Supersymmetric Field Theories on AdS_p x S^q

TL;DR

The work systematically classifies and realizes how supersymmetric field theories can live on curved non-compact spaces of the form AdS_p x S^q, identifying the allowed (p-1)-dimensional superconformal algebras as subalgebras of the corresponding flat-space algebras under Nahm-type constraints. It develops two complementary methods—background supergravity embedding and conformal mapping from flat space—to derive Killing spinors and algebras, and it applies these results to a wide range of dimensions (d = 3–6) and SUSY contents, including detailed analyses of AdS_3 x S^1 realizations for 4d N=1,2, and N=4 theories. The paper then provides concrete free-field realizations on AdS_3 x S^1, computing KK spectra, boundary conditions, and resulting 2d multiplets (e.g., (0,2), (2,0), (0,4), (2,2), (0,8), (4,4)) and showing how Chern-Simons terms can emerge on AdS_3. Overall, it demonstrates a rich landscape of SUSY-compatible subalgebras, boundary conditions, and 2d conformal structures arising from higher-dimensional theories on AdS_p x S^q, with implications for holography and decoupled sectors in string theory. The results highlight how SUSY on non-compact curved spaces is governed by conformal embeddings and boundary data, offering avenues for exact computations and brane-based realizations in AdS settings.

Abstract

In this paper we study supersymmetric field theories on an AdS_p x S^q space-time that preserves their full supersymmetry. This is an interesting example of supersymmetry on a non-compact curved space. The supersymmetry algebra on such a space is a (p-1)-dimensional superconformal algebra, and we classify all possible algebras that can arise for p >= 3. In some AdS_3 cases more than one superconformal algebra can arise from the same field theory. We discuss in detail the special case of four dimensional field theories with N=1 and N=2 supersymmetry on AdS_3 x S^1.