Higher Spins from Nambu–Chern–Simons Theory

Abstract

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.

Leibniz–Chern–Simons Theory and Phases of Exceptional Field Theory

“…It starts from a genuine Lie algebra g (that we can view as the global symmetry of the 'ungauged phase') and an embedding tensor, that in 3D is a symmetric tensor Θ on the dual space g ˚. 9 There is another 3D Chern-Simons-type theory with an infinite-dimensional extension of the AdS algebra that does not factorize [49]. This is based on the algebra of volume preserving diffeomorphisms on S 3 [50], which is a genuine Lie algebra that, however, does not have an invariant quadratic form.…”

On matter coupled to the higher spin square

“…In most known examples, the higher spin Lie algebra h can be embedded in an associative algebra a, such that the Lie bracket in h arises from the commutator in a. If this is the case, we can write down linearized equations 3 for two scalar matter fields 2 See however [11] for an example of a higher spin theory based on a Lie algebra which is simple, rather than a product of two isomorphic copies. 3 In the original work [10], see also [16], the equations were written more succinctly by introducing an extra Grassmann element φ, satisfying φ 2 = 1, and the associated projection operators P± = 1±φ 2 .…”

W1+∞ constraints for the hermitian one-matrix model

“…The q-Virasoro constraints for this matrix model have been derived by the insertion of the q-Virasoro generators under the contour integral [15], where the q-Virasoro generators are constructed in terms of q-derivatives within the q-calculus and the corresponding q-Virasoro algebra is a special case of a more general elliptic deformation of the Virasoro algebra [16]. Since 3-algebra has recently been found useful in the Bagger-Lambert-Gustavsson (BLG) theory of M2-branes [17,18], the applications of n-algebra have aroused much interest [19]- [25]. More recently it was found that there are the generalized q-W ∞ constraints for the elliptic matrix model [26].…”