An Introduction to Gauge Theory and its Applications Publicações Matemáticas An Introduction to Gauge Theory and its Applications Marcos Jardim UNICAMP impa o 25 Colóquio Brasileiro de Matemática Copyright ã 2005 by Marcos Jardim Direitos reservados, 2005 pela Associação Instituto Nacional de Matemática Pura e Aplicada - IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ Impresso no Brasil / Printed in Brazil Capa: Noni Geiger / Sérgio R. Vaz 25o Colóquio Brasileiro de Matemática · A Short Introduction to Numerical Analysis of Stochastic Differential Equations - Luis José Roman · An Introduction to Gauge Theory and its Applications - Marcos Jardim · Aplicações da Análise Combinatória à Mecânica Estatística - Domingos H. U. Marchetti · Dynamics of Infinite-dimensional Groups and Ramsey-type Phenomena - Vladimir Pestov · Elementos de Estatística Computacional usando Plataformas de Software Livre/Gratuito - Alejandro C. 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Pedrosa Distribuição: IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ E-mail: [email protected] - http://www.impa.br ISBN: 85-244-0235-0 “cursoNew” i i 2005/5/17 page i i Preface For almost a century, classical and quantum field theory have occu- pied center stage in theoretical physics, whose main object of study are the so-called gauge fields. Examples are Weyl’s description of classical electromagnetism as a U(1) gauge theory, Yang & Mills’ model of isotopic spin and Weinberg & Salam’s standard model of particle physics. The theory of fiber bundles, connections and characteristic classes was being developed at around the same time, with little relation to the physicists’ work. It was only in the early 70’s that physicists and mathematicians realized that the gauge fields were best described as the curvature of a connection on some principal or vector bundle, and that characteristic classes were topological “charges”, akin to the usual electric charge. Once made, this observation led to a hurricane of activity in math- ematical physics during the 70’s and 80’s. I believe it is no exagger- ation to say that the interplay between pure mathematics and the- oretical physics during those two decades have revolutionized both fields. The goal of these notes is to provide the reader with one aspect of this revolution. We will study anti-self-dual connections on 4- dimensional Riemannian manifolds and some closely related objects in lower dimensions. More precisely, we will discuss the two most fundamental results of the subject, in my opinion: Donaldson’s non- existence theorems in differential topology and the Hitchin-Kobayashi correspondence between stable holomorphic bundles and anti-self- dual connections. Rather than focusing on the (very hard) techniques involved, we will concentrate on the ideas and basic definitions, trying to give a flavor of how these results are proved. While it is probably true that these (and other) results could have been proved without any input from physics, such input provided important motivation and inspiration. It is hard to imagine how one could even dream of these two fundamental results without the physics input. Another important feature of these two Theorems is that they provide true examples of the unity within mathematics. To prove a result in differential topology, Donaldson uses the geometry of con- i i i i “cursoNew” i i 2005/5/17 page i i nections and deep analytical tools together with the more traditional methods of algebraic topology; and it is worth noting that most of the examples that have been studied actually come from algebraic geometry. The Hitchin-Kobayashi correspondence is a bridge from differential geometry to algebraic geometry through the analysis of partial differential equations. These notes are organized as follows. In Chapter 1 we revise the basic notions of vector bundles, connections and their curvature and characteristic classes. The anti-self-duality equations are introduced in chapter 2, where we also explain the construction of the instan- ton moduli space as a Riemannian manifold. This manifold is then used in Chapter 3 to sketch the proof of Donaldson’s non-existence theorems and the existence of fake R4’s. Chapter 4 is dedicated to complex geometry and the second main result, the Hitchin-Kobayashi correspondence. We conclude in Chapter 5 by examining the simplest dimensional reductions of the anti-self-duality equations to dimension 3 (Bogomolny equations), dimension 2 (Hitchin’s equations) and di- mension 1 (Nahm’s equations). We focus on Hitchin’s equations to emphasize how gauge theory provides a bridge between algebraic ge- ometry and the theory of integrable systems. Donaldson’s non-existence theorems links the differential geome- try of vector bundles to the topology of smooth 4-dimensional mani- folds, while the Hitchin-Kobayashi correspondence links the differen- tial geometry of vector bundles to algebraic geometry. Gauge theory has also found equally profound links and applications to other tra- ditional fields of mathematics, like representation theory, symplectic geometry and differential geometry itself, which unfortunately are outside the scope of these notes. Moreover, the infiltration of gauge theory within other mathematical subjects is by no means over; in fact, gauge theory has evolved into a set of tools available for use in a wide variety of problems. It is our hope that these notes will motivate the reader to further explore such links, which place gauge theory as a keystone of 21st century mathematics. i i i i “cursoNew” i i 2005/5/17 page i i Contents 1 Vector bundles with connections 1 1.1 Vector bundles . 1 1.2 Connections and curvature . 3 1.3 Chern classes . 5 1.4 What is Gauge Theory? . 6 2 Yang-Mills equation in dimension 4 7 2.1 The Yang-Mills and anti-self-duality equations . 8 2.2 The moduli space of instantons . 9 2.3 Instantons on Euclidean space . 14 3 Topology of smooth 4-manifolds 18 3.1 Donaldson’s theorems . 18 3.2 Exotic R4’s . 21 4 The connection with algebraic geometry 24 4.1 Stable holomorphic bundles . 24 4.2 The Hitchin-Kobayashi correspondence . 26 5 Hitchin’s equations and integrable systems 29 5.1 Hitchin’s equations . 31 i i i i “cursoNew” i i 2005/5/17 page 1 i i Chapter 1 Vector bundles with connections We start with a brief review of the central objects of study in gauge theory: vector bundles with connections. A more detailed exposition can be found at [14, 16, 17]. 1.1 Vector bundles Let X be a smooth manifold of dimension d. Recall that a rank r complex vector bundle over X is a smooth manifold E together with a smooth map π : E → X satisfying the following conditions: 1. π is surjective; −1 2. for each point x ∈ X, Ex = π (x) is a complex vector space of dimension r; 3. for each point x ∈ X, there is a neighborhood U of x and a −1 r r smooth map φU : π (U) → U ×C such that φU |Ex : Ex → C is an isomorphism. The vector space Ex is called the fiber of E over x, while the map φU is called a local trivialization of E. i i i i “cursoNew” i i 2005/5/17 page 2 i i 2 [CAP. 1: VECTOR BUNDLES WITH CONNECTIONS In other words, a complex vector bundle E → X is a family of complex vector spaces parameterized by X which is locally a product of the parameter space with a fixed complex vector space. Clearly, one can define real vector bundles in the same manner simply by substituting Cr for Rr in the definition. Every complex vector bundle defines a set of transition functions. More precisely, consider the complex vector bundle π : E → X, and −1 r let {Uα} be an open covering of X such that φα : π (Uα) → Uα ×C are local trivializations. If Uα ∩ Uβ 6= ∅, we can consider the maps −1 r r gαβ = φβφα : Uα ∩ Uβ × C → Uα ∩ Uβ × C which are called the transition functions of E with respect to the covering {Uα}. It is easy to see that they must satisfy the following relation: gαβgβγ gγα = 1 , on Uα ∩ Uβ ∩ Uγ . Conversely, given the set of transition functions, one can reconstruct the bundle E. Proposition 1. Let X be a smooth manifold, let {Uα} be an open covering of X, and let ταβ be a collection of smooth functions r ταβ : Uα ∩ Uβ → GL(C ) , satisfying ταβτβγ τγα = 1 , on Uα ∩ Uβ ∩ Uγ . Then there exist a rank r vector bundle E → X with local trivializa- tions −1 r φα : π (Uα) → Uα × C −1 satisfying φβφα (x, v) = (x, ταβ(x)v). The proposition above helps us to form new vector bundles out of two vector bundles E and F over the same base space X. Assume r that E and F are given by transition functions gαβ ∈ GL(C ) and s hαβ ∈ GL(C ), respectively. Then one can define the bundles: • E ⊕ F is the bundle given by transition functions µ ¶ gαβ 0 r+s kαβ = ∈ GL(C ); 0 hαβ i i i i “cursoNew” i i 2005/5/17 page 3 i i [SEC.