Spinorial representation of SpinC submanifolds in Rn

Orientador: Rafael de Freitas LeãoTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Dois tópicos de pesquisa bem conhecidos no século XX, na área de geometria diferencial, são a equação de Dirac e imersões mínimais de superfícies. Em 1998, Thomas Friedrich, elucidou a relação entre imersões isométricas de superfícies com uma dada curvatura média e soluções da equação de Dirac. Na literatura, outros autores abordaram o problema da relação entre soluções da equação de Dirac e imersões isométricas para outras variedades Spin em espaços de dimensões maiores. O objetivo desta tese é apresentar uma caracterização spinorial de imersões isométricas de variedades que possuem uma estrutura SpinC de dimensão arbitrária em Rn para algum n, já que essas estruturas são mais naturais no contexto de variedades quase-complexas. Para alcançarmos nosso objetivo, resolvemos esse problema por meio de duas abordagens. Na primeira, mostramos a equivalência entre imersões isométricas de variedades SpinC em Rn e uma solução de uma equação tipo-Killing para spinores do tipo Spin-Clifford, que por serem inversíveis nos permitem uma manipulação mais simples. Na segunda, utilizando ideais minimais na álgebra de Clifford complexa, traduzimos a solução encontrada anteriormente para a linguagem de spinores clássicos, isto é, spinores provenientes da restrição de uma representação irredutível complexa da álgebra de Clifford complexa. Observe que nossa contribuição é mais geral do que as apresentadas na literatura, pois consideramos variedades e fibrados vetoriais SpinC que não são necessariamente SpinAbstract: Two well known research topics in 20th century in differential geometry are the Dirac equation and minimal immersion of surfaces. In 1998, Thomas Friedrich, elucidated the relationship between isometric immersions of surfaces and solutions of the Dirac equation. In literature, some authors have addressed the problem about the relationship between solutions of Dirac equation and isometric immersions to different Spin manifolds in spaces with higher dimensions. This thesis presents a spinorial characterization of isometric immersions of Riemannian SpinC manifolds with arbitrary dimension in Rn for some n. Note that these structures are more natural in the context of almost complex manifolds. To achieve our goal, we solve this problem through two approaches. Initially, we show the equivalence between isometric immersions of Riemannian SpinC manifolds in Rn and a solution of a Killing-like equation for Spin-Clifford spinors, which, because of their invertible property, allows simpler manipulation. In our second approach, we translate the solution found previously, employing minimal ideals in complex Clifford algebra, to classical spinors language, that is, spinors that came from the restriction of a irreducible complex Clifford algebra representation. The description by ideals elucidates the advantage of the description by Spin-Clifford spinors due to its simplicity. It is important to point out that our contribution is more general than those presented in the literature, since we consider SpinC manifolds and vector bundles that are not necessarily SpinDoutoradoMatematicaDoutor em Matemática140098/2016-0CAPESCNP

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements