Gravitational formulation of stress-tensor deformed field theories

TL;DR

This paper develops a gravitational framework for stress-tensor deformed field theories, revealing a universal bilocal deformation at leading order and constructing local formulations on deformed backgrounds, with applications to various models.

Contribution

It introduces a gravitational approach to stress-tensor deformations, providing a systematic expansion and local formulations for a broad class of theories, including scalar, gauge, and higher-dimensional models.

Findings

Universal bilocal deformation kernel derived from graviton Green's function.

Local flow equations on deformed backgrounds via $f(R)$ gravity and eigenvalue methods.

Quantum corrections generate Einstein-Hilbert terms with controlled higher derivatives.

Abstract

Stress-tensor deformations suggest a geometric origin of emergent gravity but are typically non-local for $d>2$. We couple a seed QFT to Einstein gravity with deformation parameter $\lambda$ and evaluate the gravitational path integral at the metric saddle. Around a fixed reference background, the leading deformation is universal: a bilocal term quadratic in the stress tensor with kernel set by the graviton Green's function, plus a systematic higher-order expansion. Expressed on the saddle-point (deformed) metric, the flow becomes local. We then provide two constructive completions on deformed backgrounds--Palatini $f(R)$ gravity and an eigenvalue method for general Ricci-based theories--and apply them to scalar generalized Nambu-Goto and $\det T$ deformations (arbitrary $d$), two-dimensional multi-scalar ModMax and Born-Infeld models, and four-dimensional root-$T\bar T$ and $T\bar T$ flows of Maxwell theory yielding ModMax and Born-Infeld electrodynamics. In free field theory, an off-shell analysis further shows that the leading quantum correction generates an Einstein-Hilbert term with controlled higher-derivative terms.