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?


First of all, let me say that there is nothing perculiar about 3d HS theories - I mean the Vasiliev equations look exactly the same as in any other dimension.

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.

Some explanation comes from gravity which, as is known, can be reformulated as a gauge theory of Poincare, de Sitter or anti-de Sitter algebra. Again, vielbein and spin-connection are particular components of a connection of one of the above mentioned algebras.

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.

Therefore, to both of your questions I would answer no and no.

  • $\begingroup$ Thanks for the answer! However, my questions were related to sl(n) Chern-Simons theories that look quite self-consistent even if you say that they do not represent a full HS theory. I am wondering what is the physical meaning of generalized spin connection in this particular context. $\endgroup$ – Gian Jan 29 '15 at 14:23
  • $\begingroup$ Ok, then it is not exactly about HS theory and related ads/cft duality, which requires matter fields to be present in a theory and they cannot couple to sl(n) Chern-Simons. You probably need to google higher teichmuller theory, which is a way to talk about sl(n) structures $\endgroup$ – John Jan 29 '15 at 19:20

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