Deformation quantisation and Airy structures

Swaddle, Michael

Deformation quantisation and Airy structures

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Final thesis file (1.01 MB)

Authors

Swaddle, Michael

Date

2022

Affiliation

School of Mathematics and Statistics

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Document Type

PhD thesis

Access Status

Open Access

URI

https://hdl.handle.net/11343/324548

Description

Abstract

Following the work of Kontsevich and Soibelman, we study Airy structures and the relation to an algorithm known as topological recursion, which solves interesting enumerative problems. Airy structures encode the data of a Lagrangian subvariety in an infinite dimensional symplectic vector space. First, with examples, we examine the deformation quantisation of the structure sheaf of a Lagrangian sub-variety. Deformation quantisation is an algorithm that replaces a commutative algebra with a non-commutative algebra. Looking for modules over these non-commutative rings, leads to objects called wavefunctions. Wavefunctions are generators for particular cyclic modules. With some choice of initial conditions, wavefunctions encode the data of topological recursion. Next we examine how these wavefunctions behave under symplectic reduction. In particular we give an example of a particular integral formula involving wavefunctions. Finally we put these discussions into a bigger context. The infinite dimensional symplectic space required to explicitly produce the topological recursion invariants is a space of differentials over a curve in a surface. Symplectic reduction relates this to a deformation space of curves over this surface. A new interesting consequence is that some data from the Airy structure defines the variation of the period matrix, the so called Donagi-markman cubic, generalising a result of Baraglia and Huang.

Keywords

Mathematical physics, algebraic geometry, symplectic geometry, sympletic reduduction, topological recursion

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