Relation between nCob and Hilb


Next: Appendix Up: Category nCob in General Previous: Category nCob in General

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