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Symplectic geometry. --- Symplectic groups. --- Domains of holomorphy. --- Géométrie symplectique. --- Groupes symplectiques. --- Domaines d'holomorphie. --- Géométrie symplectique --- Groupes symplectiques --- Domaines d'holomorphie --- Symplectic geometry --- Symplectic groups --- Domains of holomorphy --- Holomorphy domains --- Analytic continuation --- Functions of several complex variables --- Groups, Symplectic --- Linear algebraic groups --- Geometry, Differential

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"We study the moduli space of trace-free irreducible rank 2 connections over a curve of genus 2 and the forgetful map towards the moduli space of underlying vector bundles (including unstable bundles), for which we compute a natural Lagrangian rational section. As a particularity of the genus 2 case, connections as above are invariant under the hyperelliptic involution: they descend as rank 2 logarithmic connections over the Riemann sphere. We establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical approaches by Narasimhan-Ramanan, Tyurin and Bertram. This allows us to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical (16, 6)-configuration of the Kummer surface. We also recover a Poincarape family due to Bolognesi on a degree 2 cover of the Narasimhan-Ramanan moduli space. We explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by van Geemen-Previato. We explicitly describe the isomonodromic foliation in the moduli space of vector bundles with sl2-connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles"--

Vector bundles. --- Higgs bosons. --- Fibrés vectoriels. --- Higgs, Bosons de. --- Moduli theory. --- Differential equations, Parabolic. --- Vector bundles --- Moduli theory --- Differential equations, Parabolic --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Fiber spaces (Mathematics)

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This book offers a review of the vibrant areas of geometric representation theory and gauge theory, which are characterized by a merging of traditional techniques in representation theory with the use of powerful tools from algebraic geometry, and with strong inputs from physics. The notes are based on lectures delivered at the CIME school "Geometric Representation Theory and Gauge Theory" held in Cetraro, Italy, in June 2018. They comprise three contributions, due to Alexander Braverman and Michael Finkelberg, Andrei Negut, and Alexei Oblomkov, respectively. Braverman and Finkelberg’s notes review the mathematical theory of the Coulomb branch of 3D N=4 quantum gauge theories. The purpose of Negut’s notes is to study moduli spaces of sheaves on a surface, as well as Hecke correspondences between them. Oblomkov's notes concern matrix factorizations and knot homology. This book will appeal to both mathematicians and theoretical physicists and will be a source of inspiration for PhD students and researchers.

Geometry, Algebraic. --- Algebra, Homological. --- Moduli theory. --- Gauge fields (Physics) --- Particles (Nuclear physics) --- Elementary particles (Physics) --- High energy physics --- Nuclear particles --- Nucleons --- Nuclear physics --- Fields, Gauge (Physics) --- Gage fields (Physics) --- Gauge theories (Physics) --- Field theory (Physics) --- Group theory --- Symmetry (Physics) --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Homological algebra --- Algebra, Abstract --- Homology theory --- Algebraic geometry --- Geometry --- Algebraic geometry. --- Physics. --- Category theory (Mathematics). --- Homological algebra. --- Elementary particles (Physics). --- Quantum field theory. --- Algebraic Geometry. --- Mathematical Methods in Physics. --- Category Theory, Homological Algebra. --- Elementary Particles, Quantum Field Theory. --- Relativistic quantum field theory --- Quantum theory --- Relativity (Physics) --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics