December 3, 1995

This Week's Finds in Mathematical Physics (Week 71)

John Baez

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 n-categories. (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 n-categories - 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, n-Categories, sketch of a definition,

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 4-dimensional gauge theory. Also, there is some good stuff in Thompson's review article about the Seiberg-Witten 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 electric-magnetic duality, Nuc. Phys. B455 (1995), 211-225, available as arXiv:hep-th/9506011.

George Thompson, New results in topological field theory and abelian gauge theory, 64 pages, available as arXiv:hep-th/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:gr-qc/9511049.

Thomas Thiemann, Reality conditions inducing transforms for quantum gauge field theory and quantum gravity, available as arXiv:gr-qc/9511057.

Abhay Ashtekar, A generalized Wick transform for gravity, available as arXiv:gr-qc/9511083.

Renate Loll, Making quantum gravity calculable, preprint available in LaTeX form as arXiv:gr-qc/9511080.

Rodolfo Gambini and Jorge Pullin, A rigorous solution of the quantum Einstein equations, available as arXiv:gr-qc/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 Wheeler-DeWitt 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

dt2 + dx2 + dy2 + dz2

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

-dt2 + dx2 + dy2 + dz2

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 better-understood one of Riemannian quantum gravity! Alas, I have been too distracted by n-categories 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 Alexander-Conway 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 Xiao-Song Lin, On Vassiliev knot invariants induced from finite type, available as arXiv:q-alg/9506001.

Lev Rozansky, On finite type invariants of links and rational homology spheres derived from the Jones polynomial and Witten- Reshetikhin-Turaev invariant, available as arXiv:q-alg/9511025.

Scott Axelrod, Overview and warmup example for perturbation theory with instantons, available as arXiv:hep-th/9511196.

These papers all deal with perturbative Chern-Simons theory and its spinoffs. The first two consider homology 3-spheres. In Chern-Simons theory, this makes the moduli space of flat SU(2) connections trivial, thus eliminating some subtleties in the perturbation theory. A homology 3-sphere is a 3-manifold whose homology is the same as that of the 3-sphere... the first one was discovered by Poincare when he was studying his conjecture that every 3-manifold with the homology of a 3-sphere is a 3-sphere. It turns out that you can get a counterexample if you just take an ordinary 3-sphere, 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 3-sphere that's not a 3-sphere. So Poincare had to revise his conjecture to say that every 3-manifold homotopic to a 3-sphere is (homeomorphic to) a 3-sphere. 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 3-sphere, it makes some sense to ponder these manifolds carefully.

Now, just as perturbative Chern-Simons theory gives certain special invariants of links, said to be of "finite type", the same is true for homology 3-spheres. 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 3-spheres 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:hep-th/9511151.

Alan Carey, M. K. Murray and B. L. Wang, Higher bundle gerbes and cohomology classes in gauge theories, available as arXiv:hep-th/9511169

Higher-dimensional 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) Jean-Luc Brylinski, Holomorphic gerbes and the Beilinson regulator, in Proc. Int. Conf. on K-Theory (Strasbourg, 1992), to appear in Asterisque.

Jean-Luc Brylinski, The geometry of degree-four characteristic classes and of line bundles on loop spaces I, Duke Math. Jour. 75 (1994), 603-638.

Jean-Luc Brylinski, Cech cocyles for characteristic classes, J.-L. Brylinski and D. A. McLaughlin.

7) Joe Polchinski, Recent results in string duality, available as arXiv:hep-th/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:hep-th/9511227.

Among other things, this review discusses the "holographic hypothesis" mentioned in "week57":

9) Boguslaw Broda, A gauge-field approach to 3- and 4-manifold invariants, available in TeX form as arXiv:q-alg/9511010.

This summarizes the Reshetikhin-Turaev 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, Higher-dimensional algebra I: braided monoidal 2-categories, available as arXiv:q-alg/9511013.

In this paper, we begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their applications to 4d topological quantum field theories and 2-tangles (surfaces embedded in 4-dimensional space). Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories, 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 2-category 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 4-dimensional 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 2-categories.

© 1995 John Baez