A topological quantum field theory is a functor $Z:Cob(n) \to Hilb$ mapping from the $n$-dimensional cobordisms to $Hilb$, the category of Hilbert spaces with morphisms being bounded linear operators. It's a simple theorem that every such TQFT always takes values of finite-dimensional Hilbert spaces. This set of notes (prop. 2.6) says that it is possible to extend that by restricting to a subcategory of $Cob(n)$ or allowing "volume dependence". Can anyone point me to a reference to something like that? Is there a good way to create a similar functor from some subcategory $cob(n)$, or even from $hTop$ which can take infinite-dimensional values? Is there a way to make that functor full?

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    $\begingroup$ Re: volume-dependence, one reference is this paper of Runkel-Szegedy. They show in particular that Yang-Mills theory in dimension 2 is a volume-dependent field theory and that it attaches an infinite-dimensional Hilbert space to $S^1$. $\endgroup$ Oct 10, 2021 at 13:07
  • $\begingroup$ Thanks, that's really helpful :) $\endgroup$ Oct 11, 2021 at 5:04


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