Field theories and algebraic topology

Transcription

1 Field theories and algebraic topology Tel Aviv, November 2011 Peter Teichner Max-Planck Institut für Mathematik, Bonn University of California, Berkeley

2 Mathematics as a language for physical theories 1. Relativity theory uses the notion of a Lorentzian manifold of dimension four. 2. Quantum mechanics uses operator theory and Wiener measures on path spaces. 3. Quantum field theory is not yet understood mathematically.

3 1a. Searching for a notion of space The following axioms, formulated by Felix Hausdorff, describe a space by (open) neighborhoods of points: 1. There corresponds to each point x at least one neighborhood U(x). 2. For two neighborhoods of the same point x, there must exist a neighborhood U(x) that is a subset of both. 3. If the point y lies in U(x), there must exist a neighborhood U(y) that is a subset of U(x). 4. For two distinct points, there are two corresponding neighborhoods with no points in common.

4 Felix Hausdorff Professor in Bonn from and The only cluster of excellence in Mathematics currently awarded by the Deutsche Forschungsgemeinschaft is the Hausdorff Center for Mathematics (HCM) in Bonn

5 1b. Definition of a manifold The notion of a Hausdorff space is still used today in topology and many other areas of mathematics. An important special case is the following: Definition 1: [Riemann, Poincaré, Hausdorff, Kneser] An n-dimensional manifold is a Hausdorff space that is locally isomorphic to n-space R (i.e. described locally by n real coordinates).

6 Justifications for the definition Generality: All physical systems arising as configuration spaces or phase spaces are manifolds, as long as the parameters are generic. Examples: The surface of the earth, our ambient 3- space and 4-dimensional space-time are manifolds. Classification: An understanding of all closed manifolds is possible, after fixing the fundamental group.

7 Classification of manifolds: Dimension 2 Closed 2-manifolds are classified by the intersection form on their first homology (i.e. genus + orientation)

8 Dimension 3 All closed 3-manifolds can be obtained from identifying opposite faces of a polyhedron. They were recently classified by Perelman, proving Thurston s geometrization conjecture

9 Dimension 4 Closed 4-manifolds are not fully understood. Amazingly, for a few small fundamental groups, the intersection form does classify, similarly to dimension 2 [Freedman,...]. If one requires a smooth structure, i.e. the notion of differentiable functions, then Gauge theory enters the picture and shows that much less is known: Uncountably many exotic smooth structures on [Freedman, Donaldson,...] R

10 2a. Classical mechanics Say the Riemannian manifold M is the configuration space of a mechanical system. Then a world-line is a path kinetic energy γ :[, ] M with (γ) = γ ( ) The action (γ) is given by subtracting a potential term. The principal of least action says that the minima of S give the classical evolution.

11 2b. Quantum mechanics The state of the quantum system is the wave function Ψ, a unit vector in the Hilbert space L 2 (M). The probability of finding the particle at the point x is given by Ψ( ) The quantum evolution is given by the Feynman integral over all path M from x to y: γ :[, ] (Ψ)(, ) = γ exp( (γ)) γ/

12 Examples without potential: S = E If Δ is the Laplacian on a Riemannian manifold M then we get the (Wick rotated) evolution on L 2 M via =exp( ) If D is the odd Dirac operator on a Riemannian spin manifold then a super evolution on sections of the spinor bundle comes from,θ =exp( + θ )

13 2. An abstract definition of QM Mathematically, the quantum system is thus described by the Hilbert space L 2 (M), together with the unitary 1-parameter group Ut or said differently: Definition 2: A quantum mechanical system is a Euclidean field theory of dimension 1. This is a (symmetric monoidal) functor from the bordism category of Euclidean 0- and 1-manifolds to the category of Hilbert spaces and unitary operators.

14 Why symmetric monoidal structures? In addition to giving a unitary evolution, a functor as above carries the following information: Inner product comes from an interval with 2 points on one end: monoidal structure: Independent particles are described by the tensor product of Hilbert spaces. symmetry operator distinguishes bosons and fermions in the presence of a super Hilbert space. The intervals are then super manifolds.

15 3. In search of a mathematical notion of Quantum field theory Definition 3: ?? [Atiyah, Kontsevich, Segal, refined by Stolz-Teichner] A (d δ)-dimensional quantum field theory is a (fibred) symmetric monoidal functor from a (d δ)-dimensional bordism category to a d-category of topological vector spaces.

16 Quantizations of classical field theories A classical Σ-model is given by a world-sheet Σ and a target M. The classical fields are the smooth maps Γ: Σ M. Again there is a classical action S(Γ) given by a kinetic and a potential term. Classically, only minima of S are relevant, in quantum field theory we should again average over all possible fields. If Σ is (d δ)-dimensional, this should lead to a functorial field theory in this dimension.

17 Field theories over a manifold The classical Σ-model is an example of a field theory over the manifold M. Thus we really have a (contravariant) functor from manifolds to sets, or categories: M QFT(M) If we divide by a homotopy relation, this is reminiscent of a generalized cohomology theory known from algebraic topology. This analogy distinguishes our functorial definition from other mathematical approaches to QFT.

18 A deep analogy The locality of a QFT is expressed by higher categories and leads to long exact Mayer-Vietoris sequences. The degree (or twist) of a cohomology class corresponds to the central charge of the QFT. Classical and quantum field theories can be described in this language and the quantization map corresponds to a push-forward in the cohomology theory. Equivariant cohomology corresponds to gauged QFTs.

19 Main results Theorem: [Stolz-Teichner] (a) 0 1-EFT(M) is isomorphic to the set of closed differential forms on M [with Hohnhold and Kreck]. (b) Every vector bundle on M with Quillen super connection gives a point in 1 1-EFT(M). [with Dumitrescu] Moreover, the space of all functorial EFTs classifies: (d δ) = (0 1): de Rham Cohomology (d δ) = (1 1): K-Theory [with Hohnhold]

20 Outlook Conjecturally, the space of 2 1-EFTs classifies the Hopkins-Miller theory of Topological Modular Forms. Since the homotopy groups of the spectrum TMF are completely understood, this would determine all (deformation classes) of EFTs in this dimension. There are many open problems in higher dimension, in particular whether there are applications to actual physical theories.