# Real World application of Topological Quantum Field Theory

What is a "killer-app" for the formalism of topological quantum field theory in "established real world physics"?

To be more precise, I'm looking for an actual physical experiment, state of matter or something alike, together with a article that well describes it by a topological quantum field theory.

I'm, in particular, not looking for hypothesized physics like string theory, just for something that already has been observed.

• Seek out Floer Homology and Sir Michael Atiyah's seminal paper on New invariants for manifolds of dimensions 3 and 4. – Autolatry May 8 '15 at 8:20
• Can you please be a little more precise. What physical system do they talk about? As a mathematician, I'm very aware of the importance in mathematics, but it is kind of hard these days to separate established physics from pure math or hypothesized physics. Especially when the term "quantum" is involved. – Mark Neuhaus May 8 '15 at 8:27
• The physical system is spacetime since TQFT is a QFT that computes topological invariants; in this instance if you like; one can consider the topological invariants of smooth four-manifolds. – Autolatry May 8 '15 at 8:37
• @MarkNeuhaus In my opinion it is far more common to listen the words "higher category", "cobordism" and "homology" in a theoretical physics department than in the whole $($math $\setminus$ $($number theory $\cup$ algebraic geometry$))$ department... :-D – yuggib May 8 '15 at 8:58
• This question (v2) seems like a list question. – Qmechanic May 8 '15 at 10:05

There is a nice pedagogical review of the quantum Hall effect that can be found here. They explain (sketchily) how to derive an effective action describing the bulk of a quantum Hall fluid, which is a topological quantum field theory: Chern-Simons (CS) theory. The main physics which can be gleaned from this is that any defects in the bulk of the fluid could be anyonic quasiparticles. However, to my knowledge these defects are not observed in experiments; instead, the anyonic quasiparticles live on the edges of the sample. So this is hardly a "killer application", actually the CS theory just tells you that nothing interesting really happens in the bulk.

Of course, this review is pretty old and I expect that the state-of-the-art has changed considerably since then.

• Thanks! That's indeed along the line of what I'm searching for. .. Are saying that the observation of the anyons is different from the Chern-Simon prediction in this example? – Mark Neuhaus May 8 '15 at 22:35
• @MarkNeuhaus Yes, the fractional charge-carriers that are detected in transport experiments on quantum Hall samples are edge modes, they do not live in the bulk. However, I believe that the charges and statistics of the bulk theory excitations are probably related to/the same as those of the corresponding CFT on the boundary. – Mark Mitchison May 8 '15 at 22:57

Topological Quantum Field Theory (TQFT) is the low energy effective theory for the topological ordered states in real world, such as FQH states. In fact, the name "topological order" was motived from the term "Topological Quantum Field Theory". [See Topological Orders in Rigid States, Xiao-Gang Wen, Int. J. Mod. Phys. B4, 239 (1990) http://dao.mit.edu/~wen/pub/topo.pdf].

We also have a modern mathematical summary in arXiv:1405.5858 Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions which point out that:

Topological orders in $n$ space-time dimension = unitary $n$-categories with one object = $n$ dimensional fully extented TQFT = gravitational anormalies in one lower dimensions

Since topological orders describe gapped quantum phases in real world, $n$-categories, fully extented TQFT, gravitational anormalies all have connection to real word quantum states of matter, such as FQH state. The spins in 2+1D TQFT can be measured by edge tunneling IV curve or RT curve.

• Yes thanks. That is exactly what I wanted to see. Personally I find this a little more interesting then the quantum hall effect. However I mark the hall effect as the answer, since I would say, it is a little more "the" archetypical example, if that makes sense. – Mark Neuhaus May 22 '15 at 16:44
• Hello. I am looking for a reference to start studying tqft. Do you know any good pedagogical one? thanks – Oscar Jun 22 '15 at 10:35