Coset Construction for Duals of Non-relativistic CFTs

TL;DR

This work develops a coset-based approach to holographic duals of non-relativistic CFTs by treating backgrounds as $G/H$ cosets of the Schrödinger group and its variants, with metrics constructed from a non-degenerate $H$-invariant symmetric two-form on $\mathfrak{g}/\mathfrak{h}$. It demonstrates a general, non-reductive framework for defining $G$-invariant metrics via the Maurer–Cartan form and validates a KN-type correspondence between invariant metrics and $H$-invariant bilinear forms. Applying the method to the $d=2$ Schrödinger algebra reproduces the five-dimensional backgrounds proposed for fermions at unitarity and proves their uniqueness under natural physical assumptions; the same scheme also accommodates a Lifshitz gravity dual with a similar uniqueness result. The paper further discusses extensions to $z\neq 2$ Lifshitz-like geometries and comments on super-cosets, providing a principled algebraic avenue to classify NR holographic backgrounds and guide future supersymmetric generalizations.

Abstract

We systematically analyze backgrounds that are holographic duals to non-relativistic CFTs, by constructing them as cosets of the Schrodinger group and variants thereof. These cosets G/H are generically non-reductive and we discuss in generality how a metric on such spaces can be determined from a non-degenerate H-invariant symmetric two-form. Applying this to the d=2 Schrodinger algebra, we reproduce the five-dimensional backgrounds proposed as duals of fermions at unitarity, and under reasonable physical assumptions, we demonstrate uniqueness of this background. The proposed gravity dual of the Lifshitz fixed-point, for which Galileian symmetry is absent, also fits into this organizational scheme and uniqueness of this background can also be shown.