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:


$\bullet (i)$ 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 $\Sigma$.


4 Answers 4


The Atiyah-Segal axioms and generally the axioms of FQFT formalize the Schrödinger picture of quantum physics:

  • to a codimension-1 slice $M_{d-1}$ of space one assigns a vector space $Z(M_{d-1})$ -- the (Hilbert) space of quantum states over $M_{d-1}$;

  • to a spacetime manifold $M$ with boundaries $\partial M$ one assigns the quantum propagator which is the linear map $Z(M) : Z(\partial_{in} M) \to Z(\partial_{out} M)$ that takes incoming states to outgoing states via propagation along the spacetime/worldvolume $M$. This $Z(M)$ is alternatively known as the the scattering amplitude or the S-matrix for propagation from $\partial_{in}M$ to $\partial_{out}M$ along a process of shape $M$.

Now for genuine topological field theories all spaces of quantum states are finite dimensional and hence we can equivalently consider the linear dual spaces (using that finite dimensional vector spaces form a compact closed category). Doing so the propagator map

$$ Z(M) : Z(\partial_{in}M) \to Z(\partial_{out}M) $$

equivalently becomes a linear map of the form

$$ \mathbb{C} \to Z(\partial_{out}M) \otimes Z(\partial_{in}M)^\ast = Z(\partial M) \,. $$

Notice that such a linear map from the canonical 1-dimensional complex vector space $\mathbb{C}$ to some other vector space is equivalently just a choice of element in that vector space. It is in this sense that $Z(M)$ is a vector in $Z(\partial_{out}M) \otimes Z(\partial_{in}M)^\ast = Z(\partial M)$.

In this form in physics the propagator is usually called the correlator .

Segal's axioms were originally explicitly about the propagators/S-matrices, while Atiyah formulated it in terms of the correlators this way. Both perspectives go over into each other under duality as above.

Notice that this kind of discussion is not restricted to topological field theory. For instance already plain quantum mechanics is usefully formulated this way, that's the point of finite quantum mechanics in terms of dagger-compact categories. There this is being used to great effect in quantum information theory and quantum computing.

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    $\begingroup$ @ Urs Schreiber, thanks so much! +1. It is great to have experts around here to provide nice thoughts. Let me get back to you soon after digestion. $\endgroup$
    – wonderich
    Jan 12, 2014 at 0:04
  • $\begingroup$ this is related: physics.stackexchange.com/questions/116406/… $\endgroup$
    – wonderich
    Jun 2, 2014 at 3:15

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.

  • $\begingroup$ Also, one result you'll learn rather soon on is that 2D TQFTs are equivalent (in the categorical sense) to commutative Frobenius algebras. If you really want to understand TQFT from the categorical point of view, then you'll ultimately want to check out the extension of that result: extended TQFTs and their classification from Lurie. That ventures into the realm of higher categories, so this expository article might be a nice motivation for that. $\endgroup$ Jan 11, 2014 at 2:55
  • $\begingroup$ Although (unfortunately) I still don't have time to write a better answer, let me quickly respond to the above. First of all, in general something being a set doesn't mean it isn't an element. For example, for any set $X$, $X\in\mathcal{P}(X)$. At any rate, in this case the statement $Z(M)\in Z(\partial M)$ is not a typo, Atiyah knew what he was talking about -- that is the correct statement. I will add a brief explanation of this point to my answer right now, as it seems to be confusing a number of people. :) $\endgroup$ Jan 11, 2014 at 15:10
  • $\begingroup$ There, I hope that helps everyone see why the statement $Z(M)\in Z(\partial M)$ is not "nonsense." :) I'll try to write up a clearer answer soon. $\endgroup$ Jan 11, 2014 at 15:35
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    $\begingroup$ Hey Trimok, what you write is just wrong and mixing things up. Please check first if you understand what a question is about before makeing statements like this. $\endgroup$ Jan 11, 2014 at 23:38
  • $\begingroup$ @UrsSchreiber : I understand my error now. $\endgroup$
    – Trimok
    Jan 13, 2014 at 10:45

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.

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    $\begingroup$ @ Ralph, thank you so much for the nice answers! +1.+1. Let me digest it carefully and get back to you soon. Earlier I recall that for some category theory of gapped boundaries $(\partial M)^{d-1}$ for a bulk manifold $M^{d}$, indeed the boundary theory (say $Z(\partial M)$) contains more information than the bulk theory (say $Z(M)$); so in the gapped boundary cases, I may appreciate why $Z(M) \in Z(\partial M)$. $\endgroup$
    – wonderich
    Jan 11, 2014 at 20:15
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    $\begingroup$ Ref, See, e.g. Kitaev, Kong and Wang, Wen on gapped boundaries (i.e. topological boundary conditions). Do you agree this thinking, or not? How about gapless boundaries cases (i.e. those with Wess-Zumino-Witten theory on the boundary)? $\endgroup$
    – wonderich
    Jan 11, 2014 at 20:15
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    $\begingroup$ For more nitty gritty on the categorical perspective for gapped boundaries also see Fuchs, Schweigert, Valentino, Kapustin and Saulina, and according to Kong, Beigi, Shor, and Whalen. $\endgroup$ Jan 12, 2014 at 0:51
  • $\begingroup$ I'm not familiar with those ideas, so I'll have to look into the references and get back to you! $\endgroup$ Jan 12, 2014 at 11:49
  • $\begingroup$ this is related $\endgroup$
    – wonderich
    Jun 2, 2014 at 3:16

This is absolutely not a typo. The answer is simple: there is an abuse of notation ‘Z’. For every d dimensional manifold we assign a vector space, and for every d+1 dimensional manifold we assign a single vector. These are two groups of data! But unfortunately Atiyah used the same letter ‘Z’. That’s all.


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