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Scaling Symmetry and Carrollian Gravity

Hamid Afshar, Mehdi Ahmadi-Jahmani

TL;DR

This work develops a gauge-theoretic construction of matter-coupled scaling–Carroll gravity across a general dynamical exponent $z$, by gauging the scaling–Carroll algebra and employing a compensating scalar to implement local dilatations. After fixing scale, the gravity multiplet includes $( au_ ho,e_ ho{}^a,b_a,S_{ab})$, with the trace of the extrinsic curvature $K$ and the Carroll boost parameter playing central roles; depending on how the boost and the dilatation sector are treated, the framework naturally interpolates between dynamical Carroll gravity, Aristotelian geometry, and a fracton gauge theory on Aristotelian spacetime. A key result is that the extrinsic curvature need not vanish, unlike in conformal Carroll constructions, due to the presence of $b_a$ and the relaxed symmetry; this enables a unified description and transitions between regimes via gauge fixing. The paper also develops Carrollian invariant field theories, their coupling to geometry, and curvature invariants, providing explicit forms for kinetic and curvature terms up to higher time-derivative orders and identifying the fracton sector through boost-invariant combinations and Frobenius integrability. Overall, the work offers a versatile geometric framework connecting Carrollian gravity, nonrelativistic (Aristotelian) geometry, and fracton dynamics, with potential implications for flat-space holography and holographic duals of Carrollian boundary theories.

Abstract

We formulate matter-coupled scaling-Carroll gravity as a gauge theory and analyze its associated gravity multiplet. After fixing the scaling symmetry, the theory is governed by the trace of the extrinsic curvature, the Carroll boost symmetry, and a vector field descending from dilatations. We show that appropriate gauge choices and geometric constraints lead to distinct regimes, including dynamical Carroll gravity, Aristotelian gravity, and a fracton gauge theory coupled to Aristotelian geometry. In the fracton phase, the Carroll boost parameter plays the role of a vector-charge gauge symmetry.

Scaling Symmetry and Carrollian Gravity

TL;DR

This work develops a gauge-theoretic construction of matter-coupled scaling–Carroll gravity across a general dynamical exponent , by gauging the scaling–Carroll algebra and employing a compensating scalar to implement local dilatations. After fixing scale, the gravity multiplet includes , with the trace of the extrinsic curvature and the Carroll boost parameter playing central roles; depending on how the boost and the dilatation sector are treated, the framework naturally interpolates between dynamical Carroll gravity, Aristotelian geometry, and a fracton gauge theory on Aristotelian spacetime. A key result is that the extrinsic curvature need not vanish, unlike in conformal Carroll constructions, due to the presence of and the relaxed symmetry; this enables a unified description and transitions between regimes via gauge fixing. The paper also develops Carrollian invariant field theories, their coupling to geometry, and curvature invariants, providing explicit forms for kinetic and curvature terms up to higher time-derivative orders and identifying the fracton sector through boost-invariant combinations and Frobenius integrability. Overall, the work offers a versatile geometric framework connecting Carrollian gravity, nonrelativistic (Aristotelian) geometry, and fracton dynamics, with potential implications for flat-space holography and holographic duals of Carrollian boundary theories.

Abstract

We formulate matter-coupled scaling-Carroll gravity as a gauge theory and analyze its associated gravity multiplet. After fixing the scaling symmetry, the theory is governed by the trace of the extrinsic curvature, the Carroll boost symmetry, and a vector field descending from dilatations. We show that appropriate gauge choices and geometric constraints lead to distinct regimes, including dynamical Carroll gravity, Aristotelian gravity, and a fracton gauge theory coupled to Aristotelian geometry. In the fracton phase, the Carroll boost parameter plays the role of a vector-charge gauge symmetry.
Paper Structure (23 sections, 107 equations, 1 figure, 1 table)

This paper contains 23 sections, 107 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notation and convention.
  3. Carrollian extrinsic curvature
  4. Co-frame.
  5. Scaling-Carroll algebra and gauging
  6. Gauging the scaling-Carroll algebra
  7. Transformation rules
  8. Carrollian conformal construction
  9. Dynamical Carrollian gravity
  10. Scaling-Carroll invariant field theories
  11. Carrollian supertranslation.
  12. Scalar coupled Carroll gravities
  13. Ricci identities.
  14. Torsion identities.
  15. Curvature terms
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Schematic relation between different regimes arising from scaling--Carroll gauge theory. Fixing $\phi=1$ gauge fixes the dilatation and yields dynamical Carroll gravity. For $K\neq0$, imposing $b_a=0$ in the dynamical Carroll gravity, gauge fixes the boost and leads to Aristotelian gravity, while for $b_a\neq0$ imposing hypersurface orthogonality results in a fracton gauge theory coupled to Aristotelian geometry.