# Generalized spin connection and dreibein in higher spin gravity

I am studying 3D 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.

1. 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?

2. Is there also a relation with Hitchin's generalized geometry?

The starting point in HS theory is to gauge HS algebra, so the generalized spin-connection and generalized dreibeins are just particular components of a single connection of a HS algebra. Connection of a HS algebra is just usual Yang-Mills connection, but the algebra is not $su(n)$ but something usually infinite-dimensional.
HS algebras are infinite-dimensional (I ignore $sl(n)$ that was considered by many people in $3d$ since it is not a part of a consistent HS theory --- one can take $sl(n)$ Chern-Simons, of course, but it is not a full HS theor). Hence the space they act geometrically is infinite-dimensional as well and because of that it is not a well-studied topic.