Emergent strings, holography, and cosmology from four-fermion interactions: a bottom-up derivation of AdS/CFT, dS/CFT, and $w_{1+\infty}$

Abstract

We derive holographic duality from first principles starting from the $(1+1)$-dimensional Gross-Neveu (GN) model with $N$ fermion species and a local quartic interaction, without assuming any string or geometric input. Using a Bargmann-Wigner scheme, the competition between chiral condensation $Δ_0=\langle\barψψ\rangle$ and spin-1 pairing $Δ_1=\langle\barΦ_1Φ_1\rangle$ defines an emergent radial coordinate $z=m^{-1}(Δ_1/Δ_0^2-1)^{1/2}$; local fluctuations of this ratio, tracked by a comoving derivative, generate the AdS$_3$ line element via the enhanced large-$N$ species dispersion; the condensate competition \emph{is} the extra dimension. From this single mechanism the complete AdS$_3$/CFT$_2$ correspondence emerges: Newton's constant, the Virasoro algebra ($c=2N^2$), D1-branes with open strings, open/closed T-duality, the Hagedorn/BKT transition, and the BTZ black hole whose horizon circumference is quantised in Planck units by individual vortex nucleation events. Analytic continuation $z\to iζ$ across the chiral critical point realises the Strominger dS/CFT conjecture microscopically. Six constraints identify the emergent string as Type IIB on AdS$_3\times S^3\times\mathcal{M}_4$, with emergent worldsheet $\mathcal{N}=(1,1)$ supersymmetry, NS/R spectral flow, and GSO projection. Extension to the $(2+1)$d NJL model yields AdS$_4$/CFT$_3$, a dS$_4$/CFT$_3$ realisation, and a structural identification of the $w_{1+\infty}$ celestial algebra. Extension to the $(3+1)$d NJL model yields AdS$_5$/CFT$_4$ and holographic QCD with chiral symmetry breaking and linear Regge trajectories $M_s^2=4(s+1)Λ_\mathrm{QCD}^2$, capturing the correct QCD infrared physics from a four-fermion interaction.

Figures (8)

• Figure 1: The systematic sequence of bottom-up holographic dualities derived in this paper. Each row applies the same mechanism --- the fusion condition and comoving derivative --- to a four-fermion model in one higher spacetime dimension, yielding an emergent AdS bulk of one dimension higher. The $(1+1)$d Gross-Neveu model gives AdS$_3$/CFT$_2$ and Type IIB string theory (Sections \ref{['sec:higherspin']}--\ref{['sec:stringID']}); the $(2+1)$d NJL$_3$ model gives AdS$_4$/CFT$_3$, dS$_4$/CFT$_3$, and the $w_{1+\infty}$ algebra (Section \ref{['sec:4d']}); the $(3+1)$d NJL$_4$ model gives AdS$_5$/CFT$_4$ and holographic QCD (Section \ref{['sec:QCD']}).
• Figure 2: Phase diagram of the GN model in the $(T,\,\Delta_1/\Delta_0^2)$ plane. The three phase boundaries are derived from the microscopic GN parameters (Appendix \ref{['app:phase']}): the Hawking-Page boundary $T_\mathrm{HP}\propto(\Delta_1/\Delta_0^2)^{1/2}$ (bold solid), the Hagedorn/BKT boundary $T_H\propto(\Delta_1/\Delta_0^2)^{1/6}$ (dash-dotted), and the Planck temperature $T_P\sim m$ (thin dashed, flat). The dotted curve for $\Delta_1<\Delta_0^2$ is the Gibbons-Hawking temperature of the emergent dS$_3$ geometry, $T_\mathrm{dS}\propto(1-\Delta_1/\Delta_0^2)^{1/2}$. The vertical dashed line at $\Delta_1=\Delta_0^2$ is the chiral transition surface (triple boundary of Section \ref{['sec:threeregion']}).
• Figure 3: (a) The effective Schrödinger potential $V_\mathrm{eff}(z)=(4\nu^2-1)/4z^2 + \kappa^4 z^2$ for the spin-1 bulk field $\Phi_1$ in the soft-wall AdS$_3$ geometry, with $\kappa = \Lambda_\mathrm{QCD} = 0.535$ GeV and $\nu=1$. The potential combines an AdS centrifugal barrier ($\propto z^{-2}$, diverging at the boundary) with a soft confining wall ($\propto z^2$, growing into the IR). The first three eigenstates $n=0,1,2$ are shown as wave functions offset at their eigenvalue levels $M_n^2 = 4\kappa^2(n+1)$; their peak positions shift toward larger $z$ with increasing $n$, reflecting deeper penetration of higher radial excitations into the IR. (b) The BKT vortex condensate profile $\rho(r)/\rho_0 = (r/\xi)/\sqrt{(r/\xi)^2+2}$ (Bogomolny approximation), satisfying the nonlinear Ginzburg-Landau equation $\rho"+\rho'/r-\rho/r^2 = \lambda\rho(\rho^2-\rho_0^2)$ with boundary conditions $\rho(0)=0$, $\rho(\infty)=\rho_0$. The healing length $\xi = (\sqrt{\lambda}\,\rho_0)^{-1}$ marks the radius at which the condensate recovers to $1/\sqrt{3}$ of its bulk value; the vortex core ($r\ll\xi$) is the region of depleted condensate corresponding to the Hagedorn depth $\tilde{z}_H$ in Frame 2 (the deeper spin-1 transition, distinct from the BTZ horizon at $\tilde{z}_\mathrm{hor}$; see Section \ref{['sec:BHinterior']}). Above $T_H$ vortices proliferate and $\langle\rho\rangle\to 0$ everywhere, dissolving the AdS geometry entirely. Derivations of both equations are in Appendix \ref{['app:wavevortex']}.
• Figure 4: The emergent AdS$_3$ geometry in Frame 2, shown in the Poincaré disk representation. The thick boundary circle is the CFT$_2$ at $\tilde{z}=0$. The three concentric circles are D1-branes at radial positions $\tilde{z}_i<\tilde{z}_j<\tilde{z}_k$, each one a diagonal composite $\Phi_1^{(ii)}=\rho^{(i)}e^{i\theta^{(i)}}$ whose phase $\theta^{(i)}$ winds around the compact angular direction; these windings are the closed strings. The three wavy lines are the off-diagonal composites $\Phi_1^{(ij)}$, $\Phi_1^{(jk)}$, $\Phi_1^{(ik)}$ (open strings) stretched radially between all pairs of D1-branes, carrying the Chan-Paton structure of the $U(N)$ matrix model. The radial coordinate $\tilde{z}$ increases from the boundary inward; the shading lightens toward the centre, reflecting the dissolving classical geometry as $\tilde{z}\to\ell_\mathrm{P}$.
• Figure 5: Cross-section of the black hole interior in Frame 2, viewed as a spatial slice of the AdS$_3$ bulk. The boundary ($\tilde{z}=0$) is the outer circle; the Planck depth ($\tilde{z}=\ell_\mathrm{P}$) is the centre. Four concentric dashed circles mark the phase transitions at $T_\mathrm{HP}$ (BTZ horizon), $T_\mathrm{HS}$ (graviton dissolution / tensionless limit), $T_H$ (Hagedorn / string proliferation), and $T_P$ (geometry ends). The progressive lightening from boundary to centre represents the increasing density of $U(1)_2$ micro-vortices --- the transition from classical geometry to quantum gravity to no geometry. Right labels identify the bulk transitions; left labels identify the degrees of freedom in each region.