
This week I will get back to mathematical physics... but before I do, I can't resist adding that in my talk in that conference I announced that James Dolan and I had come up with a definition of weak ncategories. (For what these are supposed to be, and what they have to do with physics, see "week38", "week49", and "week53".) John Power was at the talk, and before long his collaborator Ross Street sent me some email from Australia asking about the definition. Gordon, Power, and Street have done a lot of work on ncategories  see "week29". Now, Dolan and I have been struggling for several months to put this definition onto paper in a reasonably elegant and comprehensible form, so Street's request was a good excuse to get something done quickly... sacrificing comprehensibility for terseness. The result can be found in
1) John Baez and James Dolan, nCategories, sketch of a definition, http://math.ucr.edu/home/baez/ncat.def.html
A more readable version will appear as a paper fairly soon. I should add that this will eventually be part of Dolan's thesis, and he has done most of the hard work on it; my role has largely been to push him along and get him to explain everything to me.
On to some physics...
First, it's amusing to note that Maxwell's equations are back in fashion! In the following papers you will see how the duality symmetry of Maxwell's equations (the symmetry between electric and magnetic fields) plays a new role in modern work on 4dimensional gauge theory. Also, there is some good stuff in Thompson's review article about the SeibergWitten theory, which is basically just a U(1) gauge theory like Maxwell's equations... but with some important extra twists!
2) Erik Verlinde, Global aspects of electricmagnetic duality, Nuc. Phys. B455 (1995), 211225, available as arXiv:hepth/9506011.
George Thompson, New results in topological field theory and abelian gauge theory, 64 pages, available as arXiv:hepth/9511038.
Next, it's nice to see that work on the loop representation of quantum gravity continues apace:
3) Thomas Thiemann, An account of transforms on (A/G)^bar, available as arXiv:grqc/9511049.
Thomas Thiemann, Reality conditions inducing transforms for quantum gauge field theory and quantum gravity, available as arXiv:grqc/9511057.
Abhay Ashtekar, A generalized Wick transform for gravity, available as arXiv:grqc/9511083.
Renate Loll, Making quantum gravity calculable, preprint available in LaTeX form as arXiv:grqc/9511080.
Rodolfo Gambini and Jorge Pullin, A rigorous solution of the quantum Einstein equations, available as arXiv:grqc/9511042.
The first three papers here discuss some new work tackling the "reality conditions" and "Hamiltonian constraint", two of the toughest issues in the loop representation of quantum gravity. First, the Hamiltonian constraint is another name for the WheelerDeWitt equation
H ψ = 0
that every physical state of quantum gravity must satisfy (see "week11" for why). The "reality conditions" have to do with the fact that this constraint looks different depending on whether we are working with Riemannian or Lorentzian quantum gravity. The constraint is simpler when we work with Riemannian quantum gravity. Classically, in Riemannian gravity the metric on spacetime looks like
dt^{2} + dx^{2} + dy^{2} + dz^{2}
at any point, if we use suitable local coordinates. In this Riemannian world, time is no different from space! In the real world, the world of Lorentzian gravity, the metric looks like
dt^{2} + dx^{2} + dy^{2} + dz^{2}
at any point, in suitable coordinates. Folks often call the Riemannian world the world of "imaginary time", since in some vague sense we can get from the Lorentzian world to the Riemannian world by making the transformation
t → it,
called a "Wick transform". It looks simple the way I have just written it, in local coordinates. But a priori it's far from clear that this Wick transform makes any sense globally. Apparently, however, there is something we can do along these lines, which transforms the Hamiltonian for Lorentzian quantum gravity to the betterunderstood one of Riemannian quantum gravity! Alas, I have been too distracted by ncategories to keep up with the latest work on this, but I'll catch up in a bit. Maybe over Christmas I can relax a bit, lounge in front of a nice warm fire, and read these papers.
The goal of all these machinations, of course, is to find some equations that make mathematical sense, have a good right to be called a "quantized version of Einstein's equation", and let one compute answers to some physics problems. We don't expect that quantum gravity is enough to describe what's really going on in interesting problems... there are lots of other forces and particles out there. Indeed, string theory is founded on the premise that quantum gravity is completely inseparable from the quantum theories of everything else. But here we are taking a different tack, treating quantum gravity by itself in as simple a way as possible, expecting that the predictions of theory will be qualitatively of great interest even if they are quantitatively inaccurate.
As described in earlier Finds ("week55", "week68"), Loll has been working to make quantum gravity "calculable", by working on a discretized version of the theory called lattice quantum gravity. If one does it carefully, it's not too bad to treat space as a lattice in the loop representation of quantum gravity, because even in the continuum the theory is discrete in a certain sense, since the states are described by "spin networks", certain graphs embedded in space. (See "week43" for more on these.) Her latest paper is an introduction to some of these issues.
In a somewhat different vein, Gambini and Pullin have been working on relating the loop representation to knot theory. One of their most intriguing results is that the second coefficient of the AlexanderConway knot polynomial is a solution of the Hamiltonian constraint. Here they argue for this result using a lattice regularization of the theory.
Now let me turn to a variety of other matters...
4) Matt Greenwood and XiaoSong Lin, On Vassiliev knot invariants induced from finite type, available as arXiv:qalg/9506001.
Lev Rozansky, On finite type invariants of links and rational homology spheres derived from the Jones polynomial and Witten ReshetikhinTuraev invariant, available as arXiv:qalg/9511025.
Scott Axelrod, Overview and warmup example for perturbation theory with instantons, available as arXiv:hepth/9511196.
These papers all deal with perturbative ChernSimons theory and its spinoffs. The first two consider homology 3spheres. In ChernSimons theory, this makes the moduli space of flat SU(2) connections trivial, thus eliminating some subtleties in the perturbation theory. A homology 3sphere is a 3manifold whose homology is the same as that of the 3sphere... the first one was discovered by Poincare when he was studying his conjecture that every 3manifold with the homology of a 3sphere is a 3sphere. It turns out that you can get a counterexample if you just take an ordinary 3sphere, cut out a solid torus embedded in the shape of a trefoil knot, and stick it back in with the meridian and longitude (the short way around, and the long way around) switched  making sure they wind up pointing in the correct directions. This is called "doing Dehn surgery on the trefoil". It gives something with the homology of a 3sphere that's not a 3sphere. So Poincare had to revise his conjecture to say that every 3manifold homotopic to a 3sphere is (homeomorphic to) a 3sphere. This improved "Poincare conjecture" remains unsolved... its analog is known to be true in every dimension other than 3! Since every possible counterexample to the Poincare conjecture is a homology 3sphere, it makes some sense to ponder these manifolds carefully.
Now, just as perturbative ChernSimons theory gives certain special invariants of links, said to be of "finite type", the same is true for homology 3spheres. When we say a link invariants is of finite type, all we mean is that it satisfies a simple property described in "week3". There is a similar but subtler definition for an invariant of homology 3spheres to be of finite type; they need to transform in a nice way under Dehn surgery. (See "week60" for more references.)
The second paper concentrates precisely on those subtleties due to the moduli space of flat connections, developing perturbation theory in the presence of "instantons" (here, nontrivial flat connections).
5) Alan Carey, Jouko Mickelsson, and Michael Murray, Index theory, gerbes, and Hamiltonian quantization, available as arXiv:hepth/9511151.
Alan Carey, M. K. Murray and B. L. Wang, Higher bundle gerbes and cohomology classes in gauge theories, available as arXiv:hepth/9511169
Higherdimensional algebra is sneaking into physics in yet another way: gerbs! What's a gerb? Roughly speaking, it's a sheaf of groupoids, but there are some other ways of thinking of them that come in handy in physics. See "week25" for a review of Brylinski's excellent book on gerbs, and also:
6) JeanLuc Brylinski, Holomorphic gerbes and the Beilinson regulator, in Proc. Int. Conf. on KTheory (Strasbourg, 1992), to appear in Asterisque.
JeanLuc Brylinski, The geometry of degreefour characteristic classes and of line bundles on loop spaces I, Duke Math. Jour. 75 (1994), 603638.
JeanLuc Brylinski, Cech cocyles for characteristic classes, J.L. Brylinski and D. A. McLaughlin.
7) Joe Polchinski, Recent results in string duality, available as arXiv:hepth/9511157.
This should help folks keep up with the ongoing burst of work on dualities relating superficially different string theories.
8) Leonard Susskind and John Uglum, String physics and black holes, available as arXiv:hepth/9511227.
Among other things, this review discusses the "holographic hypothesis" mentioned in "week57":
9) Boguslaw Broda, A gaugefield approach to 3 and 4manifold invariants, available in TeX form as arXiv:qalg/9511010.
This summarizes the ReshetikhinTuraev construction of 3d topological quantum field theories from quantum groups, and Broda's own closely related approach to 4d topological quantum field theories.
10) John Baez and Martin Neuchl, Higherdimensional algebra I: braided monoidal 2categories, available as arXiv:qalg/9511013.
In this paper, we begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their applications to 4d topological quantum field theories and 2tangles (surfaces embedded in 4dimensional space). Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2category Z(C) as the `center' of a semistrict monoidal category C. This is analogous to the construction of a braided monoidal category as the center, or `quantum double', of a monoidal category. The idea is to develop algebra that will do for 4dimensional topology what quantum groups and braided monoidal categories did for 3d topology. As a corollary of the center construction, we prove a strictification theorem for braided monoidal 2categories.
© 1995 John Baez
baez@math.removethis.ucr.andthis.edu
