Mirror symmetry for Nahm branes

The Dirac-Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We extend this construction to the case of arbitrary rank $n$ and degree $0$, studying the associated connection and curvature. We then generalize to the case of rank $n > 1$ the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli spaces of rank $1$ Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle over the moduli space of stable Higgs bundles of rank $n$ and degree $0$, twisted by the gerbe of liftings of the projective universal bundle. Our hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as $(BBB)$-branes twisted by the above mentioned gerbe. We then use the Fourier-Mukai and Fourier-Mukai-Nahm transforms to describe the corresponding dual branes restricted to the smooth locus of the Hitchin fibration. The dual branes are checked to be $(BAA)$-branes supported on a complex Lagrangian multisection of the Hitchin fibration.Comment: 34 pages. Exposition greatly improve

Peter háberle «versus» isis: tensions between utopia and constitutional reality in the tumultuous «contemporary islamic world» in light of Háberle's legalphilosophical thinking. Quick reflections

Peter HÁBERLE reflexiona en numerosas ocasiones y a lo largo de toda su vida de iusfilósofo constitucionalista sobre la temática jurídica básica del «Mundo islámico». En los últimos años asume incluso valores constitucionales compartidos con Occidente, excluyendo en todo caso los integrismos. Así puede llegar a constituirse un Derecho Constitucional Común Islámico compatible con el Derecho Constitucional Común Occidental. Un decálogo de conclusiones condensa esfuerzos futuros en base a su «Teoría de la Constitución como Ciencia de la Cultura» ampliada en general al mundo islámico progresista y laico.We find a lot of topics in the writings of P. HÁBERLE concerning the «islamic constitutional World». Some islamic values agreed with the West ones, with exclusion of every kind of integrism. Finally the so-called «commum islamic constitutional Law» may be very similar to the homologous systems usual in the West countries. The futur of such islamic systems could be drawn upon the «Constitutional cultur Theory» of Háberle, but only if a large degree of laicism could be globally accepted

Higgs bundles over elliptic curves

In this paper we study $G$-Higgs bundles over an elliptic curve when the structure group $G$ is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable $G$-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable $G$-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of $G$-Higgs bundles and we study the generic and non-generic fibres