I decided to include this as a separate answer, rather than mess with the above. Sorry in advance for the length. I still wholeheartedly suggest that you check out:

(1) Quantum Quandaries, by Baez;

(2) Frobenius Algebras and 2D Topological Quantum Field Theories, by Koch (a portion of it is here, and there is a "short version" here);

(3) An Introduction to Topological Quantum Field Theories, by Atiyah.

Some additional (mostly standard) references include:

(4) Higher-dimensional Algebra and Topological Quantum Field Theory, by Baez-Dolan (the more detailed version of the aforementioned expository article);

(5) Categorical Aspects of Topological Quantum Field Theories, by Bartlett;

(6) Topological Field Theory, Higher Categories, and Their Applications, by Kapustin;

(7) Lectures on Tensor Categories and Modular Functor, by Bakalov and Kirillov (in particular, Chapter 3);

(8) Quantum Invariants of Knots and 3-manifolds, by Turaev;

(9) Dirichlet Branes and Mirror Symmetry, by Aspinwall et al (especially chapters 2 and 3).

That being said, however, I will attempt to explain it as well as I can below.

(i) You asked if $Z(M)\in Z(\partial M)$ is correct, and why. I think the confusion here boils down to thinking of $Z$ as a *function between two manifolds*, in which case $Z(\partial M)\subset Z(M)$ would make more sense. In this categorical approach to TQFT, however, *$Z$ is not a function, but a ***functor**. A functor can be understood as the categorical analogue of a function, but it is not the same thing -- in fact, this sort of TQFT functor $Z$ is one of the standard examples given of a functor which is not a function, so understanding more about $Z$ will help you understand a bit more about category theory. Perhaps a little background will help clarify the picture.

So, a **category** is a bit different from a set: in set theory one speaks of *elements* $x$ belonging to a set $X$, but, a priori, there is no relationship between any two elements of a given set. In contrast to this, in category theory one speaks of *objects* $A$ belonging to a category $\mathcal{C}$, but *there is a relationship between any two given objects*! Indeed, given two objects $A,B\in\mathcal{C}$, there is a class (typically a set) $\text{Hom}(A,B)$ of morphisms from $A$ to $B$. Just as a function between sets can be thought of as a relation between elements of those sets (e.g. for $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by $f(x)=x^3+1$, $f$ provides a relation between elements of the domain and elements of the range: for example, 28 is related to 3 via the rule $f(3)=28$), a morphism between objects of a category is a relation between them. It is in this way that categories are different than sets.

Consequently, if you want to talk about *maps between categories*, you cannot simply worry about where you send objects of one category into the other, but you *also need to worry about where the morphisms between objects are being sent*. So, in a sense, a **functor** $F$ between two categories $\mathcal{C}$ and $\mathcal{D}$ is both a relationship between objects *and a relationship between relationships between objects* (cue obnoxious inception horn). Now we have set the stage for my previous answer.

Again, the idea here is that $Z$ is a functor between categories: its "domain" is a geometric category and its "range" is an algebraic category. In this case, the geometric category **dCob** consists of:

**--Objects =** $d$-dimensional, closed manifolds $\Sigma$,

**--Morphisms between two objects $\Sigma_1$ and $\Sigma_2$ =** cobordisms from $\Sigma_1$ to $\Sigma_2$, i.e. $(d+1)$-dimensional manifolds $M$ such that $\partial M=\Sigma_1\cup\Sigma_2$.

The algebraic category **$\Lambda$-Mod** (following your original question) in this case consists of:

**--Objects =** finitely generated $\Lambda$-modules $R$,

**--Morphisms between two objects $R_1$ and $R_2$ =** $\Lambda$-module homomorphisms (i.e. linear maps) $f:R_1\rightarrow R_2$.

As was implicit in my previous answer, one often works with the category **Vec** of vector spaces instead of more general modules.

So, **$Z:$dCob$\rightarrow \Lambda$-Mod** is a *functor*, so you need to think about where is sends *both* objects (closed manifolds) and morphisms (cobordisms). To each closed $d$-dimensional manifold $\Sigma$, $Z$ assigns a $\Lambda$-module $Z(\Sigma)$. To each cobordism $M$ between $\Sigma_1$ and $\Sigma_2$ (don't forget: $\partial M=\Sigma_1\cup\Sigma_2$), $Z$ assigns a $\Lambda$-module homomorphism $Z(M):Z(\Sigma_1)\rightarrow Z(\Sigma_2)$. Using the multiplicative property $Z$ is assumed to satisfy, we find that

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

hence $Z(\partial M)$ in this context stands for the collection of all $\Lambda$-module homomorphisms between $Z(\Sigma_1)$ and $Z(\Sigma_2)$. Since $Z(M)$ is such a map, as we saw above, we have that $Z(M)\in Z(\partial M)$.

(ii) Although I'm not a physicist, so I might not be the best person for this answer, I will give it a go. This is how I think of it (heavily inspired by Baez's expository article): a $d$-dimensional closed manifold can be thought of a the geometry of space at a given time slice. A cobordism between two such manifolds $\Sigma_1$ and $\Sigma_2$ can be thought of as a *process* in which the geometry of space is (smoothly) changed from that of $\Sigma_1$ to that of $\Sigma_2$. At "time 0" you have the geometry of $\Sigma_1$ and as "time" goes on (along the cobordism $M$ from the boundary component $\Sigma_1$ to the other boundary component $\Sigma_2$) the geometry is changed a little bit, until it ultimately gets changed into that of $\Sigma_2$. So the geometric category in this case can be thought of as describing processes whereby the geometry of spacetime is transformed.

Now, in quantum physics, one doesn't really deal with spacetime so much anymore, but rather with vectors in a Hilbert space: each vector is a state that a quantum mechanical system can be in. A linear map between two vector spaces, then, can be thought of as a process taking one system (i.e. one collection of states) to another system (the other Hilbert space of states). Of course, both of these are a little different from the situation Atiyah is considering: in GR one is more interesting in a specific type of manifolds (pseudo-Riemannian, i.e. a metric, curvature, etc. are also involved) and in QFT one is interested in these Hilbert spaces. They simply work with generic (smooth) manifolds and $\Lambda$-modules to make things more tractable.

The interpretation, then, of the functor $Z$ is defining some sort of correspondence between states and processes in one description and states and processes in the other description. In other words, $Z$ is a way of codifying how the quantum mechanical analogue of processes that change spacetime geometry change.

What about the homotopy axiom and the addition axiom?

The homotopy axiom is really where the *topological* part of the name comes from: it is saying that two cobordisms that are homotopically equivalent will give the same $\Lambda$-module homomorphisms on the algebraic side. Physically, this is just saying that any two physical processes that change space from $\Sigma_1$ to $\Sigma_2$ that are the same in terms of topology (to be clear: the topologies of the "processes, i.e. the cobordisms, are the same, not necessarily the topologies of the $\Sigma_i$'s) will give the same results on the QFT end, i.e. the theory only cares about topological differences so it is a topological QFT.

The additive axiom, in this context, is the multiplicative property I mentioned above: $Z(\Sigma_1\cup\Sigma_2)=Z(\Sigma_1)\otimes Z(\Sigma_2)$. Physically, as I tried to mention in my previous answer, this corresponds to when you have two separate processes between systems, and you consider them as one process running in parallel. On the quantum side, as one knows from basic QM, the Hilbert spaces giving the states of each system don't combine so simply: one needs to consider the tensor product of them in order to get the correct results. So the additive axiom can really be thought of as encoded the quantum part of "topological quantum field theory."

I hope this helps give some insight into what is going on physically. I strongly suggest you read what I said above with the references given at the beginning so that you can see nice images illustrating what I'm trying to say here.

(iii) In the case where $\Sigma$ is a $d$-dimensional closed manifold, yes $Z(\Sigma)$ is interpreted as the TQFT partition function.

This post imported from StackExchange Physics at 2014-04-04 16:41 (UCT), posted by SE-user Ralph Mellish