mathematical physics - about the Atiyah-Segal axioms on topological quantum field theory

Trying to go through the page on Topological quantum field theory - The original Atiyah-Segal axioms - "Let $\Lambda$ be a commutative ring with 1, Atiyah originally proposed the axioms of a topological quantum field theory (TQFT) in dimension d defined over a ground ring $\Lambda$ as following":

$(1):$ A finitely generated $\Lambda$-module $Z(\Sigma)$ associated to each oriented closed smooth d-dimensional manifold $\Sigma$ (corresponding to the homotopy axiom).

$(2):$ An element $$Z(M) \in Z(\partial M)$$ associated to each oriented smooth (d+1)-dimensional manifold (with boundary) $M$ (corresponding to an additive axiom).

Pardon my question is really stupid, if not just naive:

Questions:

$\bullet (i)$ How do I see this axiom $(2)$: $Z(M) \in Z(\partial M)$ is correct? Instead of $Z(M) \ni Z(\partial M)$? It seems $M$ is one higher dimension than its boundary $\partial M$, so why not more intuitively $Z(M) \ni Z(\partial M)$? Or is that a misleading typo in Wiki, instead we have $$Z(\Sigma) \in Z(\partial M)$$ with $M=M^{d+1}$ being one higher dimensional than $\Sigma=\Sigma^{d}$?

$\bullet (ii)$ How do I physically intuitively digest (1) as a homotopy axiom and (2) as an additive axiom?

ps. I suppose we shall view $Z(\Sigma)$ as a TQFT partition function on the manifold $\Sigma$.

Answer

EDIT #3: My other answer gives a more detailed and structured account (I hope).

(I would leave this as a comment, but I don't have enough reputation so…)

You should check out Atiyah's paper itself. He makes attempts to explain at least some of these things. Unfortunately, I need to get going at the moment (but I'll come back and edit this with a more complete answer, unless someone else wants to in the meantime), but I can say a few things:

(1) If you check out Atiyah's paper, then you will find that he writes $Z(M)\in Z(\partial M)$ in multiple different places, so it's a safe bet to assume it is not a typo. ;) I'll do my best to explain the details when I get back.

(2) A nice expository article that explains somewhat the physical interpretation of the homotopy axiom and the additive axiom can be found here (in particular $\S$2), by mathematician John Baez. Again, I'll say more myself when I get a chance later.

(3) $Z(\Sigma)$ is interpreted as the space of states of a system. The overall idea is that to each geometric object, i.e. a manifold $\Sigma$, one associates with it an algebraic objects, i.e. a vector space $Z(\Sigma)$ (or, eventually, one would want a Hilbert space since this is about quantum physics). And for each "process" taking one manifold $\Sigma_1$ to another manifold $\Sigma_2$ (interpreted as a manifold $M$ with $\partial M=\Sigma_1\cup\Sigma_2$), one gets a "process" taking the state space $Z(\Sigma_1)$ of $\Sigma_1$ to the state space $Z(\Sigma_2)$ of $\Sigma_2$, i.e. a (bounded) linear maps between the vector spaces.

Another great resource is this book and this other article by Atiyah. Hope this helps, at least until I can give a better answer.

---

EDIT: Here is what the expression $Z(M)\in Z(\partial M)$ means: Let $M$ be a manifold so that $\partial M=\Sigma_1\cup\Sigma_2$, as above. Let's say $M$ is $(d+1)$-dimensional. Then, as I was alluding to above, the idea of the $Z$ you're asking about is that it is a functor from a geometric category to an algebraic one. The geometric category has as objects $d$-dimensional closed manifolds (these are the $\Sigma_i$'s), and its morphisms are given by cobordisms between closed $d$-manifolds, i.e. the morphisms are $(d+1)$-dimensional manifolds whose boundary is made up of a disjoint union of closed $d$-dimensional manifolds ($M$ is the cobordism for us here). The algebraic category in this case has as objects finite-dimensinoal vector spaces, and the morphisms are (bounded) linear maps between vector spaces.

So, a functor between two categories is a map that sends objects to objects and morphisms to morphisms. In this case, the functor $Z$ sends a closed $d$-dimensional manifold $\Sigma$ to a vector space $Z(\Sigma)$ and it sends a cobordism $M$, as above, between two of the objects $\Sigma_1$ and $\Sigma_2$ (the ones making up its boundary) to a linear map $Z(M):Z(\Sigma_1)\rightarrow Z(\Sigma_2)$. That is the context, now let us try to understand the statement $Z(M)\in Z(\partial M)$.

The key to understanding this point is that the functor $Z$ is what Atiyah calls multiplicative. What this means is that $Z$ sends disjoint unions to tensor products (this is just like in quantum mechanics when you are dealing with two systems: the states of the product system are not simply products of states in each system, but they are given by tensor products, and the Clebsch-Gordon coefficients come in, etc.). In other words

$$Z(X_1\cup X_2)=Z(X_1)\otimes Z(X_2).$$

So, let's look at what this means for $M$. Since $\partial M=\Sigma_1\cup\Sigma_2$, we have that

$$Z(\partial M)=Z(\Sigma_1)\otimes Z(\Sigma_2).$$

But a standard result in algebra yields:

$$Z(\partial M)=Z(\Sigma_1)\otimes Z(\Sigma_2)\cong \text{Hom}(Z(\Sigma_1),Z(\Sigma_2)).$$

In other words, $Z(\partial M)$ can be thought of as the collection of all linear maps from $Z(\Sigma_1)$ to $Z(\Sigma_2)$. As mentioned above, since $M$ is a cobordism, a morphism in the geometric category, $Z(M)$ is a morphism in the algebraic category, i.e. $Z(M)$ is a linear map $Z(M):Z(\Sigma_1)\rightarrow Z(\Sigma)_2$. So, what have we found, $Z(M)$ is an element of the collection of all linear maps between those two vector spaces, i.e. $Z(M)\in Z(\partial M)$.

---

EDIT #2: I just wanted to add that I'm being (intentionally) vague about the orientations above. Technically speaking (if I'm remembering Atiyah's notation correctly) one should write $M=\Sigma_1^{\ast}\cup\Sigma_2$, where the $\ast$ means it has the opposite orientation. More on that once I write up something better.