Impressions of Topological field theories in mathematics

Question

There have been recent results in mathematics regarding Conformal field theories and topological field theories. I am curious about the reaction to such results in the physics community. So I guess a starting place for a question would be something like: Did Atiyah's axiomatization of TQFT capture what physicists feel are important in QFT's? How many more parameters are there to what a physicist would call a QFT or FT in general?

Perhaps I should mention that mathematicians (the ones who dont know any phsyics) think about various Field Theories as things that assign invariants to manifolds that should be sensitive to certain data (depending on the word proceeding field). A lot has been done when you only care about the basic topology of the Manifold you are evalutating your field theory at. Have mathematicians gotten rid of too much for physicists to care? What types of things do physicists want a Field theory to keep track of, what kind of structure on the manifold that is?

Answer 1

The work being done on TQFT by physicists and mathematicians is wonderful, but in no way should you think it somehow captures what is important in QFT to physicists trying to explain the real world. QFT as applied to the real world has particle-like excitations, non-trivial correlation functions depending on the spacetime distance between operators, spontaneous symmetry breaking, non-trivial renormalization group flows and so on and calculations are done in 4d flat Minkoswki space because this is an excellent approximation to the local spacetime metric. TQFT, as the name implies, captures information about the topology of manifolds and would be completely boring on $R^{3,1}$. You can't use it to study pion scattering, or compute the short distance interaction between quarks, or the production cross-section for the Higgs boson or really anything that particle theorists do with QFT. So the answer to your first question is no. The axioms of TQFT do not capture what physicists feel is important in QFT. It isn't really a question of adding more parameters, they are just very different beasts. To your question of whether mathematicians have gotten rid of too much for physicists to care, the answer is yes, for most physicists. As I said earlier, for all practical purposes on can do particle theory on flat Minkowski space and the only structure that really matters is the causal structure. None of this is meant to denigrate work on TQFT of course.

Answer 2

Hold on there now: don't be too hasteful in declaring TQFT's some artificial construct only benificial for abstract mathematics. This is simply not true, as there are examples of physical systems exhibiting behavior which is encaptured by the phyics of a TQFT!

First things firsts: it's true that the model which physicists use for elementary particle physics does not use any topological field theories. This is the standard model, and it is based on a large construct of 4D QF Theories, such as QED, Yang-Mills theory, the Higgs boson, Dirac Field theory and so on. You can obtain properties of single particles, or the scattering amplitidues of a bunch of particles, etc.

However, in condensed matter physics, such as metals, insulators, cold atom gases, lattice systems, you deal with not one or a few particles, but with many, many (>10^10 --- 10^26) particles. Due to the impossibility of even attempting to write something that looks like a solution of the theory, starting from first principles, is simply impossible and highly ineffecient. Instead, one tries to come up with an effective model which encompasses the essence of the many-body problem. This is standard practice in condensed matter physics and frequently leads to an effective quantum field theory, which has little resemblence with the basic interactions of the original constituents (try to apply QED to an ionic lattice with electrons flowing around. Can you derive the electronic transport properties of such a system? The answer is: no, it's far too complex).

What does this elaboration have anything to do with topological quantum field theories? The reason is simple: some systems are effectively described by a TQFT! Namely, the fractional quantum Hall effect. This effect arises as follows: two types of material are grown on top of each other. An interface between the two layers arises, and this interface "confines" the motion electrons. The electrons can only move in two out of three spatial dimensions. They are effectively living in a two-dimensional world.

Without explaning what the quantum Hall effect actually is, I will instead tell you that the effective field theory associated with this effect is a Chern-Simons theory -- an example of a TQFT.

Other examples exist as well. Topological insulators are likely to be connected to TQFT's as well (BF theories). I'm sure other examples follow.

The various links with CFT, quantum groups, knot theory, etc all have left footprints in this field. I'm sure there is plenty of "translation" of mathematical results to the physical implications left undone. The "invariants" you are talking about pop up as very physical observables. For instance, it's known since the 80's that the so-called conductivity of the fractional quantum Hall effect is nothing more but a certain Chern number.

Answer 3

I think that, as always, there is a significant disconnect between the math and physics camps. However, physicists probably realize that the topological partition function of the B-model at genus g is not easy to calculate and doesn't have a good mathematical definition, in the holomorphic limit or not. That is, they would be likely to agree that mathematicians like Costello have done valuable work in trying to nail down this elusive quantity.

Physicists working in topological field theory are pretty comfortable with the Atiyah-Segal axioms and with theories of various depths/levels. Kapustin, for example, has spoken widely about these points (categories and 2-categories of branes).