## Relation between nCob and Hilb

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### Relation between nCob and Hilb

Given the category nCob and Hilb, it is possible to create a covariant functor $Z:nCob\rightarrow Hilb$ such that, for any (n-1)-manifold S it assigns a Hilbert space of states Z(S) and, given a cobordism $M:S\rightarrow S_1$ we obtain the corresponding function $Z(M):Z(S)\rightarrow Z(S_1)$.
Z(M) is such that the following conditions are satisfied:
• given $M:S\rightarrow S_1$ and $M^1:S_1\rightarrow S_2$ then
$Z(M^1 o M)= Z(M^1) o Z(M)$
• $Z(1_S)=1_{Z(S)}$ where S=(n-1)-dimensional manifold.
The functor Z is such that it preserves the *-Category structure i.e. $Z(M^*)=Z(M)^*$
J. Baez identified this functor as a representation of a Topological Field Theory (For details see [1])

Cecilia Flori 2006-11-21