The M-Theory Two-Brane in AdS_4 x S^7 and AdS_7 x S^4

TL;DR

This work constructs the full superspace geometry and the complete M-theory two-brane action in the highly supersymmetric backgrounds $AdS_4\times S^7$ and $AdS_7\times S^4$ to all orders in the fermionic coordinates $\theta$. By formulating the coset models of the isometry superalgebras $OSp(8|4)$ and $OSp(6,2|4)$, and solving the Maurer–Cartan equations with a factorized coset representative, the authors obtain all-order expressions for the supervielbein $E$, spin connection $\Omega$, and the four-form field strength $F_{(4)}$, confirming agreement with low-order gauge-completion results. The M2-brane action is written as $S=-\int d^3\zeta\sqrt{-\det g_{ij}}+\int_{M_3} B$ using the derived superspace data, providing a solid foundation for studying branes in these AdS backgrounds and enabling potential extensions to M5-branes and matrix-model descriptions. These results offer a robust, explicit framework for exploring holography and supersymmetric dynamics in maximally supersymmetric AdS geometries.

Abstract

We construct the supermembrane action in an AdS_4 x S^7 and AdS_7 x S^4 background to all orders in anticommuting coordinates. The result is compared to and agrees completely with results obtained earlier for generic supergravity backgrounds through gauge completion at low orders in θ.