Abstract
We propose a new theory of higher spin gravity in three spacetime dimensions. This is defined by what we will call a Nambu-Chern-Simons (NCS) action; this is to a Nambu 3-algebra as an ordinary Chern-Simons (CS) action is to a Lie (2-)algebra. The novelty is that the gauge group of this theory is simple; this stands in contrast to previously understood interacting 3D higher spin theories in the frame-like formalism. We also consider the $N=8$ supersymmetric NCS-matter model (BLG theory), where the NCS action originated: Its fully supersymmetric M2 brane configurations are interpreted as Hopf fibrations, the homotopy type of the (infinite) gauge group is calculated and its instantons are classified.
Highlights
We have demonstrated that SDiff(S3) NCS theory is a higher-spin gauge theory on Euclidean de Sitter space in dimension three, whose gauge group is simple, and which may be seen as a unification of all su(n) ⊕ su(n) theories at once
While that might be an interesting fact in itself, one wonders whether a generalisation to (Euclidean) anti de Sitter exists
The obvious candidate is the NCS theory associated to hyperbolic 3-space H3 ∼= S O(3, 1)/S O(3)
Summary
The codimension is high enough that one can remove any self-intersections by pushing coincident points slightly apart in the transverse directions, and any cusp-like singularities can presumably be obtained as limits of smooth spherical embeddings. The essential step appears to involve carefully integrating out the unbroken gauge field AH (where the abelian H ⊂ G acts trivially on scalars in M) in the low-energy effective action describing motion in the moduli space. This is complicated because the gauge field associated to the broken, residual gauge group K (where K = O(2) for G = S O(4) or SU (2) × SU (2) and K = Diff(S2) for G = SDiff(S3)) couples to it through a B F type term: AH ∧ FK. The infrared limit (where S1 blows up to R) fails to match any of the generic sectors (nonvanishing Hopf invariant) of the SDiff(S3) BLG vacuum moduli space
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