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Rodrigo Barbosa: Lectures

October 28, 2020
TITLE: Coassociative ALE Fibrations and Spectral Covers of Riemannian Higgs Bundles

ABSTRACT: I will explain a general setup for constructing G_2-orbifolds that are coassociative fibrations by ADE singularities M \to Q. There is a family of smoothings of M whose fibers also admit G_2-structures, and these are parametrized by spectral covers of “Riemannian” Higgs Bundles on the base Q. The G_2-structure on a smoothing M_s is closed if the spectral cover – viewed as a one-form with values in a flat bundle – is closed. I will describe two examples of this setup involving a certain flat 3-manifold Q. Time permitting, I will comment on the co-closed condition and how the setup relates to Kovalev-Lefschetz fibrations and the work of Joyce and Karigiannis.
Slides of Lecture

April 10, 2019
TITLE: Deformations of G2-structures, String Dualities and Flat Higgs Bundles

ABSTRACT: We study M-theory compactifications on (resolutions of) G2-orbifolds given by total spaces of ALE-fibrations over a compact flat Riemannian 3-manifold Q. The flatness condition allows an explicit description of the moduli space of supersymmetric vacua: it is parametrized by flat sections of a bundle of Brieskorn-Grothendieck resolutions over Q. Moreover, when instanton corrections are neglected, we have an explicit description of the moduli space for the IIA dual compactification in terms of flat Higgs bundles on Q. We explain how it suggests a new interpretation of SYZ mirror symmetry, while also providing a description of G2-structures in terms of IIB-branes. The net result is two algebro-geometric descriptions of the moduli space of complexified G2-structures: one as a character variety, and another as a Hilbert scheme of points on a threefold. We show the moduli spaces match in an important example. This is joint work with Tony Pantev.
Slides of Lecture