# What if I change field in a Topological Quantum Field Theory?

Of course I'm talking about the algebraic notion of field.

In a few words, if a TQFT consists of a functor $Z\colon Cob(n)\to \mathbf{Vec}_k$, I'm wondering if there are sensible relations among different choices of $k$... Some example coming to my mind:

1. Fix a real valued TQFT. Can I turn the functor sending objects to $Z(M)\otimes_\mathbb R\mathbb C$ (the "complexification" of the real vector space associated to $M$) into a complex valued TQFT?

2. What if I consider, in general, a $k$-valued TQFT and a $K$-valued TQFT (with $K$ an extension of $k$)? What if $K$ is finite, infinite, algebraic, Galois or not, positive charateristic or not, etc etc.?

(Feel free to retag the question).

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The answer to your first question is trivial: you have a functor Cob(n)-> Vec_k and you compose it with the base extension functor Vec_k -> Vec_K. I don't think it is so trivial: a TQFT must have some special properties the simple composition $F=(-\otimes_k K)\circ Z_k\colon Cob(n)\to Vec_k\to Vec_K$ does not have; for example what's happen if i want to compute $F$ on the disjoint union of two manifolds? –  tetrapharmakon May 21 '11 at 7:22