Field theories and algebraic topology

Peter Teichner

Max-Planck Institut für Mathematik, Bonn University of California, Berkeley

Tel Aviv, November 2011

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.

Mathematics as a language for physical theories

1a. Searching for a notion of space

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).

The following axioms, formulated by Felix Hausdorff, describe a space by (open) neighborhoods of points:

4. For two distinct points, there are two corresponding neighborhoods with no points in common.

Felix Hausdorff 1868 - 1942

Professor in Bonn from 1910 -13 and 1921- 35

The only cluster of excellence in Mathematics currently awarded by the

Deutsche Forschungsgemeinschaft is the

Hausdorff Center for Mathematics (HCM)

in Bonn

1b. Definition of a manifold

Definition 1: 1854-1926 [Riemann, Poincaré, Hausdorff, Kneser]

An n-dimensional manifold is a Hausdorff space that is locally isomorphic to n-space (i.e. described locally by n real coordinates).

RR

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:

Justifications for the definition

• Generality: All physical systems arising as configuration spaces or phase spaces are manifolds, as long as the parameters are generic.

• Classification: An understanding of all closed manifolds is possible, after fixing the fundamental group.

• Examples: The surface of the earth, our ambient 3-space and 4-dimensional space-time are manifolds.

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

Classification of manifolds: Dimension 2

All closed 3-manifolds can be obtained from identifying opposite faces of a polyhedron.

Dimension 3

They were recently classified by Perelman, proving Thurston’s geometrization conjecture

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�

Dimension 4

2a. Classical mechanicsSay the Riemannian manifold M is the configuration space of a mechanical system. Then a world-line is a path M with kinetic energy

γ : [�, X] →

)(γ) =

! X

�||γ!(W)||�HW

The action is given by subtracting a potential term. The principal of least action says that the minima of S give the classical evolution.

7(γ)

2b. Quantum mechanics

The quantum evolution is given by the Feynman integral over all path M from x to y:γ : [�, X] →

9X(Ψ)(\, ]) =

!

γ

exp(M7(γ))(γ/>X

The state of the quantum system is the wave function , a unit vector in the Hilbert space L2(M). The probability of finding the particle at the point x is given by ||Ψ(\)||�

Ψ

Examples without potential: S = E

• If Δ is the Laplacian on a Riemannian manifold M then we get the (Wick rotated) evolution on L2M via

• If D is the odd Dirac operator on a Riemannian spin manifold then a super evolution on sections of the spinor bundle comes from

9X = exp(−X∆)

9X,θ = exp(−X(� + θ()

Mathematically, the quantum system is thus described by the Hilbert space L2(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.

2. An abstract definition of QM

Why symmetric monoidal structures ?

• Inner product comes from an interval with 2 points on one end: ∪

In addition to giving a unitary evolution, a functor as above carries the following information:

• symmetry operator distinguishes bosons and fermions in the presence of a super Hilbert space. The intervals are then super manifolds.

• monoidal structure: Independent particles are described by the tensor product of Hilbert spaces.

3. In search of a mathematical notion of Quantum field theory

Definition 3: 1988 - 20?? [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.

Quantizations of classical field theoriesA 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.

Field theories over a manifoldThe 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.

• Classical and quantum field theories can be described in this language and the quantization map corresponds to a push-forward in the cohomology theory.

A deep analogy

• The locality of a QFT is expressed by higher categories and leads to long exact Mayer-Vietoris sequences.

• Equivariant cohomology corresponds to gauged QFTs.

• The degree (or twist) of a cohomology class corresponds to the central charge of the QFT.

Main resultsTheorem: [Stolz-Teichner]

(a) 0|1-EFT(M) is isomorphic to the set of closed differential forms on M [with Hohnhold and Kreck].

Moreover, the space of all functorial EFTs classifies: (d|δ) = (0|1): de Rham Cohomology

(d|δ) = (1|1): K-Theory [with Hohnhold]

(b) Every vector bundle on M with Quillen super connection gives a point in 1|1-EFT(M). [with Dumitrescu]

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.