Resum interactions to obtain the full non‑linear Einstein–Hilbert action and identify the Chern–Simons structure

Develop a resummation of the full tower of interaction terms (Φ1Φ1)^n in the composite-field Lagrangian L′1 to recover the complete non‑linear Einstein–Hilbert action (including R and Λ terms at all orders in the metric fluctuation hμν) and identify the associated SL(2, R) × SL(2, R) Chern–Simons formulation of the emergent 2+1‑dimensional gravity.

Background

The paper shows that the spin‑2 sector of the composite field reproduces the linearised Einstein–Hilbert action in AdS3 and fixes G3 consistently with Brown–Henneaux. This establishes linearised gravity from the Gross–Neveu construction.

Extending beyond the quadratic approximation requires accounting for higher‑order interactions among composites. The authors state that resumming these interactions to obtain the full non‑linear theory, and matching it to the Chern–Simons description of AdS3 gravity, remains to be done.

References

We note that the full non-linear Einstein-Hilbert action, including the R and Λ terms at all orders in the hμν, would require going beyond the quadratic approximation and resumming the full tower of (Φ1Φ1)n interaction terms in L′1. This resummation, together with the identification of the SL(2, R) × SL(2, R) Chern-Simons structure of the full non-linear theory, is left for future work.

— Emergent strings, holography, and cosmology from four-fermion interactions: a bottom-up derivation of AdS/CFT, dS/CFT, and $w_{1+\infty}$  (2603.27824 - Haddad, 29 Mar 2026) in Section 9.5