## TQFT Axioms – 1

The sources here will be Kock’s book, “Frobenius Algebras and 2D Topological Quantum Field Theories.”  I’d like to do a separate post on Michael Atiyah’s 1988 article “Topological Quantum Field Theories” where he set down the axioms (not to be confused with a similar entitled article with the word “introduction” in the title).

How we describe TQFTs will depend on the level of our understanding of the concepts. For example, Kock, by page 55, can define a TQFT as follows:

Definition An n-dimensional topological quantum field theory is a symmetric monoidal functor from $(\mbox{\textbf{nCob}},\sqcup, \emptyset, T)$ to $(\mbox{\textbf{Vect}}_K, \bigotimes, \mathbb{K}, \sigma)$.

That’s pretty short, but it requires a lot of unpacking.  In fact I still need to pack it up in the first place!  So we’ll pass on that definition and look at Kock’s more verbose definition (page 30).

Definition: An n-dimensional topological quantum field theory is a rule $\mathcal{A}$ which to each closed oriented (n-1)-manifold $\Sigma$ associates a vector space $\mathcal{A}(\Sigma)$, and  to each oriented coboridism $M:\Sigma_0\to\Sigma_1$ associates a linear map $\mathcal{A}(M): \mathcal{A}(\Sigma_0)\to \mathcal{A}(\Sigma_1)$.  This rule must satisfy the following five axioms.

(1) Two equivalent cobordisms must have the same image:

$M\cong M' \Rightarrow \mathcal{A}(M) =\mathcal{A} (M')$

(2) The cylinder $\Sigma\times I$, thought of as a cobordism from $\Sigma$ to itself, must be sent to the identity map of $\mathcal{A} ( \Sigma )$.

(3) Given a decomposition $M=M'M''$ then

$\mathcal{A} (M) = \mathcal{A} (M') \mathcal{A} (M'')$

(4) Disjoint union goes to tensor product: if $\Sigma=\Sigma' \sqcup\Sigma''$ then $\mathcal{A} (\Sigma ) = \mathcal{A} (\Sigma ') \bigotimes \mathcal{A} ( \Sigma '')$. This must also hold for cobordisms: if $M:\Sigma_0\to\Sigma_1$ is the disjoint union of $M':\Sigma'_0\to\Sigma'_1$ and $M'':\Sigma''_0\to\Sigma''_1$ then  $\mathcal{A} (M) = \mathcal{A} (M') \bigotimes \mathcal{A} (M'')$

(5) The empty manifold must be sent to the base field $\mathbb{K}$. (It follows that the empty cobordism (which is the cylinder $\Sigma=\varnothing$) is sent to the identity map on $\mathbb{K}$.)

It’ll be quite a while before we talk about how to construct such a rule.  That’s where all the physics lies which is entirely above my head.  There are a range of techniques for constructing such a functor and I have some articles I hope to chat about in the future.

I used the word functor above and I can’t remember if I defined it before, but this is essentially a function between categories.  It takes objects and arrows of one category to objects and arrows of another category in a way that makes sense.  This is common practice in mathematics as we can shuffle problems from one area of math to another.  This is the case with homotopy groups, homology/cohomology groups, and Lie algebras for example.  In each case we can an object (pointed topological spaces, topological spaces, Lie groups) and send them to another structure that we can study.

So a TQFT is a functor from the category of nCob to the category Vect of finite dimensional vector spaces.  This functor obeys certain rules, given above.  Any functor that obeys these rules is technically a TQFT.  The examples that will follow in future posts won’t be physical (or at least I don’t think they are) but frankly the physical ones are highly non-trivial and I don’t want to try tackling quite yet.

Coming up: a discussion of Atiyah’s paper and a couple examples from Kock.