I am studying higher spin gravity and I would like to know the mathematical and physical meaning of generalized spin connection and generalized dreibein that appear in this theory. It is well known that there are important connections between higher spin theories and string theory. For this reason I am wondering if the generalized spin connection is related or not to the parallel transport of extended objects (strings, membranes, etc.) on compact manifolds. Is there also a relation with Hitchin's generalized geometry?

## 1 Answer

Higher spin symmetries can be interpreted as generalized symmetries of free field differential equations. "Generalized symmetries" means here symmetries generated by differential operators of order larger than one. (Amusingly, some mathematicians call such symmetries "higher symmetries" and this dates back from before the link with higher spin theories was recognized.) In a suitable sense, the higher spin fields are gauging these generalized symmetries.

A general discussion of this fact appears in Bekaert's paper

http://arxiv.org/abs/0807.4223

I showed in

http://arxiv.org/abs/1402.4486

Section 4.2 how the family of higher spin Lie algebras in 3-dimension (usually called ${\rm hs}(\lambda)$) can be recovered from the generalized symmetries of the Klein-Gordon equation on AdS3, with the parameter $\lambda$ being related to the mass term.

Good references on generalized symmetries include the book *Application of Lie groups to differential equations* by Olver and *Symmetries and conservation laws for differential equations of mathematical physics* by Krasilshchik and Vinogradov.