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 $Δ0=\langle\barψψ\rangle$ 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.

• The paper demonstrates that emergent gravitational and string-theoretic structures arise dynamically from four-fermion interactions in large-N field theories.
• It constructs the AdS/CFT framework, black hole microphysics, and de Sitter holography by using condensate competition as the emergent bulk coordinate.
• The work provides quantitative matches with known physical observables, offering novel computational tools for exploring quantum gravity and QCD-like infrared physics.

Emergent Holography and Quantum Gravity from Four-Fermion Interactions

Introduction

The paper "Emergent strings, holography, and cosmology from four-fermion interactions: a bottom-up derivation of AdS/CFT, dS/CFT, and $w_{1+\infty}$" (2603.27824) develops a unified, first-principles derivation of holographic duality, string theory, and aspects of quantum gravity, starting from solvable non-gravitational quantum field theories with four-fermion interactions. The work systematically reconstructs the full machinery of AdS/CFT and its generalizations—open/closed string duality, black hole microphysics, and even de Sitter holography—using the (1+1)-dimensional Gross-Neveu (GN) model and its higher-dimensional extensions, entirely within the framework of large-$N$ fermionic theories. The approach does not assume any geometric or string-theoretic input; instead, all gravitational and string-theoretic structures emerge dynamically from the large-$N$ collective behavior of composite operators.

Mechanism of Emergence: Condensate Competition and Radial Dimension

The central technical innovation is the identification of competing condensates in the GN model as the dynamical origin of the emergent AdS radial direction. In (1+1) dimensions, the four-fermion interaction generates both a chiral (spin-0) condensate and a hierarchy of higher-spin condensates. Specifically, the ratio of the spin-1 to spin-0 condensates, $z = m^{-1}(A_1/A_0 - 1)^{1/2}$, is shown to play the role of an emergent spatial (bulk) coordinate. Local fluctuations in the condensate ratio, tracked using a comoving (material) derivative, define the bulk kinetic structure and lead to a dynamical AdS$_3$ metric with correct curvature and scaling [Sections 2, 3].

This condensate competition intrinsically generates several physically distinct regimes, reflected in a hierarchy of emergent length and energy scales: the Planck length ($l_p$), the AdS radius ($L_{\text{AdS}}$), and the string length ($l_s$), leading to transitions analogous to Hawking-Page (thermal AdS/black hole), Hagedorn/BKT (string proliferation), and ultimately the restoration of chiral symmetry at the Planck scale. This construction is not kinematic—unlike earlier holographic models—but entirely dynamical and constructive, with the bulk direction realized by collective quantum order parameters.

Bottom-Up Derivation of AdS/CFT and Extensions

The full AdS$_3$/CFT$_2$ correspondence, including Newton’s constant, the Virasoro algebra (with $c=2N^2$), D1-brane worldvolumes, open/closed string duality, the modular bootstrap, and the Hagedorn transition, emerges from the composite operator structure of the GN model. The approach generalizes systematically: the (2+1)-dimensional Nambu–Jona-Lasinio (NJL$_3$) model yields AdS$_4$/CFT$_3$ and celestial w$_{1+\infty}$ symmetry; the (3+1)-dimensional NJL$_4$ model produces AdS$_5$/CFT$_4$, holographic QCD with chiral symmetry breaking and linear Regge trajectories $M^2 = 4(s+1)\Lambda_{\text{QCD}}^2$, reproducing known IR properties of QCD [Sections 11, 12]. The analytic continuation of the emergent coordinate across the chiral transition gives direct access to de Sitter holography (dS/CFT) in both three and four bulk dimensions, providing a concrete, microscopic realization of the Strominger conjecture.

Black Hole Microphysics and Topological Phase Structure

A major achievement is the explicit, microscopic construction of black hole thermodynamics and quantum structure. Decoherence of the spin-2 (graviton) condensate yields a dual BTZ black hole, with the horizon realized as a macroscopic quantum vortex in the condensate. The entropy arises from counting the composite off-diagonal modes, matching the Cardy formula via the same modular transformation responsible for open/closed string duality. Each vortex nucleation event increments the horizon’s circumference by one Planck unit; the Bekenstein-Hawking entropy is thus built combinatorially from Planck-scale microstates [Sections 7, 8]. The black hole interior exhibits a stratified structure with successive phase transitions: horizon formation, higher-spin dissolution, the Hagedorn string phase, and finally full chiral restoration. Information is preserved via the topological winding of the condensate phase, providing a field-theoretic mechanism for black hole microstate conservation.

Holographic Duality Web, Modular Structure, and Emergent Supersymmetry

The duality structure discovered (Z$_2$ frame symmetry, open/closed S-duality, BKT/Kosterlitz-Thouless self-duality) is identified as facets of a deeper modular SL(2,$\mathbb{Z}$) structure linking field-theoretic condensate variables to bulk string duals. The model gives rise to a complete higher-spin tower with correct Regge scaling; the emergence of Type IIB string theory on AdS$_3 \times S^3 \times M_4$, including worldsheet N=(1,1) supersymmetry, GSO projection, and spectral flow, follows from consistent constraints in the large-$N$ limit, without presupposing supersymmetry at the microscopic level [Section 11.9].

A field-theoretic interpretation of the w$_{1+\infty}$ algebra and its relation to the BMS symmetry and celestial holography emerges naturally from the composite operator spectrum, directly constructing the celestial algebra from a universal quantum field theoretical starting point.

Numerical Results, Claims, and Notable Features

• Central charges: $c=2N^2$ (GN matrix model) matches Brown–Henneaux and Virasoro calculations; reproduces the expected scaling for large-$N$ CFT duals.
• Newton’s constant: $G_3 \sim l_{\text{AdS}}/N^2$, $G_5 \sim 1/(N^2 N_f)l_{\text{AdS}}^3$; these relations are not postulated externally but derived from the large-$N$ combinatorics.
• Hagedorn/BKT transition temperature: $T_H \approx (2 \pi l_s)^{-1}$, coinciding with the BKT self-dual point, and associated with modular invariance and vortex unbinding.
• BTZ entropy and microstate count: $S_{\text{BTZ}} \sim N^2$; entropy derived combinatorially from the composite field basis.
• Meson mass trajectory: $M^2 = 4(s+1)\Lambda_{\text{QCD}}^2$ for the NJL$_4$ model; matches QCD Regge phenomenology.
• Cosmological constant problem: The model provides a mechanism by which small positive cosmological constant emerges naturally as a property of deep condensate-dominated phases.

Theoretical and Practical Implications

On the theoretical side, this bottom-up approach provides a self-consistent, non-perturbative, large-$N$ completion of the emergent gravity paradigm, filling the gap between thermodynamic/hydrodynamic analogies and direct field-theoretic constructions; it circumvents the Weinberg-Witten theorem by leveraging emergent extra dimensions and species labeling. It gives a concrete field-theoretic realization of the Dvali-Gomez graviton BEC/spacetime foam paradigm, with black hole horizons as self-sustaining topological defects (vortices) in the spin-2 condensate.

Practically, these results imply that QCD-like infrared physics and black hole microphysics can, in principle, be accessed quantitatively from condensed-matter–style field theories with four-fermion interactions, equipped with condensate-competition mechanisms and large-$N$ expansions. The connection of BKT transitions, Regge trajectories, and vortex dynamics in the boundary field theory to black hole thermodynamics and bulk geometry provides new computational tools for both gauge theory and gravitational systems.

Outlook and Future Developments

This construction suggests several future directions:

• Application of the condensate competition and large-$N$ fusion framework to higher-dimensional and gauge theories, and to theories with more realistic (non-integrable) spectra.
• Quantitative studies of the cosmological constant, de Sitter entropy, and inflationary correlators derived from the underlying four-fermion models in cosmological settings.
• Investigation into the role of phase topological defects (micro-vortices) as information carriers for black hole interiors and the possible realization of firewall/fuzzball scenarios.
• Exploration of emergent supersymmetry and the RG structure at phase boundaries to connect with bootstrapped or asymptotically safe completions.

Conclusion

This work establishes a new paradigm for emergent holographic dualities and string theory, demonstrating that the core structural features of AdS/CFT, string/black hole microphysics, and de Sitter/CFT holography are not exclusive to special UV-complete string constructions, but can be systematically derived from universal, solvable models of interacting fermions. The key mechanisms—large-$N$ compositeness, condensate phase competition, and the associated dynamical emergence of extra dimensions and topology—yield a comprehensive, non-geometric foundation for quantum gravity, duality networks, and holographic field theory.

Reference:

Laith H. Haddad, "Emergent strings, holography, and cosmology from four-fermion interactions: a bottom-up derivation of AdS/CFT, dS/CFT, and $w_{1+\infty}$" (2603.27824)