(How) Does TQFT actually describe a physical theory?

  • #1

Main Question or Discussion Point

So I have been reading '2D Topological Quantum Field Theory and Frobenius Algebras' by Joachim Kock recently and I couldn't help but wonder, how is this related to physics? I'm currently in the first chapter and he defined a TQFT as a monoidal functor. Now this seems somewhat abstract (which I appreciate, since I'm a math major) but I was left wondering, how does this tie back to physics?
I understand that topological quantum field theory is now a 'mathematical' theory but I've discussed about QFT with physicists and it looks nothing like what I've been learning. I want to become a mathematical physicist and I was wondering if someone could give me a book/paper to get some insight in the more physical side of things? I don't know much physics beyond lagrangian mechanics and basic quantum mechanics so maybe this is out of reach. A simple explanation would go a long way too!
Thanks.
 

Answers and Replies

  • #2
king vitamin
Science Advisor
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In many condensed matter systems, the low-energy part of the theory may be described by an effective quantum field theory. Even though these systems are made up of a bunch of electrons and ions and the only interaction at play between them is electromagnetism, depending on the detailed interactions you can get a huge variety of interesting field theories describing the physics at low energy.

As I'm sure you can guess I'm going with this, given your question, it is actually possible for the low-energy field theory of a condensed matter system to be a TQFT. This is one of the most remarkable possibilities for the low-energy physics of a system, because having a QFT involving some "conventional" matter (like electrons) coupled to the gauge fields of a TQFT means that the actual charge/spin/statistics of the excitations are fractionalized, meaning they do not have the same quantum numbers as the "conventional" fields if they were not coupled to the gauge field.

The most celebrated material of this type is the fractional quantum Hall systems. The most simple of these are the Laughlin states, which are described by a level-k Chern-Simons theory (where k must be an odd integer for reasons having to do with fractional quantum Hall systems being made up of electrons). Even though these systems are essentially just electrons confined to 2D with a strong magnetic field pointed perpendicular to the plane, the lowest-energy excitations of these systems ("Laughlin quasiparticles/quasiholes") have electric charge e/k, and the wave function picks up a phase of pi/k under the exchange of two of these particles (recall that bosons and fermions pick up phases of 2pi and pi respectively.). But there are many interesting models (which may be related to materials, or perhaps simulated with cold atoms) which realize various other TQFTs.

It's a little tough to recommend a book, because most books I know which cover this stuff are many-body physics books which might be hard to jump into. Xiao-Gang Wen was the founder of explicitly using TQFTs in describing these phases, so his many-body textbook might be helpful. I also cannot resist linking one of my favorite papers, "Superconductors are topologically ordered" (https://arxiv.org/abs/cond-mat/0404327), which shows that actually the superconducting state seen in experiments since the early 1900s are actually also described by a TQFT at low energies. (The term "topological order," coined by Wen, means emergent gauge fields in the low energy QFT. This includes TQFTs as a subset.) I love this paper because it is very pedagogical and beautifully written; I've really learned a lot from it.
 

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