The geometry, topology and physics of moduli spaces of Higgs bundles: ICM Satelite...
| Grant number: | 15/06696-2 |
| Support Opportunities: | Scholarships abroad - Research Internship - Post-doctor |
| Start date: | July 20, 2015 |
| End date: | July 19, 2016 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Geometry and Topology |
| Principal Investigator: | Marcos Benevenuto Jardim |
| Grantee: | Emilio Franco Gómez |
| Supervisor: | Richard Paul Winsley Thomas |
| Host Institution: | Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil |
| Institution abroad: | Imperial College London, England |
| Associated to the scholarship: | 12/16356-6 - Higgs bundles over elliptic curves, BP.PD |
Abstract Branes are certain kind of submanifolds of hyperkahler manifolds, that have great importance in Mathematical Physics, more concretely in String Theory. Mirror Symmetry is a largely studied duality that conjectures the equivalence of the String Theories II-A and II-B, which have different mathematical formulations. The behaviour of branes under Mirror Symmetry has physical implications according with [KW].The moduli space of Higgs bundles is a hyperkahler manifold where the Mirror Symmetry can be understood as the composition of a Fourier-Mukai transform and a hyperkahler rotation.In the last two years, the study of branes inside the moduli space of Higgs bundles has lead to a description of a large number of them in terms of fixed point subvarieties of certain involutions. The moduli space of Higgs bundles over an elliptic curve is the first non-trivial case of this kind of moduli spaces. The description of this manifold was achieved during the phD and subsequent work of the applicant, including the study (and explicit description) of branes inside this moduli space. The existence of an explicit description of the moduli space of Higgs bundles is specific of the elliptic curve case. In no other (non-trivial) situation has been achieved a description with such a level of expliciteness.Our project consists in the following goals: (1) The explicit description of the moduli space of Higgs bundles over an elliptic curve could lead to a precise understanding of the Mirror Symmetry transformation in this case. (2) Once we have described the transformation associated to Mirror Symmetry, we would study what occurs with the branes (recall that those are certain subvarieties) under such transformation. (3) Check if one obtains what was conjectured by Theoretical Physics. (4) Try to generalize this study to other situations, such as certain moduli stacks studied by [Gr].---- [KP] A. Kapustin & E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program. [Gr] M. Groechenig, Hilbert schemes as moduli of Higgs bundles and local systems. (AU) | |
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