Why do we define TQFTs as functors to vector spaces instead of Hilbert spaces?

Question

Let

• $\mathrm{Cob}_n$ be the category with objects closed oriented $n-1$-manifolds and morphisms being cobordisms identified upto boundary preserving diffeomorphism
• $\mathrm{Vect}_\mathbb C$ be the category of complex vector spaces and linear maps
• $\mathrm{Hilb}_\mathbb C$ be the category of complex Hilbert spaces and continuous linear maps
• $\mathrm{Hilb}_\mathbb C^\mathrm{unit}$ be the category of complex Hilbert spaces and unitary maps

Then my questions are:

• Why do we define a TQFT to be a functor $Z\colon\mathrm{Cob}_n\to\mathrm{Vect}_\mathbb C$? Isn't it closer to quantum mechanics to define $Z\colon\mathrm{Cob}_n\to\mathrm{Hilb}_\mathbb C^\mathrm{unit}$?
• Even if we want to consider nonunitary theories, then shouldn't we consider $Z\colon\to\mathrm{Cob}_n\to\mathrm{Hilb}_\mathbb C$?

What are the reasons we don't choose the other two categories?

Answer 1

Why do we define a TQFT to be a functor $Z:\mathrm{Cob}_n→\mathrm{Vect}_\mathbb C$? Isn't it closer to quantum mechanics to define $Z:\mathrm{Cob}_n→\mathrm{Hilb}^\mathrm{unit}_\mathbb C$?

Quantum mechanics is not a TQFT, but merely a functorial field theory.

TQFTs have fully dualizable images, which in the case of Hilbert spaces amounts to being finite-dimensional.

One way to encode time evolution in quantum mechanics is via a one-parameter semigroup of (say) unitary operators on a Hilbert space (possibly infinite-dimensional).

In terms of functorial field theories, this is encoded by a functor from the category of 1-dimensional manifolds equipped with a metric (i.e., length) to the category of Hilbert spaces and unitary maps. We send the point to the Hilbert space under consideration, and to an interval of length T we assign the value of the one-parameter semigroup at T. This assignment is functorial precisely because of the semigroup condition. Such a field theory is clearly nontopological, since its values depend on lengths of intervals.

Even if we want to consider nonunitary theories, then shouldn't we consider $Z:\mathrm{Cob}_n→\mathrm{Hilb}_\mathbb C$?

If nonunitary field theories are your goal, then certainly Hilbert spaces and bounded linear maps is a legitimate target category.