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Integrable systems on the moduli space of flat connections Adam Artymowicz Master of Science Department of Mathematics and Statistics McGill University Montreal,Quebec 2018-12-13 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Masters of Science. c Adam Artymowicz ACKNOWLEDGEMENTS My sincere thanks go out to my advisor, professor Jacques Hurtubise, for his support and direction throughout the writing of this thesis, and to professor Niky Kamran for his faith and encouragement during my Masters. Lastly, I am grateful to all of my friends at Oˆ Claf for supporting me throughout my time at McGill. ii ABSTRACT In this thesis we treat the moduli space of flat SU(n) connections over a surface, as well as the various related spaces that appear when the surface is allowed to have boundary. The properties of these spaces and the relationships between them are used to construct an integrable system on the moduli space over a closed surface which generalizes the toric moment map of Jeffrey and Weitsman [10]. iii ABREG´ E´ Dans cette th`ese, on s’int´eresse `al’espace de modules des connexions plates sur une surface avec groupe de structure SU(n), ainsi qu’aux espaces associ´es qui aparaissent quand on permet `ala surface d’avoir une fronti`ere. Les propri´et´es de ces espaces et la relation entre eux nous sert a construire un syst`eme int´egrable sur l’espace de modules sur une surface ferm´ee qui g´en´eralise l’application moment torique de Jeffrey et Weitsman [10]. iv TABLE OF CONTENTS ACKNOWLEDGEMENTS ............................ ii ABSTRACT .................................... iii ABREG´ E......................................´ iv 1 Introduction.................................. 1 2 Symplecticgeometry ............................. 4 2.1 LagrangianandHamiltonianformalism . 4 2.2 Symplecticandhamiltoniangroupactions . 8 2.3 Coadjointorbits............................ 10 2.4 Symplecticreduction ......................... 15 2.5 Integrablesystems........................... 17 2.6 TheGelfand-Cetlinintegrablesystem . 18 3 Themodulispaceofflatconnections . 25 3.1 Gauge-theoreticorigins . 25 3.2 Flatconnections............................ 29 3.3 Themodulispaceoveraclosedsurface . 31 3.4 Themodulispaceoverasurfacewithboundary . 35 3.5 Extendedandframedmoduli. 37 3.6 Gluing ................................. 45 3.7 TheJeffrey-Weitsmansystem . 46 3.8 The integrable system on symplectic leaves of S .......... 50 References...................................... 52 v CHAPTER 1 Introduction There are many reasons to study moduli spaces of bundles over a surface. The first, coming from algebraic geometry, is perhaps the most direct: it is to understand the parameter space of all holomorphic bundles of a given rank on a surface – one might describe this as the theory of higher-rank analogues of the Picard variety of the surface. Another motivation comes from the theory, again classical, of systems of holomorphic ODEs on a Riemann surface. On a disc such systems admit unique solutions, but these solutions become multi-valued when the domain is not simply connected, and this multi-valued behaviour manifests itself as a representation of the fundamental group of the surface (the monodromy representation) [5]. Surfaces with boundary enter naturally when the ODEs are permitted to have singularities – if these singularities are sufficiently well-behaved, the classical Riemann-Hilbert correspondence equates such a system of ODEs with its monodromy representation, up to a suitable notion of isomorphism. Relaxing the assumption that these objects be holomorphic, we enter the world of principal bundles with connection, for which we have a similar correspondence with fundamental group representations. Here we make substantial contact with physics, as principal bundles with connection are the fundamental objects of study in gauge theory [4]. 1 The examples given above by no means constitute a full description of the zoo that exists of moduli spaces of bundles with some extra structure. They are only intended to indicate the richness of the theory, and the moral that when we study one space, we are indirectly studying them all at once. Each carries a different flavour, but a recurring theme is that of correspondences which arise when one views the same space from different viewpoints. Such bridges between algebra, geometry, and analysis are part of what make these moduli spaces interesting objects of study. The subject of this thesis is the moduli space of flat connections on a surface with structure group SU(n), which is is a smooth symplectic manifold on an open dense set. The main result is the construction of a densely-defined integrable Hamiltonian system on this space, which generalizes the toric integrable system on the SU(2) moduli space due to Jeffrey and Weitsman [10] [9]. The Jeffrey-Weitsman system on SU(2) has allowed for a computation of certain invariants of the space, in particular its symplectic volume and the dimension of its geometric quantization. Both were calculated using the image of the toric moment map, which is a polytope: the first by computing the volume of the polytope with respect to the pushforward of the Liouville measure, and the second by counting the points in the polytope belonging to a particular lattice (the Bohr-Sommerfeld points). The motivation behind con- structing the integrable system in this thesis is to eventually carry out analogous calculations for the SU(n) moduli space. In the first chapter we introduce some of the notions used in the thesis. There is nothing original in these sections, save for some aspects of the presentation of the Gelfand-Cetlin integrable system in section 2.6. Specifically, we introduce a slight 2 generalization to the usual system which will be needed in the following sections. The second chapter is devoted to describing several versions of the moduli space of flat connections on a surface with boundary. The extended moduli spaces described in section 3.5 are due to Lisa Jeffrey [8]. The framed spaces of section 3.5 are also well-known, but to the author’s knowledge the presentation of the local calculations on these spaces, and their descriptions via representations of relative fundemantal groups is novel. Finally, in sections 3.6, 3.7, and 3.8 the technology presented in the preceding sections is used to construct the SU(n) integrable system. 3 CHAPTER 2 Symplectic geometry In this chapter we will recall a few of the standard notions and theorems from sym- plectic geometry which will be used throughout this thesis. The aim of the chapter is not to act as an introduction to the subject; instead the goal is to present the point of view where homogeneous symplectic manifolds play a central role. We begin by motivating their study from a physical point of view. 2.1 Lagrangian and Hamiltonian formalism Symplectic geometry has its roots in the study of classical mechanics by the likes of Lagrange, Hamilton, and Poisson. The classical-mechanical point of view of symplec- tic geometry will be indispensable in what follows, so the basic tenets are outlined in this section. The references for this section are [1] and [15]. Classical mechanics deals with the motion of a system whose configuration is given as a point on an n-dimensional manifold X, the configuration space. For instance, a system of N noninteracting particles in 3-dimensional space has configuration space R3N , a pendulum has configuration space the circle S1, a double pendulum S1 S1, × and so on. To solve a classical mechanical system is to solve the equations that govern the motion of the system in its configuration space. 4 It is a basic fact of classical mechanics that the equations of motion are second order ordinary differential equations in the coordinates x1,...,xn of configuration space X1 . The usual trick from ODE theory is to turn a system of n coupled second order equations into 2n coupled first-order equations by introducing auxiliary dxi variables y1,...,yn, with the extra relations dt = yi. Geometrically, this amounts to rewriting our equation as a first-order ODE on the tangent bundle TX – the variable yi is just the tangent coordinate corresponding to xi. A first order ODE is a precisely a vector field v, and the flow of this vector field gives the dynamics of our system. With this point of view, one may understand a classical mechanical system as a manifold X of configurations, together with a particular vector field v on TX. The solution to the system is the one-parameter group of diffeomorphisms generated by v. Mathematics alone can only take us this far – to give this description any physical meaning, we must appeal to some experimentally verified physical principles to tell us what the vector field on TX is. There is no unique way to formulate such a ‘fundamental principle’ or set of principles, and different but (more or less) equivalent ways of phrasing the same principle give rise to different but (more or less) equivalent perspectives or ‘formalisms’ in mechanics. One such principle is the principle of least action, which states that there is a function on TX (possibly depending on time), which for a point particle is the difference L 1 Consider the prototypical example F = mx¨ 5 between the kinetic and potential energies of the particle, such that for any interval of time [t0,t1] the trajectories of the system in TX minimize the action integral t1 S := (x, x,t˙ )dt, L Zt0 where the minimization is taken over all paths with fixed beginning and end points (x, x˙)(t0) and (x, x˙)(t1). The minimizer for this variational problem is given by the d ∂ ∂ Euler-Lagrange equation L = L , which, if is convex on fibres, can be solved dt ∂x˙ ∂x˙ L to give the vector field on TX . There is a geometric interpretation of the least action principle. A trajectory of the system between times t and t can be thought of in terms of its graph γ :[t ,t ] 0 1 0 1 → TX [t ,t ].

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