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Interacting Conformal Carrollian Theories: Cues from Electrodynamics

Kinjal Banerjee, Rudranil Basu, Aditya Mehra, Akhila Mohan, Aditya Sharma

TL;DR

This work constructs a gauge-free action for the magnetic limit of Carrollian electrodynamics by appending new fields and applying Helmholtz integrability to ensure a consistent Lagrangian. It then implements conformal Carrollian symmetry to identify marginal quartic interactions that preserve the full infinite-dimensional symmetry, yielding an interacting Carrollian field theory with exact charge realization. The authors demonstrate that the Carrollian algebra closes on the Noether charges without central extensions, confirming a consistent dynamical realization in the interacting model. They also show that this magnetic Carrollian theory does not arise from a straightforward Minkowski ancestor, highlighting novel ultra-relativistic dynamics beyond simple ultra-relativistic limits. Overall, the paper provides a concrete, symmetry-rich framework for studying interacting Carrollian CFTs and sets the stage for quantum analyses, propagator studies, and potential applications in ultra-relativistic or Dirac-material contexts.

Abstract

We construct the free Lagrangian of the magnetic sector of Carrollian electrodynamics. The construction relies on Helmholtz integrability condition for differential equations in a self consistent algorithm, working hand in hand with imposing invariance under infinite dimensional Conformal Carroll algebra. It requires inclusion of new fields in the dynamics and the system is free of gauge redundancies. We next add interaction (quartic) terms to the free Lagrangian, strictly constrained by conformal invariance and Carrollian symmetry. The dynamical realization of the non-semi simple infinite dimensional symmetry algebra at the level of charge algebra is exact and free from central terms.

Interacting Conformal Carrollian Theories: Cues from Electrodynamics

TL;DR

This work constructs a gauge-free action for the magnetic limit of Carrollian electrodynamics by appending new fields and applying Helmholtz integrability to ensure a consistent Lagrangian. It then implements conformal Carrollian symmetry to identify marginal quartic interactions that preserve the full infinite-dimensional symmetry, yielding an interacting Carrollian field theory with exact charge realization. The authors demonstrate that the Carrollian algebra closes on the Noether charges without central extensions, confirming a consistent dynamical realization in the interacting model. They also show that this magnetic Carrollian theory does not arise from a straightforward Minkowski ancestor, highlighting novel ultra-relativistic dynamics beyond simple ultra-relativistic limits. Overall, the paper provides a concrete, symmetry-rich framework for studying interacting Carrollian CFTs and sets the stage for quantum analyses, propagator studies, and potential applications in ultra-relativistic or Dirac-material contexts.

Abstract

We construct the free Lagrangian of the magnetic sector of Carrollian electrodynamics. The construction relies on Helmholtz integrability condition for differential equations in a self consistent algorithm, working hand in hand with imposing invariance under infinite dimensional Conformal Carroll algebra. It requires inclusion of new fields in the dynamics and the system is free of gauge redundancies. We next add interaction (quartic) terms to the free Lagrangian, strictly constrained by conformal invariance and Carrollian symmetry. The dynamical realization of the non-semi simple infinite dimensional symmetry algebra at the level of charge algebra is exact and free from central terms.

Paper Structure

This paper contains 17 sections, 71 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Conformal Carrollian isometries and kinematics
  3. The isometry algebra
  4. Transformation of fields under conformal Carroll isometries
  5. Correlation Functions
  6. Towards a Lagrangian formulation for the magnetic limit
  7. Brief review on Carrollian electrodynamics
  8. The Helmholtz Conditions and the Consistency Algorithm
  9. The Helmholtz Conditions and the Consistency Algorithm
  10. Strong invariance check and Lagrangian
  11. Looking for a Minkowski ascendant
  12. Extension to an interacting theory
  13. Addition of Interactions
  14. Dynamical Realization of the Carrollian algebra
  15. Conclusions
  16. ...and 2 more sections

Figures (2)

  • Figure 1: In the panel (a) above, the points $A$ and $C$ are causally connected, whereas the interval $A$ - $B$ is acausal, in Minkowski space. To get Carroll space-time as the $c \rightarrow 0$ limit, one should collapse the light cone to a single line, as in panel (b). In Carroll space-time, none of the points $A, B$ and $C$ are causally connected, unless the events take place exactly at the same spatial points.
  • Figure 2: The process of taking Carrollian (ultra-relativistic) limit and applying variational derivative on action functional do not commute.