Fibraciones de Cardy
This work is focused on the study of families of open-closed topological fieldtheories parameterized by a manifold with multiplication and their relationshipswith twisted vector bundles. Open-closed field theories were axiomatized by G. Moore and G. Segal in [51]. The study of families of such theor...
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| Formato: | Tesis doctoral publishedVersion |
| Lenguaje: | Inglés |
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Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales
2015
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| Acceso en línea: | https://hdl.handle.net/20.500.12110/tesis_n5831_Amoreo |
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| Sumario: | This work is focused on the study of families of open-closed topological fieldtheories parameterized by a manifold with multiplication and their relationshipswith twisted vector bundles. Open-closed field theories were axiomatized by G. Moore and G. Segal in [51]. The study of families of such theories led us to the definition of Calabi-Yau and Cardy fibrations; these are fibred categories (in fact stacks) over the base manifoldwith multiplication which generalize the definition of Moore and Segal in thesense that when the base manifold is a one-point space, we recover the originaldefinition. A careful study of their properties (that is, a detailed proof showingthat these categories are additive, pseudo-abelian and enjoy an action of the categoryof locally free modules) led us to a relationship between these families ofopen-closed field theories and 2-vector bundles (as defined by Baas, Dundas and Rognes in [10]), thus providing an affirmative answer to a suggestion given by G. Segal in [57]. Moreover, we also found a relationship between the transitionhomomorphisms of Cardy fibrations and Higgs bundles. The last part deals with global objects (that is, objects of the category over thewhole base space). A functorial link between the category of modules over thespectral cover and the category of modules over the tangent sheaf of the manifoldis obtained. We also show that Azumaya algebras, in the sense of A. Grothendieck [29], appear naturally in the study of Cardy fibrations: given an object a of thefibred category defined over the whole base space, the space of arrows a!a can bedefined as the pushout of a certain Azumaya algebra along the spectral projection S ! M. On the other hand, as was proved by M. Karoubi in [35], twisted vectorbundles are closely related to these Azumaya algebras. This facts led us to acharacterization of global objects in the fibred category in terms of twisted vectorbundles over the spectral cover of the base manifold. Keywords: Open-closed field theory, twisted vector bundle, manifold withmultiplication, spectral cover, 2-vector bundle. |
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