Functorial Field Theories and Factorization Algebras September 3, 2014 Contents 1 Introduction 3 2 Classical field theories 4 2.1 Classical mechanics . .5 2.1.1 Digression: Riemannian manifolds and connections . .8 2.2 Electromagnetism . 13 2.2.1 Digression: The de Rham complex . 13 2.2.2 The homogeneous Maxwell equations . 16 2.2.3 Digression: Hodge theory . 17 2.2.4 The inhomogeneous Maxwell equations . 20 2.2.5 The Maxwell equations as Euler-Lagrange equations . 21 2.2.6 Digression: curvature . 23 2.2.7 Digression: Connections and their curvature on complex vector bundles 26 2.2.8 Digression: Classification of vector bundles . 33 2.3 Gravity (Einstein's theory of general relativity) . 35 2.3.1 Digression: Geometric interpretation of scalar and Ricci curvature . 37 2.3.2 Einstein field equations as Euler Lagrange equations . 38 2.4 Yang-Mills Theory . 43 2.4.1 Digression: Principal G-bundles . 43 2.4.2 Digression: Connections on principal G-bundles . 45 2.5 Chern-Simons Theory . 47 2.5.1 Digression: classifying spaces for principal G-bundles . 47 2.5.2 Digression: characteristic classes . 48 2.5.3 Digression: the Chern-Weil construction . 51 2.5.4 Digression: Chern-Simons form and Chern-Simons invariant . 54 2.5.5 The classical Chern-Simons field theory . 55 2.6 Non-linear σ-models . 55 1 CONTENTS 2 2.7 Summarizing classical field theories . 55 3 Functorial field theories 57 3.1 From classical mechanics to quantum mechanics: the Feynman-Kac formula . 57 3.2 Heuristic quantization of the non-linear σ-model . 60 3.3 Definition of a topological quantum field theory . 62 3.3.1 Digression: dualizable objects in symmetric monoidal categories . 71 3.4 Riemannian field theories . 73 4 Factorization algebras 75 4.1 Strict factorization algebras . 75 4.1.1 Digression: sheaves . 75 4.1.2 Definition of strict factorization algebras . 79 4.1.3 Examples of strict factorization algebras . 84 4.2 Factorization algebras . 86 4.2.1 Digression: Homotopy colimits . 87 4.2.2 Definition of factorization algebras . 87 4.2.3 A factorization algebra build from an algebra . 87 4.3 Free field theories . 87 4.3.1 Digression: Symplectic manifolds . 88 4.3.2 The density line bundle . 88 4.3.3 Definition of free BV-field theories . 89 4.3.4 The factorization algebra of classical observables of a free BV-theory 91 4.3.5 The factorization algebra of quantum observables of a free BV-theory 91 5 Solutions to selected homework problems 93 1 INTRODUCTION 3 1 Introduction Here is a cartoon map of the area of mathematics this course will be exploring. factorization algebras vertex operator algebras quantization ? classical quantization quantum ? ? field theories field theories ? quantization functorial field theories As the picture shows, the elephant in the middle is quantum field theory, a theory that so far has evaded a mathematical rigorous definition/construction, a fact that is indicated pic- torially by squiggly lines. There are many competing mathematical approaches to quantum field theories, among them • Functorial field theories, pioneered by Segal [Se2], Atiyah [At] and Kontsevich in the 1980's; • Vertex operator algebras, first introduced by Borcherds in 1986 [Bo]; • Factorization algebras, defined by Beilinson and Drinfeld in 2004 [BD], is a more geometric and much broader notion than vertex operator algebras. Factorization algebras and functorial field theories should be thought of as mathematical approximations to what a physicist might mean by a \quantum field theory"; in particular, 2 CLASSICAL FIELD THEORIES 4 such a field theory should give rise to a factorization algebra and a functorial field theory as indicated by the unlabeled squiggly arrows. Moreover, there are mathematically well-defined objects called classical field theories which are expected to lead to quantum field theories by a process called quantization, a process that is even less understood than the formal definition of quantum field theories. One important benchmark for the quality of a mathematical theory that hopes to be a contender for a mathematical definition of quantum field theory is the success it has in constructing objects in that theory that should be considered quantizations of classical field theories. This has only been partially done for factorization algebras and functorial field theories which is indicated by the gray arrows originating from classical field theories. In this course we start out discussing various classical field theories. Then we will motivate the definition of functorial field theories by the heuristic discussion of a particular classical field theory known as the non-linear σ-model. Next will be the discussion of functorial field theories that will be based on the survey paper [ST]. Then we will be talking about factorization algebras, drawing from [Gw] and [CG]. The final topic will be a construction producing a functorial field theory from a factorization algebra. This is based on ongoing joint work with Bill Dwyer and Peter Teichner. 2 Classical field theories Definition 2.1. A classical field theory is given by the following data: • a manifold M, typically infinite dimensional, which is called the space of fields; • a smooth map S : M! R called the action. This is a first, very preliminary definition of what a classical field theory is. As we will see in the examples we will discuss, the space space of fields M is typically a space of smooth maps on a manifold M or, more generally, the space of smooth sections of a bundle over M. Moreover, for φ 2 M, the action S(φ) is given as an integral over M. This implies that the condition for φ to be a critical point of S amounts to a differential equation for φ, which is called the Euler-Lagrange equation for the action functional S. In classical physics, we often describe physical quantities in terms of functions which satisfy differential equations, for example • the path of a classical point particle satisfies Newton's equation F = ma, where F is the net force acting on the particle, m is the mass of the particle, and a is its acceleration. • the electromagnetic field satisfies Maxwell's equation. • the gravitational field satisfies Einstein's equation. 2 CLASSICAL FIELD THEORIES 5 To go from these differential equations to an action function S : M! R, we need to do a bit of reverse engineering: in each of these cases, we need to construct a suitable action function such that the associated Euler-Lagrange equation characterizing the critical points of S is the differential equation we start with. 2.1 Classical mechanics For simplicity we restrict to a particular simple classical mechanical system, namely a point particle moving in R3, or more generally in a Riemannian manifold X, under the influence of a force field F . Now, physicists don't typically call classical mechanics a “field theory", but from a mathematical perspective it seems ok to do that, the same way that a scalar can be thought of as a 1 × 1-matrix. Mathematically, the movement of a point particle in the manifold X is described by a smooth map φ:[a; b] ! X with φ(t) 2 X giving the position of the particle at time t 2 [a; b]. According to Newton's law it satisfies the differential equation F = ma, where F is the net force acting on the particle, m is the mass of the particle, and a is its acceleration. Written more carefully, Newton's law is the second order differential equation F (φ(t)) = mφ¨(t): (2.2) Here both sides are vectors in the tangent space Tφ(t)X. It is clear how to define the acceler- ¨ n ¨ ation vector φ(t) 2 Tφ(t)X if φ is a path in R . The definition of the acceleration vector φ(t) for a path in a general Riemannian manifold X requires a discussion of connections, which we will do in the next subsection. That's fine, but what is the space of fields M and the action functional S : M! R? The first question is easy to answer: M := C1([a; b];X) := fφ:[a; b] ! X j φ is smoothg: The action functional S : M! R emerges in a reformulation of Newton's law called the Lagrangian formulation of classical mechanics. This requires the assumption that the force vector field F can be written in the form F = − grad V for some function V : X ! R that physicists call the potential. Here grad V is the gradient vector field of V (see Definition 2.12 for the definition of the gradient vector field of a function on a general Riemannian manifold). The Riemannian metric and the potential allow us to define the function m L: TX −! by L(x; v) := jjvjj2 − V (x) R 2 2 that physicists call the Lagrangian. Here x 2 X, v 2 TxX, and jjvjj = hv; vix is the norm- squared of the tangent vector v as measured in the Riemannian metric on X (see Definition m 2 2.10 for the definition of a (pseudo) Riemannian metric). Physicists call 2 jjvjj the kinetic 2 CLASSICAL FIELD THEORIES 6 energy and V (x) the potential energy. The Lagrangian in turn can be used to define an action functional S : M! R on the space M of all smooth maps φ:[a; b] ! X by setting Z b S(φ) := L(φ(t); φ_(t))dt: (2.3) a According to the main result of the Lagrangian formulation of classical mechanics, solu- tions of Newton's differential equation (2.2) correspond essentially to critical points of the action functional S.