Luis J. Boya

The Berry connection and Born–Oppenheimer method

By performing the most general Born–Oppenheimer procedure, the (non‐Abelian) Berry connection for quantum systems in a quantum environment is derived. This method is then applied to the rapid rotation of a particle about a slowly changing axis, as exemplified by the electronic motion of a diatomic molecule. The angular part of the resulting dynamics for the quantum environment is equivalent to that of a monopole.

The geometry of compact Lie groups

Abstract We explain the sense in which the compact forms of simple Lie groups behave as products of odd-dimensional spheres; the results are obtained by extensive use of homotopy and fibre bundle theory. Applications include topological properties, the study of Casimir operators and some results in representation theory.

Symplectic structure of the Aharonov-Anandan geometric phase

Abstract The symplectic structure of the Hilbert space of a quantum system is used to derive a natural expression for the geometric (Aharonov-Anandan) phase present when the system performs a cyclic evolution.

Volumes of compact manifolds

Abstract We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius condition for spheres. For this aim we first describe the manifold with some parameters, set up a metric, which induces a volume element, and perform the integration for the adequate range of the parameters; in most cases our manifolds will be either spheres or (twisted) products of spheres, or quotients of spheres (homogeneous spaces). Our results should be useful in several physical instances, as instanton calculations, propagators in curved spaces, sigma models, geometric scattering in homogeneous manifolds, density matrices for entangled states, etc. Some flag manifolds have also appeared recently as exceptional holonomy manifolds; the volumes of compact Einstein manifolds appear in string theory.

ON F-THEORY QUIVER MODELS AND KAC–MOODY ALGEBRAS

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four-dimensional base space. We focus on the base geometry which consists of intersecting F0 = CP1 × CP1 Hirzebruch complex surfaces arranged as Dynkin graphs classified by three kinds of Kac–Moody (KM) algebras: ordinary, i.e. finite-dimensional, affine and indefinite, in particular hyperbolic. We interpret the equations defining these three classes of generalized Lie algebras as the anomaly cancelation condition of the corresponding N = 1 F-theory quivers in four dimensions. We analyze in some detail hyperbolic geometries obtained from the affine base geometry by adding a node, and we find that it can be used to incorporate fundamental fields to a product of SU-type gauge groups and fields.

On Local F-theory Geometries and Intersecting D7-branes

We discuss local F-theory geometries and their gauge theory dualities in terms of intersecting D7-branes wrapped four-cycles in Type IIB superstring. The manifolds are built as elliptic K3 surface fibrations over intersecting F0 = CP1 × CP1 base geometry according to ADE Dynkin Diagrams. The base is obtained by blowing up the extended ADE hyper-Kahler singularities of eight-dimensional manifolds considered as sigma model target spaces with eight supercharges. The resulting gauge theory of such local F-theory models are given in terms of Type IIB D7-branes wrapped intersecting F0. The four-dimensional N = 1 anomaly cancelation requirement translates into a condition on the associated affine Lie algebras.

Geometry of density matrix states

in chambers,the orbits of the states under the projective group PU(N + 1). The type of states correlates withthe vertices, edges, faces, etc. of the polytope, with the vertices making up a base of orthogonalpure states. The entropy function as a measure of the purity of these states is also easily calculable;we draw and consider some isentropic surfaces. The Casimir invariants acquire then also a moretransparent interpretation.

On Hexagonal Structures in Higher Dimensional Theories

We analyze the geometrical background under which many Lie groups relevant to particle physics are endowed with a (possibly multiple) hexagonal structure. There are several groups appearing, either as special holonomy groups on the compactification process from higher dimensions, or as dynamical string gauge groups; this includes groups like SU(2), SU(3), G2, Spin(7), O(8) as well as E8 and O(32). We emphasize also the relation of these hexagonal structures with the octonion division algebra, as we expect as well eventually some role for octonions in the interpretation of symmetries in High Energy Physics.

Covariant representations in a fibre bundle framework

Canonical and covariant representations of Lie groups of the semidirect product form G = N⊙K with N Abelian, are analyzed in a fibre bundle framework. We exhibit first the relationship between both kinds of representations in such framework. Two complementary methods of selecting irreducible representations from the covariant ones are developed. The first one proceeds by restriction to an invariant subspace and is exemplified in the case of massive integer spin representations of the Poincare group. The second method takes quotients and is particularly useful when we deal with reducible but indecomposable representations. A family of stepped gauge transformations is generated when the method is used to obtain the covariant massless integer helicity representations of the Poincare group; the electromagnetic and gravitational gauge transformations are just the first two cases of such a family.

On the continuity of the boosts for each orbit

The possibility of a continuous choice of the boost for each orbit is studied by making use of some mathematical theorems on fibre bundles and it is shown that this choice is possible only for massive particles.