Hyperbolic band theory through Higgs bundles

Higgs Bundles and Non-abelian Hodge Theory

Morse Theory and Hyperkähler Kirwan Surjectivity for Higgs Bundles

This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott’s original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperkähler Kirwan map is surjective for the non-fixed determinant case. CONTENTS

Morse Theory for the Space of Higgs Bundles

Here we prove the necessary analytic results to construct a Morse theory for the YangMills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions (A, φ) converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the HarderNarasimhan-Seshadri...

Moduli of Higgs Bundles

2 Local symplectic, complex and Kähler geometry: a quick review 10 2.1 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Symplectic quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Complex manifolds . . . . . . . . . . . . . . ....

Introduction to Higgs Bundles

H(X,C) = H(X)⊕H(X) = H(X,OX)⊕H(X,ΩX). Therefore, homomorphisms π1(X) → C are the same as an element of H(X,OX), i.e. a holomorphic line bundle of degree 0, and an element of H(X,ΩX), i.e. a holomorphic 1-form. In these notes, we describe an analogous correspondence for the case of represenations into a nonabelian Lie Group G, focusing in particular on the case G = GL(n,C) [5]. Theorems are give...