Parabolic Higgs Bundles for Real Reductive Lie Groups

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter describes the objects involved in this correspondence, taking some time to motivate them by recalling the definitions of G-Higgs bundles without parabolic structure and of parabolic vector bundles. Finally, it explains the relevant polystability condition and the correspondence between local systems and Higgs bundles.

Spectral Curves for the Triple Reduced Product of Coadjoint Orbits for SU(3)

This chapter gives an identification of the triple reduced product of three coadjoint orbits in SU(3) with a space of Hitchin pairs over a genus zero curve with three punctures, where the residues of the Higgs field at the punctures are constrained to lie in fixed coadjoint orbits. Using spectral curves for the corresponding Hitchin system, the chapter identifies the moment map for a Hamiltonian circle action on the reduced product. Finally, the chapter makes use of results from Adams, Harnad and Hurtubise to find Darboux coordinates and a differential equation for the Hamiltonian.

Holonomic Poisson Manifolds and Deformations of Elliptic Algebras

This chapter introduces a natural non-degeneracy condition for Poisson structures, called holonomicity, which is closely related to the notion of a log symplectic form. Holonomic Poisson manifolds are privileged by the fact that their deformation spaces are as finite-dimensional as one could ever hope: the corresponding derived deformation complex is a perverse sheaf. The chapter develops some basic structural features of these manifolds, highlighting the role played by the divergence of Hamiltonian vector fields. As an application, it establishes the deformation invariance of certain families of Poisson manifolds defined by Feigin and Odesskii, along with the ‘elliptic algebras’ that quantize them.

Involutions of Rank 2 Higgs Bundle Moduli Spaces

This chapter considers the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over ℂ, and studies involutions defined by tensoring the vector bundle with an element α‎ of order 2 in the Jacobian of the curve, combined with multiplication of the Higgs field by ±1. It describes the fixed points of these involutions in terms of the Prym variety of the covering of X defined by α‎, and gives an interpretation in terms of the moduli space of representations of the fundamental group.

Various Generalizations and Deformations of PSL(2,ℝ) Surface Group Representations and their Higgs Bundles

The goal of this chapter is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus g into PSL(2, R) and their associated Higgs bundles generalize to the higher-rank groups PSL(n, R), PSp(2n, R), SO0(2, n), SO0(n,n+1) and PU(n, n). For the SO0(n,n+1)-character variety, it parameterises n(2g−2) new connected components as the total spaces of vector bundles over appropriate symmetric powers of the surface, and shows how these components deform in the character variety. This generalizes results of Hitchin for PSL(2, R).

Threefold Extremal Contractions of Type (IIA)

Let (X, C) be a germ of a threefold X with terminal singularities along an irreducible reduced complete curve C with a contraction f:(X,C)→(Z,o) such that C=f−1(o)red and −K X is f-ample. Assume that (X, C) contains a point of type (IIA). This chapter continues the study of such germs containing a point of type (IIA), started in our previous paper.

SL(∞,ℝ), Higgs Bundles and Quantization

Nigel Hitchin recently proposed a theory of SL(∞, R)-Higgs bundles which should parametrize a Hitchin component of representations of surface groups into SL(∞, R). This chapter discusses some properties and propose a formal approximation of SL(∞, R) representations by SL(n, R) representations in the large N limit, where n goes to infinity.

Brauer Group of Moduli of Higgs Bundles and Connections

Given a compact Riemann surface X and a semi-simple affine algebraic group G defined over C, there are moduli spaces of Higgs bundles and of connections associated to (X, G). The chapter computes the Brauer group of the smooth locus of these varieties.

Restrictions of Heterotic G2 Structures and Instanton Connections

This chapter revisits recent results regarding the geometry and moduli of solutions of the heterotic string on manifolds Y with a G 2 structure. In particular, such heterotic G 2 systems can be rephrased in terms of a differential Ď acting on a complex Ωˇ∗(Y,Q), where Ωˇ=T∗Y⊕End(TY)⊕End(V), and Ď is an appropriate projection of an exterior covariant derivative D which satisfies an instanton condition. The infinitesimal moduli are further parametrized by the first cohomology HDˇ1(Y,Q). The chapter proceeds to restrict this system to manifolds X with an SU(3) structure corresponding to supersymmetric compactifications to four-dimensional Minkowski space, often referred to as Strominger–Hull solutions. In doing so, the chapter derives a new result: the Strominger–Hull system is equivalent to a particular holomorphic Yang–Mills covariant derivative on Q|X=T∗X⊕End(TX)⊕End(V).

Classification of Boundary Lefschetz Fibrations over the Disc

This chapter shows that a 4-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to S1×S3#nCP¯2,#mCP2#nCP¯2 or #m(S2×S2). Given the relation between boundary Lefschetz fibrations and stable generalized complex structures, the chapter concludes that the 4-manifolds S1×S3#nCP¯2,#(2m+1)CP2#nCP¯2 and #(2m+1)S2×S2 admit stable generalized complex structures whose type change locus has a single component and are the only 4-manifolds whose stable structure arises from boundary Lefschetz fibrations over the disc.