As a mathematics graduate student whose research area lies in low-dimensional topology (more precisely, invariants of 3-dimensional topological manifolds), I heard that there exist multiple applications of this theory to theoretical physics, and moreover, that many mathematical problems in the field actually arise from physical ones.

I would appreciate any examples of such applications/motivation, and references to textbooks (or articles which don't require a deep knowledge of physics) where I could read more on the subject.

  • $\begingroup$ An interesting application of topology in physics arises when studying anomalies, i.e. symmetries of a quantum field theory broken by quantum corrections due to quantization. For cases in string theory, I recommend the superb volumes on Superstring theory by Witten, Schwarz, et al. I also recommend the papers by Witten regarding topological quantum field theory, c.f. the Jones polynomial. $\endgroup$
    – JamalS
    Apr 13, 2014 at 12:21
  • $\begingroup$ Related: physics.stackexchange.com/q/1603/2451 , physics.stackexchange.com/q/41589/2451 , and links therein. $\endgroup$
    – Qmechanic
    Apr 13, 2014 at 12:33
  • $\begingroup$ Hi @Pandora, list and recommendation questions are restricted on Phys.SE, cf. various meta posts. This question also tends to be too broad. I'm closing this as a duplicate, not because it is an exact duplicate, but to guide you in the right direction. $\endgroup$
    – Qmechanic
    Apr 13, 2014 at 14:00
  • $\begingroup$ Low-dimensional topology is an extremely specific sub-niche of algebraic topology and I do not see how the linked "duplicate" could adequately address the relevant themes. It's like someone asking about Rick and Morty and being directed to a conversations about cartoons. $\endgroup$
    – j0equ1nn
    Dec 10, 2016 at 5:05