Essential Self-Adjointness of the Symplectic Dirac Operators by A. Nita B.A., University of California, Irvine, 2007 M.A., University of Colorado, Boulder, 2010 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics 2016 This thesis entitled: Essential Self-Adjointness of the Symplectic Dirac Operators written by A. Nita has been approved for the Department of Mathematics Prof. Alexander Gorokhovsky Prof. Carla Farsi Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Nita, A. (Ph.D., Mathematics) Essential Self-Adjointness of the Symplectic Dirac Operators Thesis directed by Prof. Alexander Gorokhovsky The main problem we consider in this thesis is the essential self-adjointness of the symplectic Dirac operators D and D~ constructed by Katharina Habermann in the mid 1990s. Her construc- tions run parallel to those of the well-known Riemannian Dirac operators, and show that in the symplectic setting many of the same properties hold. For example, the symplectic Dirac operators are also unbounded and symmetric, as in the Riemannian case, with one important difference: the bundle of symplectic spinors is now infinite-dimensional, and in fact a Hilbert bundle. This infinite dimensionality makes the classical proofs of essential self-adjointness fail at a crucial step, 2 n namely in local coordinates the coefficients are now seen to be unbounded operators on L (R ). A new approach is needed, and that is the content of these notes. We use the decomposition of the spinor bundle into countably many finite-dimensional subbundles, the eigenbundles of the harmonic oscillator, along with the simple behavior of D and D~ with respect to this decomposition, to con- struct an inductive argument for their essential self-adjointness. This requires the use of ancillary operators, constructed out of the symplectic Dirac operators, whose behavior with respect to the decomposition is transparent. By an analysis of their kernels we manage to deduce the main result one eigensection at a time. Dedication I would like to dedicate this thesis to my parents, Alexander and Maria Nita, who were angels throughout the entire project, giving me the support and encouragement without which I could not have made it. Their selfless dedication to me throughout this difficult intellectual endeavor will never be forgotten. v Acknowledgements I would like to thank first and foremost my advisor Alexander Gorokhovsky, whose patience and guidance saw this project through to its conclusion through the many hurdles I faced along the way. Sebastian Casailina-Martin also provided much encouragement and opportunities to flesh out much of the material on analysis on manifolds, through talks I gave in his complex geometry seminar. I would also like to thank Carla Farsi for giving me the opportunity to talk about K- theory and algebraic topology, and Jeffrey Fox for several important discussions on the relationship between physics and index theory. There are many others I should thank, particularly Marty Walter and Pedro Berrizbeitia, who took valuable time away from their own work to come to my aid in the final stages of the thesis. Contents Chapter 1 Introduction 1 2 Symplectic Manifolds 1 2.1 Preliminaries on Bilinear Spaces.............................1 2.1.1 Matrix Representations of Bilinear Forms....................1 2.1.2 Orthogonality and Isotropy............................2 2.1.3 Symmetry and Skew-Symmetry..........................4 2.1.4 Reflexivity and Degeneracy............................5 2.1.5 Symmetric Spaces.................................9 2.1.6 Skew-Symmetric Spaces.............................. 13 2.1.7 Morphisms of Bilinear Spaces........................... 18 2.2 Grassmannians....................................... 21 2.2.1 Topological Manifold Structure of Grassmanians................ 21 2.2.2 Smooth Manifold Structure of Grassmanians.................. 27 2.2.3 Homogeneous Space Structure of Grassmannians................ 31 2.3 Symplectic Vector Spaces................................. 37 2.3.1 Complexifications and Complex Structures................... 37 2.3.2 Symplectic Forms.................................. 47 2n 2.3.3 The Symplectic Space (R ;!0).......................... 48 vii 2.3.4 The Symplectic Group Sp(n; R) and the Symplectic Lie Algebra sp(n; R).. 52 2.3.5 The Interaction between Inner Products, Symplectic Forms, and Complex Structures...................................... 66 2.3.6 Sums and Subspaces of Symplectic Vector Spaces................ 96 2.4 Symplectic Manifolds.................................... 107 2.4.1 Symplectic Vector Bundles and Chern Classes.................. 107 2.4.2 Symplectic Manifolds: Definitions and Examples................ 114 2.4.3 Darboux's Theorem................................ 131 2.4.4 The Poisson Bracket................................ 133 2.4.5 Hamiltonian Vector Fields............................. 137 2.4.6 Symplectic Connections.............................. 140 3 Self-Adjoint and Elliptic Operators 162 3.1 Function Spaces and Differential Operators on Manifolds................ 162 3.1.1 Function Spaces in the Euclidean Space Setting................. 162 3.1.2 Function Spaces in the Manifold Setting..................... 180 3.1.3 Differential Operators on Manifolds....................... 188 3.2 Essentially Self-Adjoint Operators............................ 203 3.2.1 The General Theory of Essential Self-Adjointness on a Hilbert Space..... 203 3.2.2 The Position and Momentum Operators..................... 224 3.3 Elliptic Differential Operators............................... 230 4 The Metaplectic Representation 234 4.1 The Weyl Algebra..................................... 234 4.2 The Harmonic Oscillator and the Hermite Functions.................. 240 4.3 The Metaplectic Representation.............................. 248 4.4 Symplectic Clifford Multiplication............................ 255 viii 5 The Symplectic Dirac Operators 261 5.1 The Symplectic Spinor Bundle.............................. 261 5.2 The Spinor Derivative................................... 265 5.3 The Symplectic Dirac Operators and Related Operators................ 268 5.4 Properties and Relations among the Operators..................... 271 5.5 Essential Self-Adjointness of the Symplectic Dirac Operators............. 288 5.6 Local Form of the Symplectic Dirac Operators..................... 309 Bibliography 315 Chapter 1 Introduction We shall give here a proof of the essential self-adjointness of the symplectic Dirac operators D and D~ acting on sections the symplectic spinor bundle Q over a compact symplectic manifold (M; !). This is a preliminary result in the larger quest for a symplectic analog of the well-known Atiyah-Singer index theorem on a Riemannian manifold. To understand the importance of Dirac operators and their self-adjointness in this quest, it is necessary to briefly review the historical development of the classical index theorem, its generalizations, and various proofs. The classical index theorem originated in the famous papers of Atiyah, Singer and Segal of the 1960s and early 1970s, [5], [6], [7], [8], [9], and [10], and gave an equivalence between the analytic index of an elliptic operator P acting on sections of a complex vector bundle E over a Riemannian manifold (M; g) and that manifold's topological index, via the equation index(P ) = fch(P ) · T (M)g[M] The term on the left is the analytic index, defined as index(P ) = dim ker P − dim coker(P ), while the term on the right consists of topological invariants: the Chern character, a ring homomorphism • ch : K(M) ! H (M; Q), taking the symbol of P , realized as an element of the K-theory ring • K(M), into the rational cohomology ring H (M; Q) of M, the Todd class T (M) of M, and the fundamental class [M] of M. The motivation for this theorem was Gel'fand's 1960 observation [36] that the analytical index index(P ) of an elliptic operator is always homotopy invariant, and so should describe a topological invariant. It had also been proven by Borel and Hirzebruch in 2 1958, [18], that the A^-genus of a spin manifold was always an integer, even though this topological quantity was defined in terms of an infinite series. Atiyah and Singer's paper [5] explained this unusual result as the index of the manifold's Dirac operator. The first announcement of the general theorem came in that same 1963 paper [5], though its proof arrived somewhat later, in Palais' 1965 book [86]. According to Atiyah and Singer themselves, [6], this early proof used the computationally taxing and ill suited methods of cobordism and rational cohomology groups, which failed to reveal the basic underlying mechanics of the theorem's simple statement, while also making difficult the theorem's generalizations. The three papers of 1968, [6], [7], [8], replaced cobordism and homology with just K-theory, which was more in line with Grothendieck's outlook on such matters. The years 1968-1973 saw several advances, the most important of which, for us, was the introduction heat equation methods to the proof method of the index theorem. These brought with them some revealing simplifications in certain cases, and reduced the dependency on topological machinery with the introduction of more analytic tools. The first inroads were made by