PhD Advisor: Marco Gualtieri
This thesis studies an equivalence between meromorphic connections of higher rank and abelian connections.
Given a complex curve X and a spectral cover pi : Sigma \to X, we construct a functor pi^ab : Conn_X --> Conn_Sigma, called the abelianisation functor, from some category of connections on X with logarithmic singularities to some category of abelian connections on Sigma, and we prove that pi^ab is an equivalence of categories.
At the level of the corresponding moduli spaces M_X, M_Sigma, which are known to be holomorphic symplectic varieties, this equivalence recovers a symplectomorphism constructed by Gaiotto, Moore, Neitzke in their work on Spectral Networks (2013).
Moreover, the moduli space M_Sigma is a torsor for an algebraic torus, so in fact pi^ab provides a Darboux coordinate system on M_X, known as the Fock-Goncharov coordinates constructed in their work on higher Teichmuller theory (2006).
To prove that pi^ab is an equivalence of categories, we introduce a new concept called the Voros class.
It is a canonical cohomology class in H^1 of the base X with values in the nonabelian sheaf Aut (pi_\ast) of groups of natural automorphisms of the direct image functor pi_\ast.
Any 1-cocycle v representing the Voros class defines a new functor Conn_Sigma --> Conn_X by locally deforming the pushforward functor pi_\ast; the result is an explicit inverse equivalence to pi^ab, called a deabelianisation functor.
We generalise the abelianisation equivalence to the case of quantum connections: these are hbar-families of meromorphic connections restricted to a sectorial neighbourhood in hbar with prescribed asymptotic regularity.
The Schrodinger equation is a quintessential example.
The most important invariant of a quantum connection nabla is the Higgs field nabla^0 obtained by restricting nabla to hbar=0 (the so-called semiclassical limit).
Then abelianisation may be viewed as a natural extension to an hbar-family of the spectral line bundle of nabla^0.
That is, we show that for a given quantum connection (E,nabla), the line bundle E^ab obtained from E by abelianisation pi^ab restricts at hbar=0 to precisely the spectral line bundle of the Higgs field nabla^0.
Finally, in this thesis we explore the relationship between abelianisation and the WKB method, which is an asymptotic approximation technique for solving differential equations developed by physicists in the 1920s and reformulated by Voros in 1983 using the theory of Borel resummation.
We give an algebro-geometric formulation of the WKB method using vector bundle extensions and splittings.
We then show that the output of the WKB analysis is precisely the data used to construct the abelianisation functor pi^ab.