CANONICAL CONNECTIONS by Ken-Hsien Chuang B.S., National Yang-Ming University, 2000 M.S., Brandeis University, 2003 Submitted to the Graduate Faculty of the Kenneth P. Dietrich School of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2014 UNIVERSITY OF PITTSBURGH KENNETH P. DIETRICH SCHOOL OF ARTS AND SCIENCES This dissertation was presented by Ken-Hsien Chuang It was defended on February 25, 2014 and approved by Professor Bogdan Ion, Department of Mathematics, University of Pittsburgh Professor Thomas C. Hales, Department of Mathematics, University of Pittsburgh Professor George Sparling, Department of Mathematics, University of Pittsburgh Professor Robert Krafty, Department of Statistics, Temple University Dissertation Director: Professor Bogdan Ion, Department of Mathematics, University of Pittsburgh ii Copyright c by Ken-Hsien Chuang 2014 iii CANONICAL CONNECTIONS Ken-Hsien Chuang, PhD University of Pittsburgh, 2014 We study the space of canonical connections on a reductive homogeneous space. Through the investigation of lines in the space of connections invariant under par- allelism, we prove that on a compact simple Lie group, bi-invariant canonical con- nections are exactly the bi-invariant connections that are invariant under parallelism. This motivates our definition of a family of canonical connections on Lie groups that generalizes the classical (+), (0), and (−) connections studied by Cartan and Schouten. We find the horizontal lift equation of each connection in this family, as well as compute the square of the corresponding Dirac operator as the element of non-commutative Weil algebra defined by Alekseev and Meinrenken. Keywords: canonical connections, Cartan-Schouten connections, Dirac operators. iv TABLE OF CONTENTS 1.0 INTRODUCTION .............................1 2.0 PRELIMINARIES .............................3 2.1 Principal bundles.............................3 2.1.1 Fundamental vector fields.....................4 2.1.2 Associated vector bundles.....................5 2.2 Connections................................6 2.2.1 Connection forms..........................7 2.2.2 Parallel transport..........................9 2.2.3 Holonomy groups.......................... 11 2.2.4 Covariant derivatives........................ 14 2.3 Affine transformations.......................... 18 2.3.1 Invariance by parallelism...................... 19 2.3.2 Symmetric spaces.......................... 20 2.4 Invariant linear connections on homogeneous spaces.......... 22 2.4.1 Reductive homogeneous spaces.................. 25 2.4.2 Canonical connections....................... 28 3.0 LINES OF K-INVARIANT CONNECTIONS INVARIANT UN- DER PARALLELISM ........................... 30 3.1 Covariant cross-derivation of torsion and curvature tensor fields... 31 = N 3.2 Lines in HomH (m m; m)........................ 37 = 2 3.3 Lines in HomH (Λ m; m).......................... 38 = 2 3.4 Lines in HomH (S m; m).......................... 39 v 3.5 The bracket connection associated to m................. 40 4.0 CARTAN-SCHOUTEN CONNECTIONS .............. 42 4.1 Bi-invariant canonical connections on compact Lie groups...... 44 4.2 Horizontal lift equation......................... 47 4.2.1 Principal bundles.......................... 47 4.2.2 Cartan-Schouten connections for SU2............... 49 4.2.3 Cartan-Schouten connections for connected Lie groups..... 54 4.2.4 Holonomy groups of Cartan-Schouten connections........ 56 5.0 THE DIRAC OPERATOR ........................ 58 5.1 Preliminaries............................... 58 5.1.1 Clifford algebras........................... 58 5.1.2 Spin groups............................. 60 5.1.3 Spin representations........................ 61 5.1.4 Spin structures........................... 61 5.1.5 Dirac operators........................... 62 5.2 Dirac operators on reductive homogeneous spaces........... 62 5.3 Dirac operators associated to Cartan-Schouten connections...... 64 5.4 The Twisted Dirac operator....................... 69 5.4.1 Some identities........................... 70 5.4.2 Main result............................. 76 6.0 INDEX .................................... 81 7.0 INDEX OF NOTATION ......................... 84 BIBLIOGRAPHY ................................ 87 vi I would like to thank everyone who has helped me during my Ph.D. studies. vii 1.0 INTRODUCTION It is known that if M is a manifold with a linear connection that is invariant under its own parallelism (or, equivalently, whose torsion and curvature are parallel), then M is an analytic manifold and the connection is itself analytic. If, in addition, M is connected, simply connected and the connection is complete then Aff(M), the group of affine transformations, is a Lie group that acts transitively on M, endowing it with a structure of homogeneous space. This situation is familiar in the context of affine locally symmetric spaces which are manifolds with a linear, torsion-free connection whose curvature is parallel. If the affine symmetric space is simply connected and the connection is complete then it is an affine symmetric space. Manifolds with linear connections that are invariant under parallelism are there- fore a natural extension of the notion of affine locally symmetric space. Since un- der some conditions (M connected, simply connected, complete connection) such a manifold is a homogeneous space and the connection is invariant, a more amenable context is that of homogeneous spaces K=H and K-invariant linear connections that are invariant under parallelism. The set of K-invariant linear connections on K=H is parametrized by a space of linear maps, according to a theorem of Wang (Theorem 2.83). In this parametrization, the connections associated with Ad(H)-stable comple- ments of Lie(H) in Lie(K) are called canonical connections. Canonical connections are necessarily complete and invariant under parallelism. This class of connections is the subject of this dissertation. Next we will give a brief account of our main results. Let K=H be a homogeneous space. In Chapter 3 we consider the question of whether its subset of connections invariant under parallelism contains lines, and we 1 derive some algebraic necessary and sufficient conditions for this to happen (Theorem 3.23, Theorem 3.24, and Theorem 3.25) and necessary and sufficient conditions for a certain distinguished line (the bracket connections) to consists of connections invariant under parallelism (Theorem 3.28). In Chapter 4 we study in detail the homogeneous space S × S=∆(S × S) for S a connected compact Lie group. In this case all self parallel invariant connections form a line whose elements are called Cartan-Schouten connections. We also record various invariants of these connections: torsion, curvature, holonomy Lie algebra. We also explicitly work out the horizontal lift equation, which is the ODE satisfied by the horizontal lift of a curve on S with respect to such a connection (Theorem 4.22). Dirac operators were introduced into the representation theory of a real reductive group by Parthasarathy [21] with the purpose of constructing discrete series repre- sentations. Kostant [14] introduced a Dirac operator that contains terms of degree 3 (which he called a cubic Dirac operator) associated to a Lie algebra, a quadratic sub- algebra and an orthogonal complement. He used the cubic Dirac operator to obtain an algebraic version of the Borel-Weil construction that does not require the existence of complex structures, a generalization of the Weyl character formula, and to discover multiplets of representations [10]. Such an operator is a geometric Dirac operator on a homogeneous space and in the case of S × S=∆(S × S) it corresponds to a Cartan- Schouten connection. The geometric Dirac operators associated to Cartan-Schouten connections were studied by Slebarski [22] who obtained a formula for their square. The relationship with the cubic Dirac operator was pointed out in [1] and [9]. In Chapter 5 we study the twisted Dirac operator associated to a Cartan-Schouten connection. These are elements of the non-commutative Weil algebra defined in [2]. In Theorem 5.46 we obtain a formula for their square in terms of the Casimir operator on S, similar to the one obtained by Slebarski in the geometric context. 2 2.0 PRELIMINARIES For the motivation and expository purposes we present a succinct overview of the theory of connections in principal bundles and related concepts. Our main reference is Kobayashi and Nomizu [13]. 2.1 PRINCIPAL BUNDLES We fix the following notation that will be used henceforth. Let M be an n-dimensional differentiable manifold, G a Lie group, and g the Lie algebra of G. Let Ad : G ! GL(g) be the adjoint representation of G and ad : g ! End(g) the adjoint represen- tation of g. Let expG : g ! G denote the exponential map of G. Definition 2.1. Let E; B; and F be topological spaces. A continuous surjection π : E ! B is said to be locally trivial with standard fibre F if for every x 2 E, there is an open neighborhood U ⊂ B of π(x) and a homeomorphism φ : π−1U ! U × F , such that π = p1 ◦ φ where p1 : U × F ! U is the natural projection. Definition 2.2. A fibre bundle consists of the data (E; B; π; F ), where E, B, and F are smooth manifolds and π : E ! B is a locally trivial differentiable surjection with standard fibre F . The space B is called the base space of the bundle, E the total space, and F the standard fibre. Definition 2.3. A(differentiable) principal G-bundle (P; G; R) over M consists of a manifold P and a right action R : P × G ! P; (u; a) 7! u · a such that (i) R is a free action, 3 (ii) M is the quotient space of G-orbits in P , and (iii) π : P ! M is locally-trivial with standard fibre G. Notation. (1) A principal fibre bundle will be denoted by P (M; G), or by π : P ! M, or, if the other ingredients are clear from the context, simply by P . For u 2 P , the orbit u · G is denoted by Pu. (2) For x 2 G, denote by l(x): G ! G, l(x)y = xy the left multiplication, and by r(x): G ! G, r(x)y = yx the right multiplication, by x. (3) For each x 2 M, π−1(x) is a closed submanifold of M.