
I've been busy, and papers have been piling up; there are lots of interesting ones that I really should describe in detail, but I had better be terse and list them now, rather than waiting for the mythical day when I will have time to do them justice.
So:
1) B. Durhuus, H. P. Jakobsen and R. Nest, Topological quantum field theories from generalized 6jsymbols, Reviews in Math. Physics 5 (1993), 167.
In "week16" I explained a paper by Fukuma, Hosono and Kawai in which they obtained topological quantum field theories in 2 dimensions starting with a triangulation of a 2d surface. The theories were "topological" in the sense that the final answers one computed didn't depend on the triangulation. One can get between any two triangulations of a surface by using a sequence of the following two moves (and their inverses), called the (2,2) move:
O O /\ / \ /  \ / \ /  \ / \ O  O <> OO \  / \ / \  / \ / \/ \ / O O
and the (3,1) move:
O O /\ / \ /  \ / \ /  \ / \ /  \ / \ / _O_ \ <> / \ / _/ \_ \ / \ / _/ \_ \ / \ /_/ \_\ / \ OO OO
Note that in either case these moves amount to replacing one part of the surface of a tetrahedron with the other part! In fact, similar moves work in any dimension, and they are often called the Pachner moves.
The really wonderful thing is that these moves are also very significant from the point of view of algebra... and especially what I call "higherdimensional algebra" (following Ronnie Brown), in which the distinction between algebra and topology is largely erased, or, one might say, revealed for the sham it always was.
For example, as explained more carefully in "week16", the (2,2) move is really just the same as the associative law for multiplication. The idea is that we are in a 2dimensional spacetime, and a triangle represents multiplication: two "incoming states" go in two sides and their product, the "outgoing state", pops out the third side:
O / \ / \ / \ A B / \ / \ / \ / \ OABO
Then the (2,2) move represents associativity:
O O /\ / \ A  (AB)C A A(BC) /  \ / \ O AB O <> OBCO \  / \ / B  C B C \/ \ / O O
Of course, the distinction between "incoming" and "outgoing" sides of the triangle is conventional, and the more detailed explanation in "week16" shows how that fits into the formalism. Roughly speaking, what we have is not just any old algebra, but an algebra that, thought of as a vector space, is equipped with an isomorphism between it and its dual. This isomorphism allows us to forget whether we are coming or going, so to speak.
Hmm, and here I was planning on being terse! Anyway, the still more interesting point is that when we think about 3dimensional topology and "3dimensional algebra," we should no longer think of
O O /\ / \ /  \ / \ /  \ / \ O  O and OO \  / \ / \  / \ / \/ \ / O O
as representing equal operations (the 3fold multiplication of A, B, and C); instead, we should think of them as merely isomorphic, with the tetrahedron of which they are the front and back being the isomorphism. The basic philosophy is that in higherdimensional algebra, as one ascends the ladder of dimensions, certain things which had been regarded as equal are revealed to be merely isomorphic. This gets tricky, since certain isomorphisms that were regarded as equal at one level are revealed to be merely isomorphic at the next level... leading us into a subtle world of isomorphisms between isomorphisms between isomorphisms... which the theory of ncategories attempts to systematize. (I should note, however, that in the particular case of associativity this business was worked out by Jim Stasheff quite a while back: it's the homotopy theorists who were the ones with the guts to deal with such issues first.)
Now, it turns out that in 3dimensional algebra, the isomorphism corresponding to the (2,2) move is not something marvelously obscure. It is in fact precisely what physicists call the "6j symbol", a gadget they've been using to study angular momentum in quantum mechanics for a long time! In quantum mechanics, the study of angular momentum is just the study of representations of the group SU(2), and if one has representations A, B, and C of this group (or any other), the tensor products (A tensor B) tensor C and A tensor (B tensor C) are not equal, but merely isomorphic. It should come as no surprise that this isomorphism is represented by physicists as a big gadget with 6 indices dangling on it, the "6j symbol".
Quite a while back, Regge and Ponzano tried to cook up a theory of quantum gravity in 3 dimensions using the 6j symbols for SU(2). More recently, Turaev and Viro built a 3dimensional topological quantum field theory using the 6jsymbols of the quantum group SU_{q}(2), and this led to lots of work, which the above article explains in a distilled sort of way.
The original PonzanoRegge and TuraevViro papers, and various other ones clarifying the relation of the Turaev/Viro theory to quantum gravity in spacetimes of dimension 3, are listed in "week16". It's also worth checking out the paper by Barrett and Foxon listed in "week24", as well as the following paper, for which I'll just quote the abstract:
2) Timothy J. Foxon, Spin networks, TuraevViro theory and the loop representation, available as grqc/9408013.
We investigate the PonzanoRegge and TuraevViro topological field theories using spin networks and their qdeformed analogues. I propose a new description of the state space for the TuraevViro theory in terms of skein space, to which qspin networks belong, and give a similar description of the PonzanoRegge state space using spin networks. I give a definition of the inner product on the skein space and show that this corresponds to the topological inner product, defined as the manifold invariant for the union of two 3manifolds. Finally, we look at the relation with the loop representation of quantum general relativity, due to Rovelli and Smolin, and suggest that the above inner product may define an inner product on the loop state space.
(Concerning the last point I cannot resist mentioning my own paper on knot theory and the inner product in quantum gravity, available as tang.tex.)
In addition to the papers by TuraevViro and FukumaShapere listed in "week16", there are some other papers on Hopf algebras and 3d topological quantum field theories that I should list:
3) Greg Kuperberg, Involutory Hopf algebras and threemanifold invariants, Internat. Jour. Math 2 (1991), 4166.
A definition of #(M,H) in the noninvolutory case, by Greg Kuperberg, unpublished.
Greg Kuperberg is one of the few experts on this subject who is often found on the net; he is frequently known to counteract my rhetorical excesses with a dose of precise information. The above papers, one of which is sadly still unpublished, make it beautifully clear how "algebra knows more about topology than we do", since various basic structures on Hopf algebras have a pleasant tendency to interact just as needed to give 3d topological quantum field theories.
4) John W. Barrett and Bruce W. Westbury, Spherical categories, available as hepth/9310164.
John W. Barrett and Bruce W. Westbury, Invariants of piecewiselinear 3manifolds, Trans. Amer. Math. Soc. 348 (1996), 39974022. Also available as hepth/9311155.
John W. Barrett and Bruce W. Westbury, The equality of 3manifold invariants, available as hepth/9406019.
Let me quote the abstract for the first one; the second one gives a construction of 3manifold invariants, and the third shows that the authors' 3manifold invariants agree with Kuperberg's when both are defined.
This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6jsymbols as used in topological field theories.We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963). In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical.
In the third section we define spherical Hopf algebras so that the category of representations is spherical. Examples of spherical Hopf algebras are involutory Hopf algebras and ribbon Hopf algebras. Finally we study the natural quotient in these cases and show it is semisimple.
5) Louis H. Kauffman and David E. Radford, Invariants of 3Manifolds derived from finite dimensional Hopf algebras, by available as hepth/9406065.
This is paper also relates 3d topology and certain finitedimensional Hopf algebras, and it shows they give 3manifold invariants distinct from the more famous ones due to Witten (and a horde of mathematicians). I have not had time to think about how they relate to the above ones, but I have a hunch that they are the same, since all of them make heavy use of special grouplike elements associated to the antipode.
6) Louis Crane and Igor Frenkel, Four dimensional topological quantum field theory, Hopf categories, and the canonical bases, available as hepth/9405183.
Work in 4 dimensions is, as one expect, still more subtle than in 3, since again various things that were equalities becomes isomorphisms. In particular, this means that various things one thought were vector spaces  which are sets that have elements that you can add and multiply by numbers, and which satisfy equations like
A + B = B + A
are now reinterpreted as "2vector spaces", which are categories that have objects that you can direct sum and tensor with vector spaces, and which have certain natural isomorphisms like the isomorphism
A ⊕ B ≅ B ⊕ A.
In particular, using Lusztig's canonical basis, Crane and Frenkel start with quantum groups (which are Hopf algebras of a certain sort) and build marvelous "Hopf categories" out of them. While they do not construct a 4d TQFT in this paper, they indicate the game plan in terms clear enough that they will probably now have to race other workers in the field to see who can get the first interesting 4d TQFT... or perhaps something a bit subtler than a 4d TQFT (e.g. Donaldson theory).
Finally, let me turn to a subject that is closely related (though unfortunately this has not yet been made sufficiently clear), namely, holonomy algebras and the loop representation of quantum gravity. Let me simply list the references now; many of these papers were discussed at my session on knots and quantum gravity at the Marcel Grossman conference, so I promise to explain at some later time (and in some papers I'm writing) a bit more about how the loop representation of a gauge theory is interesting from the viewpoint of higherdimensional algebra!
7) A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann, A manifestly gaugeinvariant approach to quantum theories of gauge fields, contribution to the Cambridge meeting proceedings, available as hepth/9408108.
Jerzy Lewandowski, Topological measure and graphdifferential geometry on the quotient space of connections, Proceedings of "Journees Relativistes 1993", available as grqc/9406025.
Abhay Ashtekar, Donald Marolf and Jose Mourao, Integration on the space of connections modulo gauge transformations, available as grqc/9403042.
A. Ashtekar and R. Loll, New loop representations for 2+1 gravity, available as grqc/9405031.
R. Loll, Independent loop invariants for 2+1 gravity, available as grqc/9408007.
R. Loll, J.M. Mourão and J.N. Tavares, Generalized coordinates on the phase space of YangMills theory, available as grqc/9404060.
C. Di Bartolo, R. Gambini and J. Griego, The extended loop representation of quantum gravity, available as grqc/9406039.
Rodolfo Gambini, Alcides Garat and Jorge Pullin, The constraint algebra of quantum gravity in the loop representation, available as grqc/9404059.
© 1994 John Baez
baez@math.removethis.ucr.andthis.edu
