## Departmental PhD Thesis Exam – Nikita Nikolaev

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Monday, November 13, 2017
3:10 p.m.
BA6183

PhD Candidate:  Nikita Nikolaev
Supervisor:  Marco Gualtieri
Thesis title:  Abelianisation of Logarithmic Connections

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Abstract:

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 $\pi^\textup{ab} : \mathsf{Conn}_X \to \mathsf{Conn}_\Sigma$, called the \textit{abelianisation functor}, from some category of connections on $X$ with logarithmic singularities to some category of abelian connections on $\Sigma$, and we prove that $\pi^\textup{ab}$ is an equivalence of categories.  At the level of the corresponding moduli spaces $\mathbb{M}_X, \mathbb{M}_\Sigma$, 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 $\mathbb{M}_\Sigma$ is a torsor for an algebraic torus, so in fact $\pi^\textup{ab}$ provides a Darboux coordinate system on $\mathbb{M}_X$, known as the \textit{Fock-Goncharov coordinates} constructed in their work on higher Teichm\”uller theory (2006).  To prove that $\pi^\textup{ab}$ is an equivalence of categories, we introduce a new concept called the \textit{Voros class}.  It is a canonical cohomology class in $H^1$ of the base $X$ with values in the nonabelian sheaf $\mathcal{Aut} (\pi_\ast)$ of groups of natural automorphisms of the direct image functor $\pi_\ast$.  Any $1$-cocycle $v$ representing the Voros class defines a new functor $\mathsf{Conn}_\Sigma \to \mathsf{Conn}_X$ by locally deforming the pushforward functor $\pi_\ast$; the result is an explicit inverse equivalence to $\pi^\textup{ab}$, called a \textit{deabelianisation functor}.

We generalise the abelianisation equivalence to the case of \textit{quantum connections}: these are $\hbar$-families of meromorphic connections restricted to a sectorial neighbourhood in $\hbar$ with prescribed asymptotic regularity.   The Schr\”odinger equation is a quintessential example. The most important invariant of a quantum connection $\nabla$ is the Higgs field $\nabla^{\tiny(0)}$ obtained by restricting $\nabla$ to $\hbar = 0$ (the so-called \textit{semiclassical limit}).  Then abelianisation may be viewed as a natural extension to an $\hbar$-family of the spectral line bundle of $\nabla^{\tiny(0)}$.  That is, we show that for a given quantum connection $(\mathcal{E}, \nabla)$, the line bundle $\mathcal{E}^\textup{ab}$ obtained from $\mathcal{E}$ by abelianisation $\pi^\textup{ab}$ restricts at $\hbar = 0$ to precisely the spectral line bundle of the Higgs field $\nabla^{\tiny(0)}$.

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 $\pi^\textup{ab}$.

A copy of the thesis can be found here: https://www.dropbox.com/s/7u89i9j27y5ivuo/PhDThesis.pdf?dl=0