# The system is topological; so what?

Lately I've been studying topological quantum field theory, mainly from mathematically oriented sources and, while I understand why a mathematician would care about TQFT's, I'm not yet quite sure why a physicist would.

In particular, I'd like to know how TQFT's are used in particle physics. Say I begin with a standard QFT consisting of some gauge and matter fields. Somehow, I'm able to argue that in the infra-red, the theory flows to a Chern-Simons theory with gauge group $G$ and level $k$. Now what? What do I learn about the system once I realise its low-energy behaviour is described by a TQFT? What measurables can I study? What "real-life" predictions can I extract from the system?

Note: here I am only interested in (relativistic) particle physics. I know that TQFT's are also used in condensed matter and quantum gravity, but that's not what I want to understand here.

• for condensed matter applications, see Real World application of Topological Quantum Field Theory. Apr 24, 2018 at 13:54
• Haha. For all the hype I hear about I'd really like to know this as well. Apr 24, 2018 at 13:55
• If you put the word "topological" in the title of your next paper, the probabilities to get it published rise because it is just cool.
– Jon
Apr 24, 2018 at 14:19
• arxiv.org/pdf/gr-qc/9711048v1.pdf Apr 24, 2018 at 15:27
• TQFT is usually defined as a QFT that is both background independent and doesn’t have local degrees of freedom. Quantum gravity is expected to be background independent, but to contain local degrees of freedom. Effectively, quantum gravity is probably a TQFT on an infinite-dimensional vector space. Associated to a “cylinder” is a projection operator on the physical sector, which is a basic ingredient of timeless quantum mechanics, an expected mathematical framework of quantum gravity. So TQFT are toy models for quantum gravity. Apr 24, 2018 at 17:57