What is the physical importance of topological quantum field theory? Apart from the fascinating mathematics of TQFTs, is there any reason that can convince a theoretical physicist to invest time and energy in it?
What are/would be the implications of TQFTs?
I mean is there at least any philosophical attitude behind it?
 A: TQFTs were not discovered by mathematicians - they were actually discovered by physicists, so one should expect there to be physical motivation for the theory. One reason why that this is difficult to discover is that mathematicians have taken over the theory so it is hard to recognise the physical motivation.
One reason to study them is as toy models for quantum gravity; for example, the Barrett-Crane model. This was first published in 1995. It quantises GR when written in the Plebanski formulation.
A: I have personally done original research in the field of TQFTs, so I can tell you the reasons I find TQFTs interesting. Some that come to mind are:

*

*Some "real life" theories are accurately described/approximated by TQFTs; for example, gauge theories such as QCD. If you have a regular QFT that is gapped, then its strongly-coupled regime is almost by definition a TQFT. The cleanest example of this phenomenon that I know is arXiv:1710.03258, where the infrared structure of QCD$_3$ is argued to be a certain TQFT. At low energies (below the gap), this TQFT is an excellent approximation to the real dynamics of the theory. So TQFTs are a tool that allows you to describe the strongly-coupled phase of (some) gauge theories, a regime where no other method really works.


*TQFTs are exactly solvable, so they are a great toy model for more complex QFTs...


*... but TQFTs exhibit all the usual features of regular QFTs. For example, TQFTs may have symmetries, which can be

*

*Continous or discrete,

*Old school zero-form or modern higher-form (cf. arXiv:1412.5148),

*Old-school invertible or modern non-invertible (cf. arXiv:2008.07567),

*Modern higher-categorical (where symmetries of different degree mix non-trivially, cf. arXiv:1802.04790),

*anomalous (in the sense of 't Hooft), non-perturbatively,

*classical or quantum (e.g., non-trivial dualities, cf. arXiv:1607.07457)

*etc.

All these features are shared with conventional QFTs, but the advantage of TQFTs is that these are exactly solvable, so one can study these properties much more reliably and explicitly.


*Also, although TQFTs can be solved exactly, they can also be solved in perturbation theory via Feynman diagrams. So you can compare the perturbative expansion to the full answer, something you cannot do for other QFTs. So TQFTs allow you to understand non-perturbative aspects of QFTs in a controlled enviroment.


*Other aspects you can study in TQFTs is e.g. the gauging of (continuous or discrete) symmetries. The general principle is the same as in generic QFTs, but here you can do the computations explicitly and cleanly, so you can learn much better what it means for a symmetry to be gauged, and why you may have obstructions ('t Hooft anomalies) or ambiguities/choices (theta parameters).


*TQFTs classify conformal blocks in one lower dimension, so if you want to want to have a proper understanding of CFT$_2$ (i.e., String Theory) then you will have to learn about TQFT$_3$ sooner or later.


*Along similar lines, TQFTs give some examples of holography so I guess they are also interesting if you care about that.


*Finally, TQFTs sometimes appear in actual experiments, I believe. The Hall effect is an oft quoted example, although I never really cared for the real world so I don't know the details.
A: In condensed matter physics, topological quantum field theories provide an effective description of (many, but not all) gapped phases of matter at low energies and long distances. A phase of matter is gapped if it costs a finite amount of energy to create any excitation above the ground state.
Examples of gapped phases of matter that admit a low-energy TQFT description include quantum Hall phases, which are described by Chern-Simons theories, and superconductors with dynamical electromagnetic fields, which are described by $BF$ theories.
Examples of gapped phases that may not have a TQFT description are so-called fracton or gapped non-liquid phases.
