July 15, 1994

This Week's Finds in Mathematical Physics (Week 36)

John Baez

I am attempting to keep my nose to the grindstone these days, in part since I'm getting ready for the Knots and Quantum Gravity session of the Marcel Grossman meeting on general relativity, which will take place at Stanford the week after next (I will report any interesting news I hear out there), and in part to make up for earlier stretches of laziness on my part.... Nonetheless, I feel I should describe a few new papers on topological quantum field theories.

The real reason for physicists' interest in topological quantum field theories (TQFTs) is that we sorely need a mathematical framework that quantum gravity might fit into. It's likely, however, that quantum gravity won't be much like the TQFTs people have studied so far. The existing TQFTs tend to be "exactly soluble" and have finite-dimensional state spaces; quantum gravity is likely to be different. Still, any experience in studying quantum field theories that don't rely on "fixed background structures" like a fixed spacetime metric is likely to be worth having. Also, quantum gravity appears to be tied mathematically to simpler TQFTs in a variety of ways. In particular, the Ashtekar formulation of quantum gravity is closely related to a 4-dimensional TQFT variously known as "B wedge F theory," "BF theory," "topological 2-form gravity" or "topological quantum gravity". This in turn is closely related to Chern-Simons theory in 3 dimensions.

Let me just say what the heck BF theories are, and then list a few references on them. The easiest way to describe them is by giving the Lagrangian. Say spacetime is an n-dimensional orientable manifold M and we have a principal G-bundle E over M, where G is a Lie group whose Lie algebra is equipped with an invariant trace on it. The two fields in BF theory are a connection A on E --- which we can think of locally as a Lie(G)-valued one-form --- and a Lie(G)-valued (n-2)-form called B. If F denotes the curvature of A, which is a Lie(G)-valued 2-form, we can take the wedge product B wedge F and get a Lie(G)-valued n-form, which gives the Lagrangian

                      tr(B wedge F),

an n-form we can integrate over M to get the action. Since we don't need any metric to integrate an n-form over our spacetime M, this action is "generally covariant". (Note also that it's gauge-invariant.) If we vary B and F to get the classical equations of motion, varying B gives us

                         F = 0,

that is, the connection A is flat, and varying A gives us

                        d_A B = 0,

that is, the exterior covariant derivative of B vanishes.

In 4 dimensions we can soup up our BF theory a bit by adding terms proportional to

                     tr(B wedge B)


                     tr(F wedge F) 

to the Lagrangian. If we take as our Lagrangian

              tr(B wedge F) + a tr(B wedge B) + b tr(F wedge F), 

the third term, when integrated over M, is proportional to an invariant called the second Chern class of E, that is, it's independent of the connection A, so it really doesn't affect the equations of motion at all! In a sense it's utterly useless. The second term does something, though; our equations of motion become

                     F = -2aB,  d_A B = 0.

Note that if we plug the first equation into the Lagrangian, we get that for solutions of the equations of motion, the action is a constant times the second Chern class (if a is nonzero).

Horowitz showed, in this four-dimensional case, that if a is nonzero, there is a single state of the quantum version of BF theory when spacetime has the form R x S (S being some oriented 3-manifold), and that this state, thought of as a wavefunction on the space of connections on S, is just the exponential of the Chern-Simons action. This is how Chern-Simons theory gets into the game.

Moreover, Baulieu and Singer showed that if you take the boring-looking "FF theory" with Lagrangian tr(F wedge F), and quantize it using the BRST approach, you get something that Witten proved was closely related to Donaldson theory --- an invariant of 4-manifolds. So there seems to be a relation between this stuff and Donaldson theory. It is a rather mysterious one as far as I'm concerned, though, because it seems you could just as well have used zero as a Lagrangian, rather than tr(F wedge F), and done the same things Baulieu and Singer did. (At least, that's how it seems to me, and I got Scott Axelrod to agree with me on that yesterday.) In other words, Donaldson theory seems to have more to do with the geometry of the space of connections on M than it does with the "FF" Lagrangian per se. But still, there are other relationships between Donaldson theory and Chern-Simons theory (which I don't understand well enough to want to discuss), so perhaps it is not too silly to say that BF theory is related to Donaldson theory in some poorly understood manner.

Now for some references: the reference that got me started on these was

Exactly soluble diffeomorphism-invariant theories, by Gary Horowitz, Comm. Math. Phys., 125 (1989) 417-437. (Listed in "week19")

I got more interested in them when I read

BF Theories and 2-knots, by Paolo Cotta-Ramusino and Maurizio Martellini, to appear in Knots and Quantum Gravity, ed. J. Baez, Oxford U. Press. (Listed in "week23")

which indicated that BF theories may give invariants of surfaces embedded in 4-dimensional manifolds, much as Chern-Simons theory gives invariants of knots in 3-dimensional manifolds. Actually, BF theories make sense in any dimension, and in dimension 3 they appear to give knot invariants, including the Alexander-Conway polynomial:

1) Three-dimensional BF theories and the Alexander--Conway invariant of knots, by A. S. Cattaneo, P. Cotta-Ramusino, and M. Martellini; 32 pages in LaTeX format (figures available upon request), available as hep-th/9407070.

Another nice-looking paper on BF theories is the following:

2) B^F theory and flat spacetimes, by Henri Waelbroeck, 21 pages in LaTeX format, available as gr-qc/9311033.

Waelbrock also has a recent paper with Zapata on a Hamiltonian formulation of the theory on a lattice:

3) A Hamiltonian formulation of topological gravity, by Henri Waelbrock and J. A. Zapata, 15 pages available in LaTeX format as gr-qc/9311035.

The paper by Baulieu and Singer relating FF theory to Donaldson theory is:

4) Topological Yang-Mills symmetry, by L. Baulieu and I. M. Singer, Nucl. Phys. (Proc. Suppl.) B5 (1988) 12-19.

BF theory in 2 dimensions is also called "topological Yang-Mills theory", and it's discussed in various places, including

5) On quantum gauge theories in two dimensions, by Edward Witten, Comm. Math. Phys. 141 (1991) 153-209.


6) Topological gauge theories of antisymmetric tensor fields, by M. Blau and G. Thompson, Ann. Phys. 205 (1991) 130-172.

© 1994 John Baez