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Item Acesso Aberto Principal bundles on 2-dimensional CW-complexes with disconnected structure group2021 - Oliveira, Andre Gama; Oliveira, Andre GamaGiven any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long-known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, it seems there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts non-trivially on its fundamental group π1G. In this note we deal with this case by obtaining the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that π0G is abelian.Item Acesso Aberto Unramified covers and branes on the Hitchin system2020-11 - Franco, Emilio; Gothen, Peter; Oliveira, Andre Gama; Peon-Nieto, AnaWe study the locus of the moduli space of GL(n,C)-Higgs bundles on a curve givenby those Higgs bundles obtained by pushforward under a connected unramified cover. We equipthese loci with a hyperholomorphic bundle so that they can beviewed as BBB-branes, and weintroduce corresponding BAA-branes which can be describedvia Hecke modifications. Wethen show how these branes are naturally dual via explicit Fourier–Mukai transform (recall that GL(n,C) is Langlands self dual). It is noteworthy that these branes lie over the singularlocus of the Hitchin fibration.As a particular case, our construction describes the behavior under mirror symmetry of thefixed loci for the action of tensorization by a line bundle of ordern. These loci play a key rolein the work of Hausel and Thaddeus on topological mirror symmetry for Higgs moduli spaces.Item Acesso Aberto Complex Lagrangians in a hyperKaehler manifold and the relative Albanese2020 - Biswas, Indranil; Gómez, Tomas; Oliveira, Andre GamaLet M be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold X, and let A→ M be the relative Albanese over M. We prove that A has a natural holomorphic symplectic structure. The projection onto M defines a completely integrable structure on the symplectic manifold A. In particular, the fibers are complex Lagrangians with respect to the symplectic form on A. We also prove analogous results for the relative Picard over M.Item Acesso Aberto SO(p,q)-Higgs bundles and Higher Teichmüller components2019 - Aparicio-Arroyo, Marta; Bradlow, Steven; Collier, Brian; García-Prada, Oscar; Gothen, Peter B.; Oliveira, AndréItem Acesso Aberto Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars2018 - Aparicio-Arroyo, Marta; Bradlow, Steven; Collier, Brian; García-Prada, Oscar; Gothen, Peter; Oliveira, AndréFor semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups SO(p,q) with 2 < p < q. These groups lie outside formerly know classes of groups associated with exotic components.Item Acesso Restrito Deformation of f-tilings versus deformation of isometric foldings2012 - Santos, Altino; Avelino, CatarinaWe present some relations between deformation of spherical isometric foldings and deformation of spherical f-tilings. The natural way to deform f-tilings is based on the Hausdorff metric on compact sets. It is conjectured that any f-tiling is (continuously) deformable in the standard f-tiling τ_s = {(x, y, z) ∈ S2 : z = 0} and it is shown that the deformation of f-tilings does not induce a continuous deformation on its associated isometric foldings.Item Acesso Aberto Representations of surface groups in the projective general linear group2011 - Oliveira, André Gama