Gauge theory in dimensions two and four

Author(s)

Marcelo Gonçalves De Martino

• Document type: Master's Dissertation
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2011-02-12

Examining board members

Marcos Benevenuto Jardim; Henrique Nogueira de Sá Earp; Elizabeth Terezinha Gasparim

• Advisor: Marcos Benevenuto Jardim

Abstract

In this work it is developed the basic knowledge required to deal with gauge theory in low dimension and it is shown some applications of this theory. Regarding the basic knowledge, apart from discussing some aspects of Hodge theory over compact manifolds, it is also covered, with a certain deal of details, the concepts of vector bundles and connections, paying close attention to the local computations regarding connections and curvature. As for the applications of the theory, we start, in dimension four, by treating the Yang-Mills equation over 4-manifolds and it is showed a solution to the anti-self-dual Yang-Mills equation, solution that is known in the literature as the 't Hooft ansatz. At last, it is given a proof, following the paper [DONALDSON, 1983], of an important theorem due to M. S. Narasimhan and C. S. Seshadri that relates the algebro-geometric notion of stability to the differential-geometric notion of existence of unitary connection whose curvature satisfies a certain condition, on vector bundles over Riemann surfaces (AU)

• FAPESP's process: 09/02621-7 - Instantons with symmetries
• Grantee: Marcelo Gonçalves de Martino
• Support Opportunities: Scholarships in Brazil - Master