January 14, 1994

This Week's Finds in Mathematical Physics (Week 29)

John Baez

I'm awfully busy this week, but feel like attempting to keep up with the pile of literature that is accumulating on my desk, so this will be a rather terse description of papers. All of these papers are related to my current obsession with "higher-dimensional algebra" and its applications to physics.

1) On algebras and triangle relations, by Ruth J. Lawrence, to appear in Proc. Top. & Geom. Methods in Field Theory (1992), eds. J. Mickelsson and O. Pekonen, World Scientific, Singapore.

A presentation for Manin and Schechtman's higher braid groups, by R. J. Lawrence, available as MSRI preprint 04129-91.

Triangulations, categories and extended topological field theories, by R. J. Lawrence, to appear in Quantum Topology, eds L. Kauffman and R. Baadtrio, World Scientific, Singapore, 1993.

Algebras and triangle relations, by R. J. Lawrence, Harvard U. preprint.

Many people are busily trying to extend the remarkable relationship between knot theory and physics, which is essentially a feature of 3 dimensions, to higher dimensions. Since the 3-dimensional case required the development of new branches of algebra (namely, quantum groups and braided tensor categories), it seems that the higher-dimensional cases will require still further "higher-dimensional algebra." One approach, which is still being born, involves the use of "n-categories," which are generalizations of braided tensor categories suited for higher- dimensional physics. (See for example the papers by Crane in "week2," by Kapranov and Voevodsky in "week4," by Fischer and Freed (separately) in week12, and the one by Gordon, Power, and Street below.) Lawrence has instead chosen to invent "n-algebras," which are vector spaces equipped with operations corresponding to the ways one can subdivide (n-1)-dimensional simplices into more such simplices. (See the paper by Chung, Fukuma and Shapere in "week16" for some of the physics motivation here.)

These alternative approaches should someday be seen as different aspects of the same thing, but there as yet I know of no theorems to this effect, so there is a lot of work to be done. Even more importantly, there is a lot of work left to be done about inventing examples of these higher-dimensional structures. For example, there may eventually be general results on "boosting" n-algebras to (n+1)-algebras, or n-categories to (n+1)-categories, which will explain how generally covariant physics in n-dimensional spacetime relates to the same thing in one higher dimension. So far, however, all we have is a few examples, which are not even clearly related to each other. For example, Crane calls this boosting process "categorification" and has done it starting with the braided tensor category of representations of a quantum group. Lawrence, on the other hand, shows how to construct some 3-algebras from quantum groups. And Freed has given a general procedure for "boosting" using path integral methods that are not yet rigorous in the most interesting cases.

2) Coherence for tricategories, by R. Gordon, A. J. Power, and R. Street, preprint, 81 pages.

An "n-category" is a kind of algebraic structure that has "objects," "morphisms" between objects, "2-morphisms" between morphisms, and so on up to "n-morphisms." However, the correct definition of an n-category for the purposes of physics is still unclear! I gave a rough explanation of the importance of 2-categories in physics in week4, where I discussed Kapranov and Voevodsky's nice definition of braided tensor 2-categories. However, it seems likely that we will need to understand the situation for larger n as well. This paper makes a big step in this direction, by defining "tricategories" (what I might call "weak 2-categories") and proving a "strictification" or "coherence" result analogous to the result that every braided tensor category is equivalent to a "strict" one. The result is, however, considerably more subtle, as it involves a special way of defining the tensor product of 2-categories due to Gray:

3) Formal Category Theory: Adjointness for 2-categories, by John W. Gray, Lecture Notes in Mathematics 391, Springer-Verlag, New York, 1974.

Coherence for the tensor product of 2-categories, and braid groups, in Algebras, Topology, and Category Theory, eds. A. Heller and M. Tierney, Academic Press, New York, 1976, pp. 63-76.

Briefly speaking, Gordon-Power-Street use a category they call "Gray," the category of all 2-categories, made into a symmetric monoidal closed category using a modified version of Gray's tensor product. Then they show that every tricategory (as defined by them) is "triequivalent" to a category enriched over Gray.

4) On pentagon and tetrahedron equations, by J. M. Maillet, preprint available in LaTeX form as hep-th/9312037.

Maillet shows how to obtain solutions of the tetrahedron equations from solutions of pentagon equations; all these geometrical equations are part of the theory of 2-categories, and this is yet another example of a "boosting" construction as alluded to above.

5) Homologically twisted invariants related to (2+1)- and (3+1)-dimensional state-sum topological quantum field theories, by David N. Yetter, preprint, 6 pages, available in LaTeX form as hep-th/9311082.

Let me simply quote the abstract: "Motivated by suggestions of Paolo Cotta-Ramusino's work at the physical level of rigor relating BF theory to the Donaldson polynomials, we provide a construction applicable to the Turaev/Viro and Crane/Yetter invariants of a priori finer invariants dependent on a choice of (co)homology class on the manifold." The dream is that this would give a state-sum formula for the Donaldson polynomials, but Yetter is careful to avoid claiming this! A while back, Crane and Yetter showed how to get 4-dimensional TQFTs from certain 3d TQFTs by another kind of "boosting" procedure related to those mentioned above, but the resulting TQFT in 4-dimensions did not by itself yield interesting new invariants of 4-manifolds. The procedure Yetter describes here generalizes the earlier work by allowing the inclusion of an embedded 2-manifold.

© 1994 John Baez