Caracterização geométrica do espaço moduli de conexões ASD do fibrado de Hopf quatérnio
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Data
2017-03-06
Autores
Maroja, Aaron Aragon
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Universidade Federal do Espírito Santo
Resumo
In the early 80’s, C.C. Taubes and K. Uhlenbeck provided the analytical foundations so that the solutions of the Yang-Mills equations, called instantons, would have a geometrical use, yet to be found in the same period. Simon Donaldson has built then a theory based on certain aspects of these solutions over 4-dimensional, oriented, closed, differentiable manifolds. In this context, one can isolate a class of connections, called anti-self-dual, that necessarily sastify the Yang-Mills equation. The collection of all such, modulo a natural equivalence relation, namely gauge equivalence, is called the moduli space M of the bundle and its study has led to astonishing insights into the structure of smooth 4- manifolds. This work is set to study in detail a particular example of Donaldson’s Theory on the Hopf bundle over the 4-dimensional manifold ?? 4 . We arrive at the BPST instantons of such bundle via the Cartan canonical 1-form on ????(2). Once these are in hand, we use the conformal invariance of anti-self-dual equations to write down a 5-parameter family of such connections. By making use of a theorem of Atiyah, Hitchin and Singer, we assert that every element of the moduli space M is uniquely represented by a connection in this family. From this we obtain a concrete realization of M as the open unit ball in R 5 . In particular, M is a 5-dimensional manifold with a natural compactification whose boundary is a copy of the base space ?? 4 .
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Moduli space , Espaço moduli , Hopf bundle , Fibrado de hopf , Yang-Mills theory , Teoria de Yang-Mills