Anti-de Sitter gravity associated with the supergroup $SU(1,1|2)\times SU(1,1|2)$

TL;DR

This work constructs three-dimensional AdS supergravity associated with the supergroup ${\cal G}= SU(1,1|2)\times SU(1,1|2)$ by treating the theory as a Chern-Simons theory. The authors derive the field content from the Maurer-Cartan equations of the supergroup, assemble a CS action from the corresponding one-forms ($e_a$, $\omega_{ab}$, gravitini, and two $SU(2)$ gauge fields), and demonstrate invariance under supersymmetry using a 1.5-order formalism. The main contribution is an explicit CS formulation for this AdS$_3$ supergravity, with discussion of potential generalizations to other AdS supergroups and prospects for coupling to matter and connecting to boundary Liouville theories in the context of AdS$_3$/CFT$_2$. This provides a clean, solvable model for exploring near-horizon geometries, spectral flow, and boundary CFT structures in the D1-D5 system. The work lays groundwork for studying alternative AdS$_3$ backgrounds and their holographic duals within a supersymmetric, gauge-theoretic framework.

Abstract

We construct the anti-de Sitter supergravity in three dimensions associated with the supergroup $SU(1,1|2)\times SU(1,1|2)$. The field content and the action are inferred using the fact that $AdS$ supergravity theories in three dimensions are Chern-Simons theories.