Gauge theories: integrable equations and self-duality

Author(s)

Henrique Malavazzi

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Física de São Carlos (IFSC/BT)
• Defense date: 2021-06-24

Examining board members

Luiz Agostinho Ferreira; Jose Francisco Gomes; Marco Aurelio Cattacin Kneipp

• Advisor: Luiz Agostinho Ferreira

Abstract

In this work, we explore two concepts of extreme importance in field theories looking for a relationship with gauge theories: self-duality and integrability. Gauge theories describe three of the four fundamental interactions that govern nature, exploring its structure can provide us a better understanding of the problems that are open in the standard model. We based on the references1–5 to seek generalizations of well-established self-dual sectors of gauge theories: the ´t Hooft-Polyakov monopoles and Instantons. Hence that, we found a generalized Yang-Mills-Higgs theory that has spatial conformal symmetry, enabling the achievement of two distinct solution behaviors: the spherical ansätz, which provides new monopole solutions (spherically symmetrical) and the conformal ansätz with toroidal symmetry. With the spherical ansätz of generalized Yang-Mills-Higgs, we constructed a self-dual sector for the ´t Hooft-Polyakov solutions.9, 10 With the conformal ansätz, we verified that the toroidal symmetry implies in vacuum solutions defined in a 3-sphere space, in addition, we show two different solutions, an abelian and a non-abelian one, with the abelian solution carrying an invariant quantity associated with the helicity of the gauge fields, however, carries an irregularity under gauge transformations. To understand the global aspects of such ansätze, we make use of the integral equations of non-abelian gauge theories6–8 to calculate their dynamic magnetic charges, then, it was possible to verify a quantization condition for the monopole solutions obtained with the spherical ansätz and we also concluded that the conformal ansätz it does not allow solutions of the monopole type. (AU)

• FAPESP's process: 18/21601-6 - Gauge Theories: integrable equations and self-duality
• Grantee: Henrique Malavazzi
• Support Opportunities: Scholarships in Brazil - Master