Here is how higher category theory (homotopy theory) arises in gravity:

First of all, the precise version of the statement that "gravity is a gauge theory" is that gravity in first order formulation is "Cartan geometry" for Minkowski spacetime regarded as the quotient of the Poincare group by the Lorentz spin-group. This statement generalizes to super-gravity, with the Poincare group replaced by the super-Poincare group ("the spacetime supersymmetry group"), see here:

https://ncatlab.org/nlab/show/super-Cartan+geometry

The corresponding Cartan-connections encode the vielbein field and the "spin connection" that are the mathematical incarnation of the field of (super-)gravity, whose quanta are the graviton and the gravitino.

But now something special happens: higher dimensional supergravity by necessity contains not just the graviton and the gravitino, but also higher degree form fields. It is the higher degree of these form fields which is the entry point of higher category theory/homotopy theory in gravity.

Namely these tensor multiplets are no longer encoded by a Cartan-connection with values in an ordinary group (the Poincare group), but they are encoded by higher Cartan connections with values in higher categorical groups!

This is a long and fascinating story, which does not fit into this comment box here. To get started you might try these lecture notes here

https://ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms

or some of the links provided there.

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user Urs Schreiber