Atiyah's definitions of Topological Quantum Field Theory

According to Atiyah, a TQFT is a functor from the category of cobordisms to the category of vector spaces.

How does this definition relate with the physics of quantum mechanics?

What does the category of cobordism in the above definition represent physically?

Solution 1:

A cobordism between two manifolds represents some kind of evolution from one state space to another state space. In quantum mechanics we have an evolution in a continuous time parameter. In order to get a TQFT that describes this, choose

• 0-dim topological manifolds, i.e. points, as objects (or "the point" as the only object)

• 1-dim cobordisms of the points as arrows,

• add a Riemannian structure to the 1-dim cobordisms, which equips them with a length.

Now a TQFT on this category will be a functor that

• associates a (finite or infinite dimensional) vector space $H$ to the point,

• associates a linear operator $U(t): H \to H$ to every (or the) interval of length $t$ such that

• $U(t s) = U(t) U(s)$ (functoriality).

For infinite dimensional vector spaces we would need to assume that they are (complex) Hilbert spaces. And we would in addition have to assume that the linear operators that are associated to cobordisms are unitary operators. Then the TQFT is identical to the Schrödinger picture of quantum mechanics. Rays in the vector space $H$ represent the state of a physical system and the operators $U(t)$ represent the evolution of the system from one state at the time 0 to the time $t$.

Edit: In the physical interpretation of TQFT the vector spaces represent state spaces of the physical systems which is not the same as physical space. A point in a state space represents all information that is necessary to completely describe the physical system at hand.

Solution 2:

John Baez has written various things about this. Briefly, cobordisms should be thought of in terms of time evolution: you have two manifolds which represent space, and a cobordism between them represents time evolution. Of course to be more physically realistic one should put a Lorentzian structure on the cobordism and make the two manifolds spacelike slices, but I guess the point of the adjective "topological" is to ignore these extra details for the sake of mathematical simplicity.

Then the functor to $\text{Vect}$ is supposed to be a simple version of a functor to $\text{Hilb}$ (the category of Hilbert spaces) assigning to a manifold the Hilbert space of states on it, and assigning to a cobordism a linear operator representing time evolution. Again, to be more physically realistic one should demand that the operator be unitary and indeed there is a notion of unitary TQFT (but many TQFTs of interest to mathematicians are not unitary).