Higgs bundles, Lagrangians and mirror symmetry

• Main Author: Branco, Lucas Castello
• Other Authors: Kirwan, Frances ; Hitchin, Nigel
• Published: University of Oxford 2017
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.736045

Let Σ be a compact Riemann surface of genus g ≥ 2. This thesis is dedicated to the study of certain loci of the Higgs bundle moduli space. After recalling basic facts in the first chapter about G-Higgs bundles for a reductive group G, we begin the first part of the work, which deals with Higgs bundles for the real forms G₀ = SU* (2m), SO* (4m), and Sp(m, m) of G = SL(2m, C), SO(4m, C) and Sp(4m, C), respectively. The second part of the thesis deals with the Gaiotto Lagrangian. Motivated by mirror symmetry, we give a detailed description of the fibres of the G-Hitchin fibration containing generic G₀-Higgs bundles, for the real groups G₀ = SU* (2m), SO* (4m) and Sp(m, m). The spectral curves associated to these fibres are examples of ribbons, i.e., non-reduced projective C-schemes of dimension one, whose reduced scheme are non-singular. Our description of these fibres is done in two different ways, each giving different and interesting insights about the fibre in question. One of the formulations is given in term of objects on the reduced curve, while the other in terms of the non-reduced spectral curve. A link is also provided between the two approaches. We use this description to give a proposal for the support of the dual BBB-brane inside the moduli space M(^LG) of Higgs bundles for the Langlands dual group ^LG of G. In the second part of the thesis we discuss the Gaiotto Lagrangian, which is a Lagrangian subvariety of the moduli spaces of G-Higgs bundles, where G is a reductive group over C. This Lagrangian is obtained from a symplectic representation of G and we discuss some of its general properties. In Chapter 7 we focus our attention to the Gaiotto Lagrangian for the standard representation of the symplectic group. This is an irreducible component of the nilpotent cone for the symplectic Hitchin fibration. We describe this component by using the usual Morse function on the Higgs bundle moduli space, which is the norm squared of the Higgs field restricted to the Lagrangian in question. Lastly, we discuss natural questions and applications of the ideas developed in this thesis. In particular, we say a few words about the hyperholomorphic bundle, how to generalize the Gaiotto Lagrangian to vector bundles which admit many sections and give an analogue of the Gaiotto Lagrangian for the orthogonal group.