to Document

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

@article{567557a0851e473da8f834e0e60755f4,

title = "Abelianisation of Logarithmic sl_2-Connections",

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 $\pi_\ast$.",

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",

author = "Nikita Nikolaev",

year = "2021",

month = aug,

day = "3",

doi = "10.1007/s00029-021-00688-5",

language = "English",

volume = "27",

journal = "Selecta Mathematica, New Series",

issn = "1022-1824",

publisher = "Birkhauser Verlag Basel",

number = "5",

}

TY - JOUR

T1 - Abelianisation of Logarithmic sl_2-Connections

AU - Nikolaev, Nikita

PY - 2021/8/3

Y1 - 2021/8/3

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

U2 - 10.1007/s00029-021-00688-5

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

Abstract

Keywords

Access to Document

Fingerprint

Cite this