Boundary Value Problems for the Lorentzian Dirac Operator

On a compact globally hyperbolic Lorentzian spin manifold with smooth space-like Cauchy boundary, the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah–Patodi–Singer boundary conditions are imposed. This chapter explores to what extent these boundary conditions can be replaced by more general ones and how the index then changes. There are some differences to the classical case of the elliptic Dirac operator on a Riemannian manifold with boundary.

Hyperfunctions, the Duistermaat–Heckman Theorem and Loop Groups

This chapter investigates the Duistermaat–Heckman theorem using the theory of hyperfunctions. In applications involving Hamiltonian torus actions on infinite-dimensional manifolds, the more general theory seems to be necessary in order to accommodate the existence of the infinite-order differential operators which arise from the isotropy representations on the tangent spaces to fixed points. The chapter quickly reviews the theory of hyperfunctions and their Fourier transforms. It then applies this theory to construct a hyperfunction analogue of the Duistermaat–Heckman distribution. The main goal will be to study the Duistermaat–Heckman distribution of the loop space of SU(2) but it will also characterize the singular locus of the moment map for the Hamiltonian action of T×S 1 on the loop space of G. The main goal of this chapter is to present a Duistermaat–Heckman hyperfunction arising from a Hamiltonian group action on an infinite-dimensional manifold.

The Deformed Hermitian–Yang–Mills Equation in Geometry and Physics

This chapter provides an introduction to the mathematics and physics of the deformed Hermitian–Yang–Mills equation, a fully non-linear geometric PDE on Kähler manifolds, which plays an important role in mirror symmetry. The chapter discusses the physical origin of the equation, and some recent progress towards its solution. In addition, in dimension 3, it proves a new Chern number inequality and discusses the relationship with algebraic stability conditions.

Quaternionic Geometry in Dimension 8

This chapter describes the 8-dimensional Wolf spaces as cohomogeneity one SU(3)-manifolds, and discover perturbations of the quaternion-kähler metric on the simply connected 8-manifold G2/SO(4) that carry a closed fundamental 4-form but are not Einstein.

Algebras of Quantum Monodromy Data and Character Varieties

This chapter examines the Poisson structure of the representation variety of the fundamental groupoid of a Riemann surface with punctures and cusps, and the associated decorated character variety.

Mirror Symmetry with Branes by Equivariant Verlinde Formulas

This chapter finds an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL2 Higgs moduli space on a Riemann surface. On one side of the agreement, components of the Lagrangian brane of U(1,1) Higgs bundles, whose mirror was proposed by Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL2 Higgs moduli space, are present. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin’s proposal.

A Hitchin Connection for a Large Class of Families of Kähler Structures

This chapter presents a Hitchin connection constructed in a setting which significantly generalizes the setting covered by the first author, which, in turn, was a generalization of the moduli space covered in the original work on the Hitchin connection. In fact, this construction provides a Hitchin connection which is a partial connection on the space of all compatible complex structures on an arbitrary but fixed prequantizable symplectic manifold which satisfies a certain Fano-type condition. The subspace of the tangent space to the space of compatible complex structures on which the constructed Hitchin connection is defined is of finite codimension if the symplectic manifold is compact. It also proves uniqueness of the Hitchin connection under a further assumption. A number of examples show that this Hitchin connection is defined in a neighbourhood of the natural families of complex structures compatible with the given symplectic form which these spaces admit.

Vertex Algebras and 4-Manifold Invariants

This chapter proposes a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d N=(0,2) theories that arise from compactification of fivebranes. Such a formulation gives a new interpretation of some known statements about Seiberg–Witten invariants, such as the basic class condition, and gives a prediction for structural properties of the multi-monopole invariants and their non-abelian generalizations.

Torsion of Elliptic Curves and Unlikely Intersections

This chapter studies effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.

Quantization of the Quantum Hitchin System and the Real Geometric Langlands Correspondence

This chapter proposes a natural quantization condition for the Hitchin system and relate this to the generating function for the variety of opers within the Hitchin space of local systems. Links with the geometric Langlands programme are investigated.