What areas of physics should a mathematician study to understand TQFT?

Question

I am studying topological quantum field theory from the view point of mathematics (axiomatic treatise). So it has no explanation about physics. I would like to know physic background of TQFT. But I have only taken several basic physics classes when I was undergraduate.

Which areas of physics are related to TQFT? (I heard QFT and conformal field theory are closely related. But I don't know how.) If I have graduate level of mathematics background, is it possible to study those related subjects directly? Or should I start studying from the basic physics first, like quantum mechanics or relativity theory?

If you can suggest any reference on TQFT that allows people with mathematics background to easily understand the physics motivation or physics origin of TQFT, that would also be helpful.

Answer 1

When Atiyah wrote down his axioms for a TQFT, he was inspired by similiar axioms that Segal came up with to describe 2 dimensional CFTs. A good explanation of the physical motivation from the axiomatic viewpoint is given in Segal's lectures (he is talking about axioms for QFTs but you will recognize parts of the axioms for TQFTs), but you can also take a look at Atiyah's original paper. Another nice reference is Baez's Prehistory of n-categorical physics or Witten's ICM address.

Topological Quantum Field theories are indeed examples of Quantum Field Theories. Their common characteristic is roughly, that the "time evolution" does only depend on changes in topology. That corresponds to the axiom $Z(M \times [0,1]) = id_{Z(M)}$. A physicist would phrase this as "the Hamiltonian vanishes".

The reason the functor is usually called $Z$ is because it should remind you of "Zustandssumme" the german term for partition function. When a physicist wants to study a problem in statistical physics or quantum field theory (they are related) he often starts by writing down a partition function (also called functional/Feynman/Pathintegral in this context)

$$Z_M[J] = \int_{C(M)} D\phi\; \exp(-S[\phi] + J\phi)$$

where $C(M)$ is some space of "fields" on a fixed manifold $M$. You can think of the axioms as properties that a reasonable partition function should have. The common language of CFT/QFT/TQFT is the language of those functional integrals.

To understand this from a physical perspective you should at least understand some quantum mechanics. I am not sure what good books there are for mathematicians, but I think there has been a question on mathoverflow about that. Then there is of course the two volume set "Quantum Fields and Strings: A course for mathematicians". The notes from which the books were made can still be found online at the ias website.

Answer 2

You can start with Chern-Simons massive electrodynamics. This is 3d electrodynamics (a very physical theory) with the Lagrangian:

$$ S = \int {1\over 4g^2} F^2 + m\epsilon^{ijk} A_i F_{jk} $$

The topological limit is $g\rightarrow\infty$ (drop the usual kinetic term). Witten's Chern-Simons theory, the Jones polynomial theory, is the natural non-Abelian generalization, again in the limit $g\rightarrow\infty$. This is the natural regulated version, and it should be more mathematically well defined than the unregulated topological version.

The toplogical version is singular, because the flux across a knot depends on the knot type, and is discontinuous when the knot moves to cross itself so as to become a different knot. The kinetic term in the above rounds out the singularity, and makes the theory physically definable by regulators. Since it is in 3d, it should have a rigorous version too, although I don't know the work done on this.

Answer 3

One book that I liked was Zee's "Quantum Field Theory In A Nutshell". It's written in a fairly informal manner (well, for a textbook anyway) and assumes that you know a decent amount of mathematics. It's also not an introductory-level quantum book, and it assumes that you've had at least one course in quantum mechanics before - bra-ket notation and such.