CHAPTER 5

(2+1)-TQFTs

In this chapter we formalize the notion of a TQFT and summarize various

examples. Our axioms are minor modifications of K. Walker’s [Wal1], which are

consistent only for (2+1)-TQFTs with trivial Frobenius-Schur indicators. The sub-

tle point of Frobenius-Schur indicators significantly complicates the axiomatization.

Axiomatic formulation of TQFTs as tensor functors goes back to M. Atiyah, G. Se-

gal, G. Moore and N. Seiberg, V. Turaev, and others.

A TQFT is a quantum field theory (QFT) whose partition functions are topo-

logically invariant. Consequently, a TQFT has a constant Hamiltonian H, which

can be normalized to H ≡ 0. Systems with constant Hamiltonians can be obtained

by restricting any Hamiltonian to its ground states, though most such theories are

either trivial or not TQFTs in our sense. In physics jargon, we integrate out higher

energy degrees of freedom. Given an initial state of a topological system |ψ0 ,

by Schr¨ odinger’s equation for H ≡ 0, the wave function |ψt will be constant on

each connected component of the evolution. But there are still choices of constants

on different connected components. For an n-particle system on the plane

R2,

connected components of n-particle worldlines returning setwise to their initial po-

sitions are braids. If the ground state is degenerate, i.e., the ground state manifold

(vector space) has dim 1, then the constants are matrices rather than numbers.

Therefore time evolution of TQFTs is given by braid group representations.

The principles of locality and unitarity are of paramount importance in for-

mulating a physical quantum theory. Locality in its most naive form follows from

special relativity: information cannot be transmitted faster than the speed of light

c, hence a point event at point x cannot affect events at other points y within

time t if the distance from x to y exceeds ct. This principle is encoded in TQFTs

by axioms arranging that the partition function Z(X) for a spacetime manifold

X can be computed from pieces of X, i.e., that we can evaluate Z(X) from a

decomposition of X into building blocks Xi such as simplices or handles if the

partition functions Z(Xi) are known and the boundaries of Xi are properly deco-

rated. It also proves fruitful to consider the theory beyond the space and spacetime

dimensions. In (2+1)-TQFTs, we may define mathematical structures for 1- and

4-dimensional manifolds, thereby tracing the framing anomaly of 3-manifold in-

variants to the anomaly of Virasoro algebras in dimension 1 and the signatures of

bounding 4-manifolds. Nothing prevents a mathematician from going even further,

defining theories for all dimensions. But substantial complications arise even for

Chern-Simons theories.

Roughly, a (2+1)-dimensional TQFT (V, Z) consists of two compatible func-

tors: a modular functor V and a partition functor Z. The modular functor V

associates a vector space V (Y ) to any compact oriented surface Y with some ex-

tra structures, takes disjoint unions to tensor products and orientation reversals

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http://dx.doi.org/10.1090/cbms/112/05