Quantized Nambu-Poisson manifolds in a 3-Lie algebra reduced model

Abstract

We consider dimensional reduction of the Bagger-Lambert-Gustavsson theory to a zero-dimensional 3-Lie algebra model and construct various stable solutions corresponding to quantized Nambu-Poisson manifolds. A recently proposed Higgs mechanism reduces this model to the IKKT matrix model. We find that in the strong coupling limit, our solutions correspond to ordinary noncommutative spaces arising as stable solutions in the IKKT model with D-brane backgrounds. In particular, this happens for S ³, ${\mathbb{R}^3}$ and five-dimensional Neveu-Schwarz Hpp-waves. We expand our model around these backgrounds and find effective noncommutative field theories with complicated interactions involving higher-derivative terms. We also describe the relation of our reduced model to a cubic supermatrix model based on an $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1\left| {32} \right.} \right)$ supersymmetry algebra.