Abelianisation of logarithmic $$\mathfrak {sl}_2$$-connections

Abstract

We prove a functorial correspondence between a category of logarithmic $$\mathfrak {sl}_2$$ sl 2 -connections on a curve $${\mathsf {X}}$$ X with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover "Equation missing". The proof is by constructing a pair of inverse functors $$\pi ^\text {ab}, \pi _\text {ab}$$ π ab , π ab , and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $$\pi _*$$ π ∗ .

Abelianisation of Logarithmic $\mathfrak{sl}_2$-Connections

“…Indeed, Proposition 3.4 shows that the abelianisation line bundle L is the correct analogue of the spectral line bundle. This article (which is an extension of the work the author completed in his thesis [Nik18]) is thus the first step in realising abelianisation from Higgs bundles to flat bundles in mathematical terms.…”

Exact WKB and Abelianization for the $$T_3$$ Equation

“…The method of [15,16] leads to a new geometric reformulation of exact WKB, both for Schr ödinger operators and their higher-order analogues. In this reformulation, the key step in exact WKB is a process of "abelianization" which replaces a flat SL(K)-connection ∇ over a surface C by a flat GL(1)-connection ∇ ab over a K-fold covering Σ → C. 1 Some aspects of this abelianization process and its relation to exact WKB have been further developed in [17,18,19,20].…”

“…Given a flat connection ∇ and a Stokes graph W, one can formulate the notion of a Wabelianization of ∇, as in [17] (see also [20] for a more recent and mathematical treatment).…”

“…These can be thought of as analogues of the bound state energies for a confining potential on the real line.Using(4.22) we can rewrite the quasiperiodicity condition (4.24) as20 …”

A geometric recipe for twisted superpotentials

“…Another proposal for SL(3, C) Darboux coordinates on the four-punctured sphere is found in[85] 15. Other accounts and examples of abelianization may be found in e.g [36,118,144]…”