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1) Michael Mueger, Galois theory for braided tensor categories and the modular closure, preprint available as math.CT/9812040.
A braided monoidal category is simple algebraic gadget that captures a bit of the essence of 3-dimensionality in its rawest form. It has a bunch of "objects" which we can draw a labelled dots like this:
x
.
So far this is just 0-dimensional. Next, given a bunch of objects
we get a new object, their "tensor product", which we can draw by setting
the dots side by side. So, for example, we can draw x tensor y like this:
x y
. .
This is 1-dimensional. But in addition we have, for any pair of
objects x and y, a bunch of "morphisms" f: x → y. We can draw a
morphism from a tensor product of objects to some other tensor
product of objects as a picture like this:
x y z
| | |
| | |
| | |
-------
| f |
-------
| |
| |
| |
u v
This picture is 2-dimensional. In addition, we require that for any
pair of objects x and y there is a "braiding", a special morphism
from x tensor y to y tensor x. We draw it like this:
x y
| |
\ /
\ /
/
/ \
/ \
| |
y x
With this crossing of strands, the picture has become 3-dimensional!
We also require that we can "compose" a morphism f: x → y and a morphism g: y → z and get a morphism fg: x → z. We draw this by sticking one picture on top of each other like this... I'll draw a fancy example where all the objects in question are themselves tensor products of other objects:
x y z
| | |
| | |
| | |
-------
| f |
-------
| |
| |
| |
-------
| g |
-------
| | | |
| | | |
| | | |
a b c d
Finally, we require that the tensor product, braiding and composition
satisfy a bunch of axioms. I won't write these down because I already
did so in "week121", but the point is that they all make geometrical
sense - or more precisely, topological sense - in terms of the above
pictures.
The pictures I've drawn should make you think about knots and tangles and circuit diagrams and Feynman diagrams and all sorts of things like that - and it's true, you can understand all these things very elegantly in terms of braided monoidal categories! Sometimes it's nice to throw in another rule:
x y x y
| | | |
\ / \ /
\ / \ /
/ = \
/ \ / \
/ \ / \
| | | |
y x y x
where we cook up the second picture using the inverse of the braiding.
This rule is good when you don't care about the difference between
overcrossings and undercrossings. If this rule holds we say our
braided monoidal category is "symmetric". Topologically, this rule
makes sense when we study 4-dimensional or higher-dimensional situations,
where we have enough room to untie all knots. For example, the
traditional theory of Feynman diagrams is based on symmetric monoidal
categories (like the category of representations of the Poincare
group), and it works very smoothly in 4-dimensional spacetime.
But 3-dimensional spacetime is a bit different. For example, when we interchange two identical particles, it really makes a difference whether we do it like this:
x y
| |
\ /
\ /
/
/ \
/ \
| |
y x
or like this:
x y
| |
\ /
\ /
\
/ \
/ \
| |
y x
Thus in 3d spacetime, besides bosons and fermions, we have other sorts
of particles that act differently when we interchange them - sometimes
people call them "anyons", and sometimes people talk about "exotic
statistics".
Now let me dig into some more technical aspects of the picture.
Starting with Reshetikhin and Turaev, people have figured out how to use braided monoidal categories to construct topological quantum field theories in 3-dimensional spacetime. But they can't do it starting from any old braided monoidal category, because quantum field theory has a lot to do with Hilbert spaces. So usually they start from a special sort called a "modular tensor category". This is a kind of hybrid of a braided monoidal category and a Hilbert space.
In fact, apart from one technical condition - which is at the heart of Mueger's work - we can get the definition of a modular tensor category by taking the definition of "Hilbert space", categorifying it once to get the definition of "2-Hilbert space", and then throwing in a tensor product and braiding that are compatible with this structure.
It's amazing that by such abstract conceptual methods we come up with almost precisely what's needed to construct topological quantum field theories in 3 dimensions! It's a great illustration of the power of category theory. It's almost like getting something for nothing! But I'll resist the temptation to tell you the details, since week99 explains a bunch of it, and the rest is in here:
2) John Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189. Also available as q-alg/9609018.
In this paper I call a 2-Hilbert space with a compatible tensor product a "2-H*-algebra", and if it also has a compatible braiding, I call it a "braided 2-H*-algebra". This terminology is bit clunky, but for consistency I'll use it again here.
Okay, great: we almost get the definition of modular tensor category by elegant conceptual methods. But there is one niggling but crucial technical condition that remains! There are lots of different ways to state this condition, but Mueger proves they're equivalent to the following very elegant one.
Let's define the "center" of a braided monoidal category to be the category consisting of all objects x such that
x y x y
| | | |
\ / \ /
\ / \ /
/ = \
/ \ / \
/ \ / \
| | | |
y x y x
for all y, and all morphisms between such objects. The center
of a braided monoidal category is obviously a symmetric monoidal
category. The term "center" is supposed to remind you of the
usual center of a monoid - the elements that commute with all
the others. And indeed, both kinds of center are special cases
of a general construction that pushes you down the columns of
the "periodic table":
k-tuply monoidal n-categories
n = 0 n = 1 n = 2
k = 0 sets categories 2-categories
k = 1 monoids monoidal monoidal
categories 2-categories
k = 2 commutative braided braided
monoids monoidal monoidal
categories 2-categories
k = 3 " " symmetric weakly
monoidal involutory
categories monoidal
2-categories
k = 4 " " " " strongly
involutory
monoidal
2-categories
k = 5 " " " " " "
I described this in "week74" and "week121", so I won't do so
again. My point here is really just that lots of this
3-dimensional stuff is part of a bigger picture that applies
to all different dimensions. For more details, including a
description of the center construction, try:
3) John Baez and James Dolan, Categorification, in Higher Category Theory, eds. Ezra Getzler and Mikhail Kapranov, Contemporary Mathematics vol. 230, AMS, Providence, 1998, pp. 1-36. Also available at math.QA/9802029.
Anyway, Mueger's elegant characterization of a modular tensor category amounts to this: it's a braided 2-H*-algebra whose center is "trivial". This means that every object in the center is a direct sum of copies of the object 1 - the unit for the tensor product.
Mueger does a lot more in his paper that I won't describe here, and he also said a lot of interesting things in his talk about the general concept of center. For example, the center of a monoidal category is a braided monoidal category. In particular, if you take the center of a 2-H*-algebra you get a braided 2-H*-algebra. But what if you then take this braided 2-H*-algebra and look at its center? Well, it turns out to be "trivial" in the above sense!
There's a bit of overlap between Mueger's paper and this one:
4) A. Bruguieres, Categories premodulaires, modularisations et invariants des varietes de dimension 3, preprint.
One especially important issue they both touch upon is this: if you have a braided 2-H*-algebra, is there any way to mess with it slightly to get a modular tensor category? The answer is yes. Thus we can really get a topological quantum field theory from any braided 2-H*-algebra. But this raises another question: can we describe this topological quantum field theory directly, without using the modular tensor category? The answer is again yes! For details see:
5) Stephen Sawin, Jones-Witten invariants for nonsimply-connected Lie groups and the geometry of the Weyl alcove, preprint available as math.QA/9905010.
This paper uses this machinery to get topological quantum field theories related to Chern-Simons theory. People have thought about this a lot, ever since Reshetikhin and Turaev, but the really great thing about this paper is that it handles the case when the gauge group isn't simply-connected. This introduces a lot of subtleties which previous papers touched upon only superficially. Sawin works it out much more thoroughly by an analysis of subsets of the Weyl alcove that are closed under tensor product. It's very pretty, and reading it is very good exercise if you want to learn more about representations of quantum groups.
Now, I said that a lot of this is part of a bigger picture that works in higher dimensions. However, a lot of this higher-dimensional stuff remains very mysterious. Here are two cool papers that make some progress in unlocking these mysteries:
6) Marco Mackaay, Finite groups, spherical 2-categories, and 4-manifold invariants, preprint available as math.QA/9903003.
7) Mikhail Khovanov, A categorification of the Jones polynomial, preprint available as math.QA/9908171.
Marco Mackaay spoke about his work in Coimbra, and I had grilled him about it in Lisbon beforehand, so I think I understand it pretty well. Basically what he's doing is categorifying the 3-dimensional topological quantum field theories studied by Dijkgraaf and Witten to get 4-dimensional theories. It fits in very nicely with his earlier work described in "week121".
People have been trying to categorify the magic of quantum groups for quite some time now, and Khovanov appears to have made a good step in that direction by describing the Jones polynomial of a link as the "graded Euler characteristic" of a chain complex of graded vector spaces. Since graded Euler characteristic is a generalization of the dimension of a vector space, and taking the dimension is a process of decategorification (i.e., vector spaces are isomorphic iff they have the same dimension), Khovanov's chain complex can be thought of as a categorified version of the Jones polynomial.
I would like to understand better the relation between Khovanov's work and the work of Crane and Frenkel on categorifying quantum groups (see "week58"). For this, I guess I should read the following papers:
8) J. Bernstein, I. Frenkel and M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl_2) by projective and Zuckerman functors, to appear in Selecta Mathematica.
9) Mikhail Khovanov, Graphical calculus, canonical bases and Kazhdan-Lusztig theory, Ph.D. thesis, Yale, 1997.
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