Supergravity dual of c-extremization

TL;DR

This work provides a holographic realization of $c$-extremization for $ ext{N}=(0,2)$ two-dimensional CFTs by identifying the exact R-symmetry and central charge with the extremization of the $T$-tensor in three-dimensional $ ext{N}=2$ gauged supergravity. By connecting the embedding tensor to ’t Hooft anomalies, the authors show the gravity trial function yields $c_R \\propto 1/T$, with the exact R-symmetry determined at supersymmetric $AdS_3$ fixed points via $ abla_i T=0$. They apply the construction to wrapped D3-branes arising from reductions of five-dimensional $U(1)^3$ gauged supergravity and reproduce the known central charge $c_R = -12 \, oldsymbol{ Eta}_{oldsymbol{ extSigma}} N^2 \frac{a_1 a_2 a_3}{oldsymbol{ Theta}}$ and the corresponding $T_R$, thereby establishing a purely 3D gravity dual to $c$-extremization. This work thus provides a compact, lower-dimensional framework for holographic $c$-extremization, with the potential to extend to wrapped M5-brane configurations and other wrapped-brane geometries. The results highlight the central role of the $T$-tensor as the gravity counterpart of the field-theory trial $c$-function and pave the way for systematic 3D realizations of R-symmetry extremization in holography.

Abstract

Recently a general principle, called c-extremization, which determines the exact R-symmetry of two-dimensional SCFTs with N = (0,2) supersymmetry was identified. In this note we show that the supergravity dual corresponds to the extremization of the T-tensor of N = 2 gauged supergravity in three-dimensions. To support this claim, we demonstrate that the expected central charge of SCFTs arising from twisted compactifications of four-dimensional N = 4 SYM on Riemann surfaces, whose gravity dual is a reduction of five-dimensional U(1)^3 gauged supergravity, is recovered in the three-dimensional framework.