The gravity dual of supersymmetric gauge theories on a squashed three-sphere

TL;DR

The authors construct a gravity dual for a class of three-dimensional N=2 Chern–Simons quiver theories on a U(1)^2-squashed S^3 with a background U(1) gauge field by solving a 1/4–BPS Euclidean AdS_4 system with a graviphoton instanton and uplifting to eleven dimensions. They compute the holographic free energy and show that it scales with the squashing parameter as F_b = (Q^2/4) F_{b=1}, where Q = b + 1/b, and b^2 = s, tying the gravity result to the large-N field theory free energy obtained from localization. On the field theory side, localization reduces the partition function to a matrix model involving the double sine function s_b and a gauge-background term; in the large-N limit for non-chiral quivers, the long-range eigenvalue forces cancel and the leading free energy reproduces the same (Q^2/4) scaling as the gravity computation. This provides an exact gauge/gravity check in a non-conformal setting and demonstrates a controlled holographic correspondence for theories on curved backgrounds. The work also paves the way for generalizations to broader Euclidean AdS_4 solutions and their M-theory uplifts on Sasaki–Einstein manifolds, strengthening connections between rigid supersymmetric field theories on curved spaces and their gravitational duals.

Abstract

We present the gravity dual to a class of three-dimensional N=2 supersymmetric gauge theories on a U(1) x U(1)-invariant squashed three-sphere, with a non-trivial background gauge field. This is described by a supersymmetric solution of four-dimensional N=2 gauged supergravity with a non-trivial instanton for the graviphoton field. The particular gauge theory in turn determines the lift to a solution of eleven-dimensional supergravity. We compute the partition function for a class of Chern-Simons quiver gauge theories on both sides of the duality, in the large N limit, finding precise agreement for the functional dependence on the squashing parameter. This constitutes an exact check of the gauge/gravity correspondence in a non-conformally invariant setting.