April 26, 1997

This Week's Finds in Mathematical Physics (Week 103)

John Baez

As I segue over from the homotopy theory conference at Northwestern University to the conference on higher-dimensional algebra and physics that took place right after that, it's a good time to mention Ronnie Brown's web page:

1) Ronald Brown, Higher-dimensional group theory, http://www.bangor.ac.uk/~mas010/home.html

Brown is the one who coined the phrase "higher-dimensional algebra", and for many years he has been developing this subject, primarily as a tool for doing homotopy theory. I wrote a bit about his ideas two years ago, in "week53". A lot has happened in higher-dimensional algebra since then, and the web page above is a good place to get an overview of it. It includes a nice bibliography on the subject. Also, if you find the math a bit strenuous, you can rest your brain and delight your eyes at the related site:

2) Symbolic sculptures and mathematics, http://www.bangor.ac.uk/SculMath/

which opens with a striking image of rotating linked tori, and includes pictures of the mathematical sculpture of John Robinson.

The Workshop on Higher Category Theory and Physics was exciting because it pulled together a lot of people working on the interface between these two subjects, many of whom had never before met. It was organized by Ezra Getzler and Mikhail Kapranov. Getzler is probably most well-known for his proof of the Atiyah-Singer index theorem. This wonderful theorem captured the imagination of mathematical physicists for many years starting in the 1960s. The reason is that it relates the topology of manifolds to the the solutions of partial differential equations on these manifolds, and thus ushered in a new age of applications of topology to physics. In the 1980s, working with ideas that Witten came up with, Getzler found a nice "supersymmetric proof" of the Atiyah-Singer theorem. Later Getzler turned to other things, such as the use of "operads" (see "week42") to study conformal field theory (which shows up naturally in string theory). Kapranov has also done a lot of work with operads and conformal field theory, and many other things, but I first learned about him through his paper with Voevodsky on "braided monoidal 2-categories" (see "week4"). This got me very excited since it turned me on to many of the main themes of n-category theory.

Alas, my description of this fascinating conference will be terse and dry in the extreme, since I am flying to Warsaw in 3 hours for a quantum gravity workshop. I'll just mention a few papers that cover some of the themes of this conference. Ross Street gave two talks on Batanin's definition of weak n-categories (and even weak ω-categories), which one can get as follows:

3) Ross Street, The role of Michael Batanin's monoidal globular categories. Lecture I: Globular categories and trees. Lecture II: Higher operads and weak ω-categories. Available in gzipped Postscript form at http://www.math.mq.edu.au/~street/Publications.html

Subsequently Batanin has written a more thorough paper on his definition:

4) Michael Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories, Adv. Math 136 (1998), 39-103, preprint available at http://www-math.mpce.mq.edu.au/~mbatanin/papers.html

I gave a talk on Dolan's and my definition of weak n-categories, which one can get as follows:

5) John Baez, An introduction to n-categories, to appear in the proceedings of Category Theory and Computer Science '97, preprint available as q-alg/9705009 or in Postscript form at http://math.ucr.edu/home/baez/ncat.ps

Unfortunately Tamsamani was not there to present his definition of weak n-categories, but at least I have learned how to get his papers electronically:

6) Zouhair Tamsamani, Sur des notions de $\infty$-categorie et $\infty$-groupoide non-strictes via des ensembles multi-simpliciaux, preprint available as alg-geom/9512006.

Zouhair Tamsamani, Equivalence de la theorie homotopique des n-groupoides et celle des espaces topologiques n-tronques, preprint available as alg-geom/9607010.

Also, Carlos Simpson has written an interesting paper using Tamsamani's definition:

7) Carlos Simpson, A closed model structure for n-categories, internal Hom, n-stacks and generalized Seifert-Van Kampen, preprint available as alg-geom/9704006.

In a different but related direction, Masahico Saito discussed a paper with Scott Carter and Joachim Rieger in which they come up with a nice purely combinatorial description of all the ways to embed 2-dimensional surfaces in 4-dimensional space:

8) J. Scott Carter, Joachim H. Rieger and Masahico Saito, A combinatorial description of knotted surfaces and their isotopies, to appear in Adv. Math., preprint available at http://www.math.usf.edu/~saito/home.html

My student Laurel Langford has translated their work into n-category theory and shown that "unframed unoriented 2-tangles form the free braided monoidal 2-category on one unframed self-dual object":

9) John Baez and Laurel Langford, 2-Tangles, preprint available as q-alg/9703033 and in Postscript form at http://math.ucr.edu/home/baez/2tang.ps

This paper summarizes the results; the proofs will appear later.

While I was there, Carter also gave me a very nice paper he'd done with Saito and Louis Kauffman. This paper discusses 4-manifolds and also 2-dimensional surfaces in 3-dimensional space, again getting a purely combinatorial description which is begging to be translated into n-category theory:

10) J. Scott Carter, Louis H. Kauffman and Masahico Saito, Diagrammatics, singularities, and their algebraic interpretations, preprint available at http://www.math.usf.edu/~saito/home.html

I am sorry not to describe these papers in more detail, but I've been painfully busy lately. (In fact, I am trying to figure out how to reform my life to give myself more spare time. I think the key is to say "no" more often.)

Thanks to Justin Roberts for pointing out an error in "week102". The phase ambiguity in conformal field theories is not necessarily a 24th root of unity; it's exp(2 π i c / 24) where c is the central charge of the associated Virasoro representation. This is a big hint as far as my puzzle goes.

Also I thank Dan Christensen for helping me understand π_4(S^2) in a simpler way, and Scott Carter for a fascinating letter on the themes of "week102". Alas, I have been too busy to reply adequately to these nice emails!

Gotta run....

© 1997 John Baez