"Okay -- shoot," replied the Wizard.
"Why are we studying topological quantum field theories? You're not claiming that quantum gravity in 4d spacetime is a TQFT, are you?"
"Good question," said the Wiz. "No, 4d quantum gravity probably isn't a topological quantum field theory. But 3d quantum gravity is -- or at least, some versions of it are! -- and it makes a good warmup for the more mysterious 4d case.
More importantly, TQFTs teach us about the connection between general relativity and quantum theory. On the one hand we have the category nCob whose objects are compact oriented (n-1)-dimensional manifolds representing possible choices of "space", and whose morphisms are oriented n-dimensional cobordisms representing possible choices of "spacetime". On the other hand we have the category Hilb whose objects are Hilbert spaces and whose morphisms are linear operators. A topological quantum field theory bridges these worlds, since it's a functor
Z: nCob -> Hilb
But there's more to it than that! nCob and Hilb have a lot more in common than merely being categories, and a TQFT should respect all these similarities. For example, we've already seen that nCob and Hilb are *-categories, so a nice TQFT, a "unitary" one, should be a *-functor.
But there's a lot more, too... and that's what I want to talk about now!
A category is all about "things" and "processes", or "ways to be" and "ways to become" -- we call the them "objects" and "morphisms". Geometrically, it's a 1-dimensional sort of thing, since we can draw the objects as dots and the morphisms as arrows:
f
x -----> y
Last quarter we used this notation instead:
|
|
x v
|
|
/ \
| f |
\_/
|
|
y v
|
|
Here it's the other way around: the objects look like arrows and the
morphisms look like dots, or blobs."
"Poincare duality!" cried Toby.
"Shhh!" said the Wiz. "Anyway, the pictures are still basically 1-dimensional, and the main fun thing to do is compose morphisms, which amounts to hooking up 1-dimensional gadgets in a line, like this:
f g
x -----> y -----> z
or like this:
|
x v
|
/ \
| f |
\_/
|
|
y v
|
|
/ \
| g |
\_/
|
z v
|
depending on how we draw it.
But in nCob and Hilb we can do more than compose processes! We can also set objects side by side and do processes on them in parallel, like this:
| |
| |
u v u' v
| |
| |
/ \ / \
| f | | f'|
\_/ \_/
| |
| |
v v v' v
| |
| |
In nCob, setting things side by side or doing processes in parallel
is called "disjoint union", and in Hilb it's called "tensoring".
For example, if we set two quantum systems side by side, and one has
Hilbert space H while the other has Hilbert space H', the combined system
has Hilbert space H (x) H'.
To make all this precise, we need the concept of a "monoidal category" --"
"Monoidal?" said Oz. "That sounds like something out of a bad science fiction story!"
The Wizard glowered, began conjuring up a fireball, but then thought better of it. "Given your sadly deficient education, perhaps you've never heard about monoids. A "monoid" is a set equipped with an associative product and a identity element for that product -- like the natural numbers together with multiplication and the element 1. A monoidal category is similar: it's a category with an associative product and an identity element. But I should just give the definition.
Hmm... I'm feeling lazy, so I'll only define a "strict" one for now!
A strict monoidal category is a category C equipped with
1) an object 1 called the "unit object".
2) for each pair of objects u and v, an object u (x) v, called their "tensor product".
3) for each pair of morphisms f: u -> v and f': u' -> v', a morphism f (x) g: u (x) v -> u' (x) v', called their "tensor product".
such that these laws hold:
4) the "left and right unit laws":
for any object u we have 1 (x) u = u (x) 1 = u.
for any morphism f we have 1_1 (x) f = f (x) 1_1 = f.
5) the "associative law":
for any objects u, v, and w we have (u (x) v) (x) w = u (x) (v (x) w).
for any morphisms f, g, and h we have (f (x) g) (x) h = f (x) (g (x) h).
6) the "interchange law":
for any morphisms f: u -> v, g: v -> w, f': u' -> v', and g': v' -> w' we have (g (x) g') (f (x) f') = gf (x) g'f'.
The last law says that this morphism is unambigously defined:
| |
u v u' v
| |
/ \ / \
| f | | f'|
\_/ \_/
| |
v v v' v
| |
/ \ / \
| g | | g'|
\_/ \_/
| |
w v w' v
| |
We can either tensor f and f', then tensor g and g', and then compose them...
or compose f and g, then compose f' and g', and then tensor them!
Either way, we get the same thing."
"You mean the same process!" said Oz.
The Wizard growled and flung a fireball at Oz. It whizzed over his head, reducing the top of his hair to ashes, and then it slammed into the rear wall of the classroom -- which, Oz noticed, was quite dented and pockmarked.
"Now I can't say "thing" anymore, because you think it's a technical term! Anyway, using these laws it's easy to prove stuff like this:
| | | |
u v | | w v
| | | |
/ \ | | / \
| f | w v u v | g |
\_/ | | \ /
| | | |
| | = | |
| | | |
| / \ / \ |
v v | g | | f | x v
| \_/ \ / |
| | | |
| x v v v |
| | | |
or in terms of equations:
(1v (x) g) (f (x) 1w) = (f (x) 1x) (1u (x) g)
The picture is prettier, no?
Now, all this is well and good, but there's a problem. Anyone know what it is?"
Toby nodded and smiled. "Equations between objects are evil!" he said.
"Right! I just defined a "strict" monoidal category, where the unit object tensored with any object x is *equal* to x, and associativity for tensor products holds as an *equation*. This is bad. For example, if you have 3 vector spaces, say U, V and W, the tensor product (U (x) V) (x) W is not really *equal* to U (x) (V (x) W) -- they're only isomorphic!
So to do things right, we should define "weak" monoidal categories, where for any 3 objects we have an isomorphism called the "associator":
au,v,w: (u (x) v) (x) w -> u (x) (v (x) w)
This lets us parenthesize a tensor product of a bunch of objects any way we like: no matter how we do it, the answer will be isomorphic. However, unless we're careful, there will be more than one isomorphism! For example, suppose we have 4 objects. Then there are 5 ways to parenthesize their tensor product, and the associator gives 5 isomorphisms like this... I'll leave out the "(x)" symbols to reduce clutter:
---------------------> (uv)(wx) ---------------------
/ \
/ v
((uv)w)x u(v(wx))
\ ^
\ /
------------> (u(vw))x ----------> u(v(wx)) ---------
This gives two isomorphisms from the object on the far left
to the object on the far right. It'd be a nuisance if these
were different, so we impose a law -- the "pentagon equation" --
saying that these isomorphisms agree. In other words, in a weak
monoidal category this diagram must commute!
But here's the cool part: the wizard Mac Lane showed that given the pentagon equation, all diagrams built from associators will commute!
Now: do you remember seeing the pentagon equation last quarter?"
"No," said Miguel. "You never talked about that."
"Yes I did!" said the Wiz. "It was in Week 10, Track 1."
"Look," said Miguel, "Here are my notes. No mention of any "pentagon equation"."
"Oh!" exclaimed the Wiz. "Wait a minute -- I forgot! Hold on a second." And with a loud BANG he disappeared.
Before the class could recover from this, there was an explosion, which sounded roughly like this:
KABLOOWIE!!!
And at the center of the explosion, there appeared the Wizard!
Unruffled, he said, "Okay, now I've told you about the pentagon
equation, and its relation to the Biedenharn-Elliot identity and
the 2-3 Pachner move. Remember?"
"Of course," said Miguel. "How could we forget? It's right here in my notes!"
"Good," said the Wiz. "So here's the point: any way of parenthesizing the tensor product of a bunch of objects can be drawn as a binary tree. It goes like this:
u v
\ /
\ / stands for u (x) v
|
|
u v w
\ / /
\/ /
\ / stands for (u (x) v) (x) w
\ /
|
|
u v w
\ \ /
\ \/
\ / stands for u (x) (v (x) w)
\ /
|
|
u v w x
\ \ / /
\ \/ /
\ / /
\/ / stands for (u (x) (v (x) w)) (x) x
\ /
\ /
|
|
and so on. See how it goes?"
"Umm, no, I don't," said Oz.
"Grr..." growled the Wiz. "It's easy! You just read the tree from top to bottom, tensoring objects as you go. Like this, if we leave out the tensor symbols:
u v w x
\ \ / /
\ \/ /
\ vw /
\ / /
\/ /
u(vw) /
\ /
|
(u(vw))x
Get it?"
"Sure, now that you actually explained it," said Oz.
The Wiz shot him a dark glance. "Okay. Now, if we use this trick to write down the associator, it looks like this:
u v w u v w
\ / / \ \ /
\/ / au,v,w \ \/
\ / --------------> \ /
\ / \ /
| |
| |
Then the pentagon equation says this diagram commutes:
u v w x
\ / \ /
\/ \/
\ /
\ /
u v w x \ / u v w x
\ / / / | \ \ \ /
\/ / / --------> | --------> \ \ \/
\ / / | \ \ /
\/ / \ \/
\ / \ /
\ / u v w x u v w x \ /
| \ \ / / \ \ / / |
| ---> \ \/ / \ \/ / ---> |
| \ / / ---> \ \ / |
\/ / \ \/
\ / \ /
\ / \ /
| |
| |
This is just like the picture I drew when I proved the Biedenharn-Elliot
identity! So that identity was really just a grungy way of stating the
pentagon equation for one particular monoidal category: the category of
representations of SU(2)."
"Hmm," said Jay. "I'm beginning to like this category theory stuff!"
"Well, wait until you hear this!" said the Wiz. "There's a definition of "weak" monoidal category, which is better than the "strict" definition I just gave you, because we never assert equations between objects. In the "weak" version, we have an "associator" isomorphism
au,v,w: (u (x) v) (x) w -> u (x) (v (x) w)
instead of the associative law
(u (x) v) (x) w = u (x) (v (x) w)
We also have a "left unit isomorphism"
Lu: 1 (x) u -> u
instead of the left unit law
1 (x) u = u,
and a "right unit isomorphism"
Ru: u (x) 1 -> u
The associator must satisfy the pentagon equation, and the left and right unit isomorphisms must make this diagram commute:
(1 (x) u) (x) 1 ----------------> 1 (x) (u (x) 1)
| |
| |
| |
v v
u (x) 1 --------> u <--------- 1 (x) u
"Hey!" said Toby. "That's a pentagon too!"
"Yes, true," said the Wizard grudgingly, "but you interrupted me before I could put the quotation mark on the end of that sentence!
Anyway, there's a bit more to the definition of a "weak" monoidal category, and I'm feeling too lazy to state it all here. But the point is this: the definition is cooked up to ensure that all diagrams built from associators, left unit isomorphisms, and right unit isomorphisms commute! This is called Mac Lane's Theorem.
One consequence of this theorem is that we can often get away with pretending weak monoidal categories are strict. Technically: any weak monoidal category is "equivalent" to a strict one. Or in other words: in many contexts, we can safely get away with pretending that tensoring is strictly associative, and ignore the associators... and pretending that tensoring any object x by the unit object gives an object equal to x.
So let's do this!
Okay. nCob is a monoidal category where the tensor product of objects is given by disjoint union of compact oriented (n-1)-manifolds. By the way, what's the unit object in this monoidal category?
"The empty set," said Miguel.
"Right! If you're careful, you can check that the empty set is a manifold of whatever dimension you like. Similarly, to tensor morphisms in nCob, we take the disjoint union of cobordisms. So if the morphism f describes a "big bang" in which a circular universe is born:
___________
/ \
/ \
| |
| |
| |
| |
| ... .. |
|' `.....' ` |
| `|
| /
\____________/
while the morphism g descibes a circular universe pinching off and
splitting in two:
___ ____
/ \___/ \
| __ \
| / \___ |
|\_/ \___/|
| |
| ___ |
| / \ |
| | | |
| | | |
| ... \ | .. |
|' `.| |' ` |
| | | `|
| | \ |
\_____/ \__/
then the morphism f (x) g looks like this:
___ ____
/ \___/ \
___________ | __ \
/ \ | / \___ |
/ \ |\_/ \___/|
| | | ___ |
| | | / \ |
| | | | | |
| | | | | |
| ... .. | | ... \ | .. |
|' `.....' ` | |' `.| |' ` |
| `| | | | `|
| / | | \ |
\____________/ \_____/ \__/
Two universes, doing their own thing, not interacting! That's what
tensoring is all about.
Actually, this "doing their own thing, not interacting" business is a bit like parallel circuits in electrical engineering. Circuits are morphisms in a certain monoidal category; you can compose them by hooking them up in series, or tensor them by putting them side by side. If I had time, someday I'd write a book explaining electrical circuits using the language of category theory...."
"Now that's sure to be a best-seller," said Oz.
At this, the Wizard began to fume. Smoke puffed from his ears, and a vein bulged dangerously on his forehead. "One more crack like that and you're history!" he said, conjuring up a size-9 blue-white fireball, the biggest Oz had ever seen, and spinning it on his fingertip for a second before hurling it against the back wall, half a millimeter from Oz's left ear. Even though it didn't actually hit Oz, the radiation was enough to give him quite a burn. He shrunk into his chair and resolved to stay quiet.
"Anyway, where was I? Okay. nCob and Hilb are both monoidal categories, so a TQFT should respect this: it should be a monoidal functor! Let me define this concept. Again, I'll only do the strict case for now:
If C and D are strict monoidal categories, a strict monoidal functor F: C -> D is a functor such that the following laws hold:
In short: F preserves the unit object, tensor products of objects, and tensor products of morphisms.1) F(1C) = 1D, where 1C is the unit object in C and 1D is the unit object in D.
2) For any pair of objects u and v in C, we have F(u (x) v) = F(u) (x) F(v).
3) For any pair of morphisms f and g in C, we have F(f (x) g) = F(f) (x) F(g).
This definition has some cool consequences for TQFTs. Since a TQFT is a monoidal functor
Z: nCob -> Hilb,
it sends the unit object of nCob to the unit object of Hilb. The unit object of nCob is the empty set; what's the unit object of Hilb?"
"The complex numbers," said Alissa.
"Right! So the Hilbert space assigned by a TQFT to the empty set is just C. Now C is one-dimensional, so it describes a system with just one state, since two unit vectors describe the same state if they differ by a phase. In short, if space is the empty set, then in a universe described by a TQFT there is only one state! How can we say this in plain English?"
"There is only one way for nothing to be," said Toby.
"Excellent. Now, consider morphism in nCob from the unit object to itself, say M: 1 -> 1. What does this amount to?"
"It's a cobordism from the empty set to itself, so it's a compact oriented n-manifold without boundary," said Miguel.
"Right. And our TQFT assigns to this... what?"
"A linear operator from C to C," said John.
"Right!" said the Wiz. And he sang out: "From C to shining C!" Then he caught himself. "Sorry. Anyway, what's a linear operator from C to C?"
"It's a number!" cried Oz, who remembered this from an earlier class, and was too excited to stay quiet.
"Right! So: if M is a compact oriented n-manifold, Z(M) is a complex number. We call this the partition function of M."
"Aha! So that's why we use the letter Z!" said Toby.
"Right, physicists always use the letter Z to mean partition function, for some weird reason. Does anyone know why?"
"It stands for Zustandsumme," said Miguel. "That's German for "sum over states" -- which is what a partition function amounts to, in statistical mechanics."
"Wow!" said the Wiz. "Zustandsumme, eh? I never knew that."
"In quantum field theory," said Jay, excitedly, "the partition function is something we calculate using a path integral: it's just the integral over all histories of the exponential of the action. Are you saying this Z(M) thing is just that?"
"Yes," said the Wizard. "Exactly!"
"So you mean all this category theory stuff is really just another way of talking about path integrals??"
"Right."
"My gosh," said Jay. "I'd like to see how that works."
"We will, we will... that's where this course is headed! We'll construct TQFTs using "state sum models", which are discretized path integrals."
"Hey!" said Miguel. "In German, "state sum" is Zustandsumme!"
"Right! Okay, now I'll finish defining a TQFT..." said the Wiz, but then he glanced at the clock. "Ulp! Time flies when you're doing time travel. I guess we should stop here. We didn't get around to defining a TQFT yet, but we're getting close: at least we know that a TQFT is a monoidal functor from nCob to Hilb, among other things. Let's take a break."