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
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