Geometric realization of stress-tensor deformed field theory

TL;DR

This paper establishes a semiclassical framework linking stress-tensor deformations of quantum field theories to gravitational actions, revealing a bidirectional relationship between stress-tensor flows and classical gravity.

Contribution

It introduces a novel approach connecting stress-tensor deformations with gravitational dynamics, including explicit examples and the emergence of an induced Newton constant.

Findings

Stress-tensor deformations can be expressed as gravitational actions evaluated at a metric saddle.

Deformed partition functions relate to gravitational path integrals at saddle points.

In a scalar theory, the one-loop effective action reveals an induced Newton constant.

Abstract

We present a semiclassical framework in which stress-tensor deformations of a quantum field theory (QFT) reorganize into a gravitational action evaluated at a metric saddle. The deformed partition function can be written as a gravitational path integral evaluated at the saddle, establishing a direct link between stress-tensor flows and gravitational dynamics. Two complementary routes arise: (i) from gravitational actions such as Einstein and Palatini, which map to stress-tensor deformations of a seed QFT; and (ii) from deformed QFTs such as generalized Nambu-Goto and $T\bar{T}$-like deformed models, which reconstruct the corresponding gravitational actions. Finally, in a free, massive scalar theory, we show that the one-loop effective action of the nonlocal deformation contains a local curvature term; its coefficient defines an induced Newton constant at a chosen renormalization scale, thereby demonstrating a bidirectional link between stress-tensor flows and classical gravity.