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Doctoral Thesis
DOI
10.11606/T.43.2018.tde-09102017-161306
Document
Author
Full name
Javier Ignacio Lorca Espiro
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2017
Supervisor
Committee
Teotonio Sobrinho, Paulo (President)
Silva, Luis Gregorio Godoy de Vasconcellos Dias da
Batista, Eliezer
Brandt, Fernando Tadeu Caldeira
Sá, Eduardo Peres Novais de
Title in Portuguese
Sobre o estado fundamental de teorias de n-gauge abelianas topológicas
Keywords in Portuguese
Cohomologia
Homologia
Mecânica quântica
Teoria de calibre
Topologia algébrica
Abstract in Portuguese
O caso finito de teorias topológicas de 1-gauge, quando nenhuma simetria global está presente, é bastante bem compreendido e classificado. Nos últimos anos, as tentativas de generalizar as teorias de 1-gauge através das chamadas teorias de 2-gauge abriram a porta para novos modelos interessantes e novas fases topológicas, as quais não são descritas pelos esquemas de classificação anteriores. Nesta tese, vamos além da construção de 2-gauge, e consideramos uma classe de modelos que vivem em maiores dimensões. Esses modelos estão inseridos em uma estrutura de complexos de cadeia de grupos abelianos, forçando a generalizar o conceito usual de configurações de gauge. A vantagem de tal abordagem é que, a ordem topológica fica manifestamente explcita. Isto é feito em ter- mos de uma cohomologia com coeficientes em um complexo de cadeia finita. Além disso, mostramos que a degenerescência do estado fundamental suporta um conjunto conve- niente de números quânticos que indexam os estados e que, além, foram completamente caracterizados. Consequentemente, nós também mostramos que muitos dos exemplos abelianos de teorias de 1 -gauge 2-gauge são recuperados como casos especiais desta construção.
Title in English
On the ground state of abelian topological higher gauge theories
Keywords in English
abelian gauge theories
cohomology
ground state degeneracy
higher gauge theories
topology
Abstract in English
The finite case of 1-gauge topological theories, when no global symmetries are present, is fairly well understood and classified. In recent years, attempts to generalize the latter situation through the so called 2-gauge theories have opened the door to interesting new models and new topological phases, not described by the previous schemes of classifica- tion. In this paper we go even beyond the 2-gauge construction by considering a class of models that live in arbitrary higher dimensions. These models are embedded in a structure of chain complexes of abelian groups, forcing to generalize the usual notion of gauge configurations. The advantage of such an approach is that, the topological order is explicitly manifest when the ground state space of these models is described. This is done in terms of a cohomology with coefficients in a finite chain complex. Furthermore, we show that the ground state degeneracy underpins a convenient set of quantum num- bers that label the states and that have been completely characterized. We also show that abelian examples of 1-gauge 2-gauge theories are recovered as special cases of this construction.
 
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Publishing Date
2018-04-03
 
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