Solé-Farré, Enric;
(2025)
Conical singularities in special holonomy and gauge theory.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
This thesis examines conical singularities in the context of special holonomy and gauge theory, with a focus on both analytical and variational aspects. In the first chapter, we study instantons on metric cones and establish a new relation between the instanton deformation operator and the Bourguignon stability operator on the corresponding link. This framework is used to study instantons with isolated conical singularities, yielding an analytical construction of their moduli spaces. As a result, we give an explicit formula for their virtual dimension. In the second chapter, we investigate generalisations of Hitchin’s functionals, whose critical points correspond to nearly Kähler and nearly parallel G2-structures. We study the gradient flow of these functionals and perform a spectral decomposition of their Hessians relative to natural indefinite inner products. This study leads to the definition of the Hitchin index, a Morse-like invariant that provides a lower bound for the Einstein co-index. We investigate the connection of this index with the deformation theory of G2 and Spin(7)-conifolds. In the third chapter, we investigate nearly Kähler manifolds under a cohomogeneity one symmetry assumption. This enables us to study and bound the cohomogeneity one contributions to the Hitchin index by reducing the PDE eigenvalue problem to an ODE eigenvalue problem. We focus our analysis on the inhomogeneous nearly Kähler structure on S3 × S3 constructed by Foscolo and Haskins, and obtain non-trivial lower bounds for both the Hitchin and Einstein indices of the manifold, thereby addressing an open question posed by Karigiannis and Lotay.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Conical singularities in special holonomy and gauge theory |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2025. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10212567 |
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