
Lately, many things give me the feeling that we're on the brink of some deeper understanding of the relations between geometry, topology, and category theory. It is very tantalizing to see the array of clues pointing towards the fact that many seemingly disparate mathematical phenomena are aspects of some underlying patterns that we don't really understand yet. Louis Crane expressed it well when he said that it's as if we are a bunch of archeologists digging away at different sites, and are all starting to find different parts of the skeleton of some gigantic prehistoric creature, the full extent of which is still unclear.
I want to keep studying the following book until I understand it, because I think it makes a lot of important connections... pardon the pun:
1) Loop Spaces, Characteristic Classes and Geometric Quantization, by JeanLuc Brylinski, Birkhauser, Boston, 1993.
The title of this book, while accurate, really does not convey the novelty of the ideas it contains. All three subjects listed have been intensively studied by many people for at least several decades, but Brylinski's book is not so much a summary of what is understood about these subjects, as a plan to raise the subjects to a whole new level.
I can't really describe the full contents of the book, since I haven't had time to really absorb some of the most interesting parts, but let me start by listing the contents, and then talk about it a bit.
1. Complexes of Sheaves and Their Hypercohomology 2. Line Bundles and Geometric Quantization 3. Kaehler Geometry of the Space of Knots 4. Degree 3 Cohomology  The DixmierDouady Theory 5. Degree 3 Cohomology  Sheaves of Groupoids 6. Line Bundles over Loop Spaces 7. The Dirac Monopole
It should be clear that while this is a very mathematical book, it is informed by ideas from physics. As usual, the physical universe is serving to goad mathematics to new heights!
The first two chapters are largely, but not entirely, "standard" material. I put the word in quotes because while Brylinski's treatment of it starts with the basics  the definition of sheaves, sheaf cohomology, Cech cohomology, deRham theory and the like  even these "basics" are rather demanding, and the slope of the ascent is rather steep. Really, the reader should already be fairly familiar with these ideas, since Brylinski is mainly introducing them in order to describe a remarkable generalization of them in the next chapters.
Let me quickly give a thumbnail sketch of the essential ideas behind this "standard" material. In classical mechanics the main stage is the phase space of a physical system. Points in this space represent physical states; smooth functions on it represent observables. Time evolution acts on this space as a oneparameter group of diffeomorphisms. The remarkable fact is that time evolution is determined by an observable, the Hamiltonian, or energy function, by means of a geometric structure on phase space called a symplectic structure. This is a nondegenerate closed 2form. The idea is that the differential of the Hamiltonian is a 1form; since the symplectic structure is nondegenerate it sets up an isomorphism of the tangent and cotangent bundles of phase space, allowing us to turn the differential of the Hamiltonian into a vector field; this vector field generates the 1parameter group of diffeomorphisms representing time evolution; and by the magic of symplectic geometry, these diffeomorphisms automatically preserve the symplectic structure.
This is the startingpoint of the beautiful approach to quantum theory known as geometric quantization, founded by Kostant in the early 1970's. His first paper is still a good place to start:
2) Quantization and unitary representations, by Bertram Kostant, in Lectures in Modern Analysis and Applications III, SpringerVerlag Lecture Notes in Mathematics 170 (1970), 87208.
Here the idea is to construct a Hilbert space of states of the quantum system corresponding to the classical system, and turn time evolution into a oneparameter group of unitary operators on this Hilbert space. Extremely roughly, the idea is to first look at the space of all L^2 complex functions on phase space, and then use a "polarization" to cut down this "prequantum" Hilbert space to "half the size," by which one means something vaguely like how L^2(R^n) is "half the size" of L^2(R^{2n})  this being the classic example. But in fact, it turns out one doesn't really want to use functions on phase space, but instead sections of a certain complex line bundle. The point is that the classification of line bundles fits in beautifully with symplectic geometry. We can equip any line bundle with a hermitian connection; the curvature of this connection is a closed 2form; this determines an element of the 2nd cohomology of phase space called the first Chern class. An important theorem says this class is necessarily an integral class, that is, it comes from an element of the 2nd cohomology with integer coefficients; moreover, isomorphism classes of line bundles over a manifold are in onetoone correspondence with elements of its 2nd cohomology with integer coefficients. The trick, then, is to try to cook up a line bundle over phase space with a connection whose curvature is the symplectic structure! This will be possible precisely when the symplectic structure defines an integral cohomology class. In fact, this integrality condition is nothing but the old BohrSommerfeld quantization condition dressed up in spiffy new clothes (and made far more precise).
So: the moral I want to convey here is just that if the symplectic structure on phase space defines an integral class in the 2nd cohomology group, then we get a line bundle over phase space which helps us get going with quantization. It then turns out that the oneparameter group of diffeomorphisms defined by any Hamiltonian on phase space lifts to a oneparameter group of transformations of this line bundle, which allows us to get a unitary operator on the space of L^2 sections of the line bundle. This is not the end of the quantization story; one still needs to chop down this "prequantum" space to half the size, etc.; but let me leave off here.
What Brylinski wants to do is to find analogs of all these phenomena involving the THIRD cohomology groups of manifolds.
At first glance, this might seem to be a very artificial desire. Note that importance of the SECOND cohomology group in the above story is twofold: 1) symplectic structures give elements of the second cohomology, 2) the curvature of a connection gives an element of the second cohomology, and in fact 2') line bundles are classified by elements of second cohomology. None of these beautiful things seem to have analogs in third cohomology! Of course, one can use the curvature of a connection to get, not just the first Chern class, but higher Chern classes. But the nth Chern class is an element of the 2nth cohomology group, so the odd cohomology groups don't play a major role here. Of course, experts will immediately reply that there are also ChernSimons "secondary characteristic classes" that live in odd cohomology, at least when one has a flat bundle around. And the same experts will immediately guess that, because ChernSimons theory has been near the epicenter of the explosion of new mathematics relating quantum groups, topological quantum field theories, conformal field theory and all that stuff, I must be leading up to something along these lines.... Well, there must be a relationship here, but actually it is not emphasized in Brylinski's book! He takes a different tack, as follows.
The basic point is that given a manifold M, the space of loops in M, say LM, is a space of great interest in its own right. It is infinite dimensional, but that should not deter us. When G is a Lie group, LG is also a group (with pointwise operations); these are the famous loop groups, which appear as groups of gauge transformations in conformal field theory. When M is a 3dimensional manifold, LM contains within it the space of all knots in M; also, we may think of LM as the configuration space for the simplest flavor of string theory in the spacetime R x M. Loops also serve to define observables called "Wilson loops" in gauge theories, and these are the basis of the loop representation of quantum gravity. So there is a lot of interesting mathematics and physics to be found in the loop space.
What does this have to do with the 3rd cohomology group of M? Well, LM is a bundle over M, so according to algebraic topology there is a natural map from the 3rd cohomology of M to the 2nd cohomology of LM! The ramifications of this are multiple.
First, every compact simple Lie group G has 3rd cohomology equal to Z. (In fact, Brylinski notes that the cohomology group is not merely isomorphic to Z, but canonically so  and this extra nuance turns out to be quite significant!) This gives rise to a special element in the 2nd cohomology of LG. This then gives a line bundle over LG. Alternatively, it gives a circle bundle over LG, in fact a central extension of LG, that is, a bigger group LG^ and an exact sequence
1 → S^1 → LG^ → LG → 1
This group is called a KacMoody group, and these are wellloved by string theorists since it turns out that when one wants to quantize a gauge theory on the string worldsheet (a kind of conformal field theory) one gets, not a representation of the gauge group LG on the Hilbert space of quantum states, but merely a projective representation, or in other words, a representation of the central extension LG^. Brylinski also notes that in some sense the canonical element in the 3rd cohomology of G is responsible for the existence of quantum groups; this is probably the deep reason for the association between quantum group representations and KacMoody group representations, but, alas, this is still quite murky to me.
Second, we can do better if we restrict ourself to knots (possibly with nice selfintersections) rather than loops. Namely, given a 3manifold M equipped with a 3form, one gets, not just an element of the 2nd cohomology of LM, but a symplectic structure on the space of knots in M, say KM. It may seem odd to think of the space of knots as a physical phase space, but Brylinski shows that this idea is related to the work of Marsden and Weinstein on "vortex filaments," an idealization of fluid dynamics in which all the fluid motion is concentrated along some curves. Brylinski also notes that if M is equipped with a Riemannian structure then KM inherits a Riemannian structure (this is easy), and that if M has a conformal structure KM has an almost complex structure. In fact, in the Riemannian case all these structures on KM fit together to make it a sort of Kaehler manifold (although one must be careful, since the almost complex structure is only integrable in a certain formal sense). Brylinski hints that all this geometry may give a nice approach to the study of knot invariants; I will have to look at the following papers sometime:
3) Vortices in He II, current algebras and quantum knots, by M. Rasetti and T. Regge, Physica 80A (1975) 217233.
4) A geometric approach to quantum vortices, by V. Penna and M. Spera, J. Math. Phys. 30 (1989), 27782784.
However, Brylinski's real goal is something much more radical! The beauty of 2nd cohomology is that integer classes in the 2nd cohomology of M correspond to line bundles on M; there is, in other words, a very nice geometrical picture of 2nd cohomology classes. What is the natural analog for 3rd cohomology? Instead of just working with LM, it would be nice to have some sort of geometrical objects on M that correspond to integer classes in 3rd cohomology. What should they be?
Brylinski gives two answers, one in Chapter 4 and another in Chapter 5. The first one, due mainly to Dixmier and Douady, is very appealing for a quantum field theorist such as myself. Just as elements of H^2(M,Z) correspond to line bundles over M, elements of H^3(M,Z) correspond to projective Hilbert space bundles over M! Recall that in physics two vectors in a Hilbert space correspond to the same physical state if one is a scalar multiple of the other; the space of equivalence classes (starting with a countabledimensional Hilbert space) is what I'm calling "projective Hilbert space," and it is bundles of such rascals that correspond to elements of H^3(M,Z). The reason is roughly this: the structure group G for such bundles is the group Aut(H)/C*, that is, invertible operators on the Hilbert space H, modulo invertible complex numbers. In other words, we have an exact sequence
1 → C* → Aut(H) → G → 1
This gives an exact sequence of sheaves on M, which, combined with the marvelous fact that Aut(H) is contractible, gives an isomorphism between H^1(M,sh(G)) (the cohomology of the sheaf of smooth Gvalued functions on M) and H^2(M,sh(C*)). But the latter is isomorphic to H^3(M,Z).
Brylinski pushes the analogy to the line bundle case further by showing how to realize the element of H^3(M,Z) starting from a connection on a projective Hilbert space bundle. But in Chapter 5 he takes a more abstract approach that I want to sketch very vaguely, since I don't understand it very well yet. This approach is exciting because it connects to recent work on 2categories (and higher ncategories), which I am convinced will play a role in unifying the wild profusion of mathematics we are seeing in this tail end of the twentieth century.
Here the best way to see the analogy to the line bundle case is through Cech cohomology. Recall that we can patch a line bundle together by covering our manifold M with charts O(i) and assigning to each intersection O(i) ^ O(j) (that's supposed to be the symbol for "intersection") a C*valued function g(ij). These "transition functions" must satisfy the compatibility condition
g(ij) g(jk) g(ki) = 1
We say then that the functions g(ij) define a 1cocycle in Cech cohomology  think of this as just jargon, if you like. Note that we will get an isomorphic line bundle if we take some C*valued functions f(i), one on each chart O(i), and multiply g(ij) by f(i)f(j)^{1}. This simply amounts to changing the trivialization of the bundle on each chart. We say that the new Cech cocycle differs by a coboundary. So line bundles are in 11 correspondence with the 1st Cech cohomology with values in sh(C*). This turns out to be the same thing as H^2(M,Z), as noted above.
Now, there is a marvelous thing called a gerbe, which is like a bundle, but is pieced together using Cech 2cocyles! These will be classified by the 2nd Cech cohomology with values in sh(C*), which is nothing but H^3(M,Z).
What are these gerbes? Well, I wish I really understood them. Let me just say what I know. The basic idea is to boost everything up a notch using categorytheoretic thinking. When we were getting ready to define bundles, we needed to have the concept of a group at our disposal (to have a structure group.) For gerbes, we need something called a category of torsors. What is a group? Well, it is a SET equipped with various MAPS satisfying various properties. What is a category of torsors? Well, it is a CATEGORY equipped with various FUNCTORS satisfying utterly analogous properties. Note how we are "categorifying" here. We have more structure, since while a set is just a bunch of naked points, a category is a bunch of points, namely objects, which are connected by arrows, namely morphisms. Given the group C* we can get a corresponding category of torsors as follows: the category of all manifolds with a simply transitive C*action (which are called torsors). A nice account of why this category looks so much like a group appears in
5) Higher algebraic structures and quantization, by Dan Freed, preprint, December 18, 1992, available as hepth/9212115.
which I already mentioned in week12.
Just as a group can act on a set, a category of torsors can act on a category. If we "sheafify" this notion, we get the concept of a gerbe. Clear? Well, part of why I am interested in these ideas is the way they make me a bit dizzy, so don't feel bad if you are a bit dizzy too now. I really think that overcoming this dizziness will be necessary for certain advances in mathematics and physics, though.
Instead of actually coming clean and defining the concept of a gerbe, let me finish by saying what Brylinski does next. He defines an analog of connections on bundles, called "connective structures" on gerbes. And he defines an analog of the curvature, the "curving" of a connective structure. This turns out to give an element of H^3(M,Z) in a natural way. He concludes in a blaze of glory by showing how the Dirac monopole gives a gerbe on S^3 whose curving is the volume form. The integrality condition turns out to be related to Dirac's original argument for quantization of electric charge. Whew!
To wrap up, let me note that the following paper, mentioned in week23, has shown up on grqc:
6) Representation Theory of Analytic Holonomy C* Algebras, by Abhay Ashtekar and Jerzy Lewandowski, to appear in Knots and Quantum Gravity, ed. J. Baez, 42 pages, currently available as grqc/9311010.
Ashtekar and Lewandowski are my friendly competitors in the business of making the loop representation of quantum gravity more rigorous by formalizing the idea of a generalized measure on the space of connections modulo gauge transformations.
© 1993 John Baez
baez@math.removethis.ucr.andthis.edu
