Instantones de grupo estructural arbitrario
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2024
Defense date
09/07/2024
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Abstract
En este trabajo de fin de máster, estudiamos un tipo especial de solución a las ecuaciones de Yang-Mills denominado instantón. Los modelos más clásicos de este tipo de solución son los monopolos, que viven en un fibrado principal con variedad base la esfera bidimensional y grupo estructural $U(1)$. Mas adelante, se generalizo este concepto a un fibrado con grupo estructural $\operatorname{SU}(2)$ y la esfera tetradimensional como variedad base, naciendo el concepto de instantón.
La teoría de instantones ha sido generalizada exitosamente en varias direcciones. Por una parte, trabajando con variedades base de holonomía especial, se han considerado variedades base de dimensión diferente de 4 pero dotada de alguna estructura geométrica modelada por grupos excepcionales ($G_2,\operatorname{Spin}(7),\ldots$), de forma que el instantón solución conviva con esta estructura. Veremos un enfoque moderno que nos permite construir este tipo de variedades e instantones con ecuaciones de evolución.
La otra dirección que nos interesará en este trabajo toma fibrados principales con cualquier grupo estructural sobre $\H^n$, y explora la construcción de instantones utilizando las ecuaciones de Nahm.
In this master thesis, we study a special type of solution to the Yang-Mills equations called instanton. The most classical models of this type of solution are monopoles, which live in a principal bundle with the bidimensional sphere as its base manifold and structural group $U(1)$. Later, this concept was generalized to a bundle with structural group $\operatorname{SU}(2)$ and with the 4-sphere as its base manifold, thus creating the concept of instanton. The theory of instantons has been successfully generalized in several directions. On the one hand, working with base varieties of special holonomy, researchers have considered base manifolds of dimension other than 4 but endowed with some geometric structure modeled by exceptional groups ($G_2,\operatorname{Spin}(7),\ldots$), so that the instanton solution coexists with this structure. We will see a modern approach that allows us to construct such varieties and instantons with evolution equations. The other direction that will interest us in this thesis takes principal bundles with any structural group over $\H^n$, and explores the construction of instantons using the Nahm equations.
In this master thesis, we study a special type of solution to the Yang-Mills equations called instanton. The most classical models of this type of solution are monopoles, which live in a principal bundle with the bidimensional sphere as its base manifold and structural group $U(1)$. Later, this concept was generalized to a bundle with structural group $\operatorname{SU}(2)$ and with the 4-sphere as its base manifold, thus creating the concept of instanton. The theory of instantons has been successfully generalized in several directions. On the one hand, working with base varieties of special holonomy, researchers have considered base manifolds of dimension other than 4 but endowed with some geometric structure modeled by exceptional groups ($G_2,\operatorname{Spin}(7),\ldots$), so that the instanton solution coexists with this structure. We will see a modern approach that allows us to construct such varieties and instantons with evolution equations. The other direction that will interest us in this thesis takes principal bundles with any structural group over $\H^n$, and explores the construction of instantons using the Nahm equations.