On the Variational Theory of Yang-Mills-Higgs Energies and the Structure of the Singular Set of $\mathbb{Z}$$_{2}$-Harmonic Spinors

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Parise, Davide 

This dissertation investigates Yang-Mills-Higgs energies and Z2-harmonic spinors, two classes of objects at the interface between analysis and geometry.

In the first part, we present joint work with Alessandro Pigati and Daniel Stern on the variational theory of the self-dual U(1)-Yang-Mills-Higgs functionals on a closed Riemannian manifold (M,g). This natural family of energies associated with sections u: M → L and metric connections ∇ of Hermitian line bundles has long been studied in differential geometry and theoretical physics. We show how its variational theory is related to the one of the (n − 2)-area functional by establishing a Γ-convergence result in the spirit of Modica and Mortola. With this in hand, we study the comparison between the corresponding min-max theories. Therefore, we relate the classical theory for C1-unctionals to the min-max theory introduced by Almgren and Pitts in the setting of geometric measure theory. In particular, we prove that min-max values for the latter always provide a lower bound for the former. En route to proving this comparison, we introduce the gradient flow of the Yang-Mills-Higgs energies and establish a Huisken-type monotonicity result along the flow. We complement this by studying the long-time existence, uniqueness and continuous dependence on the initial data of the flow.

In the second part of the dissertation, we focus on the notion of Z2-harmonic spinors. These objects were introduced in foundational work of Taubes when studying the compactification of moduli spaces of flat PSL(2, C)-connections over 3-manifolds. Their role is to abstract various limiting phenomena and they also appear in other contexts, for instance when dealing with the moduli space of solutions to the Kapustin-Witten equations, the Vafa-Witten equations, and the Seiberg-Witten equations with multiple spinors. In all of these cases, the role played by the zero loci of Z2-harmonic spinors is crucial. Based on the pioneering techniques of Simon in the setting of minimal submanifolds, we obtain structural results on the singular set of Z2-harmonic spinors, subject to the validity of frequency monotonicity (a condition implied by, for instance, an appropriate regularity assumption). More precisely, we prove uniqueness of the blow-ups for every point, excluding an exceptional set of zero 2-dimensional Hausdorff measure, hence answering a question left open in the work of Taubes. From this, we infer 2-rectifiability of the singular set and the branch set. We conclude by analysing the setting of lowest frequency value, in which case we have that, locally, the branch set is a C1,α -submanifold.

Wickramasekera, Neshan
Calculus of Variations, Differential Geometry, Geometric Measure Theory, Partial Differential Equations
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge
EPSRC (2118071)