Higher-spin composites and emergent AdS$_3$ geometry in the $(1+1)$-dimensional Gross-Neveu model

Abstract

We study the $(1+1)$-dimensional Gross-Neveu (GN) model with $N$ fermion species and derive from it an emergent AdS$_3$ geometry together with the leading entries of a holographic dictionary. We use a Bargmann-Wigner fusion scheme to generate an infinite tower of higher-spin composite fields from fundamental fermions and study their relative dynamics. A key feature is that the competition between spin-0 (chiral) condensation and spin-1 pairing defines the radial coordinate $z$ of the bulk geometry. Local fluctuations of this ratio measured by a comoving derivative generate the AdS$_3$ line element, and the large-$N$ species sum promotes $z$ from a parameter to a genuine bulk dimension. The $SO(2,2)$ bulk isometry group, whose special conformal generators mix $z$ with the boundary GN coordinates, emerges from local symmetries of the condensates. Our construction admits two dual holographic frames related by the $\mathbb{Z}_2$ exchange of the spin-0 and spin-1 condensates, covering complementary halves of the GN phase diagram. Significantly, projection onto the spin-2 sector reproduces the linearized Einstein-Hilbert action with Newton's constant $G_3 = \ell_\mathrm{AdS}/4πN^2$. We suggest throughout how the string-theoretic sector of the correspondence may emerge.