Fermions and D=11 Supergravity On Squashed Sasaki-Einstein Manifolds

TL;DR

We develop a top-down fermionic reduction of $D=11$ supergravity on warped Sasaki–Einstein seven-manifolds (the total space of a Spin$^c$ bundle over a Kähler–Einstein base) to a four-dimensional theory with $N=2$ gauged supergravity, retaining the $SU(3)$-singlet sector. The gravitino is expanded using gauge-covariantly constant spinors with definite $U(1)$ charge, after reducing the covariant derivatives and the $F_4$ flux, and the resulting four-dimensional fermions are diagonalized into the fields $\zeta_a$, $\eta$, and $\xi$, yielding an explicit effective action. The four-dimensional action organizes into the gravity, vector, and hypermultiplets of $N=2$ gauged supergravity, and is validated by matching supersymmetry variations; the framework is then specialized to a minimal gauged supergravity truncation and to a holographic superconductor embedding, revealing Majorana-type and Pauli-type couplings. This provides a UV-complete, four-dimensional description of fermions in these reductions, enabling studies of fermionic correlators and superconductivity in holographic duals of $2+1$-dimensional field theories.

Abstract

We discuss the dimensional reduction of fermionic modes in a recently found class of consistent truncations of D=11 supergravity compactified on squashed seven-dimensional Sasaki-Einstein manifolds. Such reductions are of interest, for example, in that they have $(2+1)$-dimensional holographic duals, and the fermionic content and their interactions with charged scalars are an important aspect of their applications. We derive the lower-dimensional equations of motion for the fermions and exhibit their couplings to the various bosonic modes present in the truncations under consideration, which most notably include charged scalar and form fields. We demonstrate that our results are consistent with the expected supersymmetric structure of the lower dimensional theory, and apply them to a specific example which is relevant to the study of $(2+1)$-dimensional holographic superconductors.