On the geometry of the energy operator in quantum mechanics

We analyze the different ways to define the energy operator in geometric theories of quantum mechanics. In some formulations the operator contains the scalar curvature as a multiplicative term. We show that such term can be canceled or added with an arbitrary constant factor, both in the mainstream Geometric Quantization and in the Covariant Quantum Mechanics, developed by Jadczyk and Modugno with several contributions from many authors.Comment: 18 pages; paper in honour of the 70th birthday of Luigi Mangiarotti and Marco Modugn

The geometry of real reducible polarizations in quantum mechanics

The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the class of Liouville symplectic manifolds. Our main application of this modified geometric quantization scheme is to Quantum Mechanics on Riemannian manifolds. With this method we obtain an energy operator without the scalar curvature term that appears in the standard formulation, thus agreeing with the usual expression found in the Physics literature.Comment: 29 page

Fourier-Mukai transforms for coherent systems on elliptic curves

We determine all the Fourier-Mukai transforms for coherent systems consisting of a vector bundle over an elliptic curve and a subspace of its global sections, showing that these transforms are indexed by the positive integers. We prove that the natural stability condition for coherent systems, which depends on a parameter, is preserved by these transforms for small and large values of the parameter. By means of the Fourier-Mukai transforms we prove that certain moduli spaces of coherent systems corresponding to small and large values of the parameter are isomorphic. Using these results we draw some conclusions about the possible birational type of the moduli spaces. We prove that for a given degree $d$ of the vector bundle and a given dimension of the subspace of its global sections there are at most $d$ different possible birational types for the moduli spaces.Comment: LaTeX2e, 21 pages, some proofs simplified, typos corrected. Final version to appear in Journal of the London Mathematical Societ

The meaning of the Sky of Salamanca

After presenting the main characteristics of the Sky of Salamanca, we analyse whether what is represented in it is motivated by astrological or astronomical considerations. We will see that the astrological explanation based on the system of planetary houses described in Claudius Ptolemy's Tetrabiblos does not correspond to the planetary configuration we see in the salmantine vault. Finally, we will conclude that the astronomical interpretation of the Sky of Salamanca is justified by the representation of the Universe known in the 15th century according to the Almagest, by the placement of the stars in the constellations according to that work and by the dating of the painted planetary configuration, which circumstantial evidence places in August 1475.Comment: 11 pages, 8 figure

Moduli Spaces of Semistable Sheaves on Singular Genus One Curves

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle $E_N$ of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves $\mathcal{O}(-1)$ supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank $r$ is isomorphic to the $r$-th symmetric product of the rational curve with one node.Comment: 26 pages, 4 figures. Added the structure of the biggest component of the moduli space of sheaves of degree 0 on a cycle of projective lines. Final version; to appear en IMRS (International Mathematics Research Notices 2009

The Grothendieck and Picard groups of finite rank torsion free {sl}(2)-modules

[EN] The classification problem for simple sl(2)-modules leads in a natural way to the study of the category of finite rank torsion free sl(2)-modules and its subcategory of rational sl(2)-modules. We prove that the rationalization functor induces an identification between theisomorphism classes of simple modules of these categories. This raises the question of what is the precise relationship between other invariants associated with them. We give a complete solution to this problem for the Grothendieck and Picard groups, obtaining along the way several new results regarding these categories that are interesting in their own right.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCLE