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Field Theories on Null Manifolds

Arjun Bagchi, Rudranil Basu, Aditya Mehra, Poulami Nandi

TL;DR

The paper argues that field theories on null manifolds generically acquire BMS or conformal Carrollian structures and develops the first explicit interacting conformal Carrollian field theory: Carrollian scalar electrodynamics in the electric sector. It demonstrates strong (off-shell) and weak (on-shell) invariance under conformal Carrollian symmetries, verifies Helmholtz conditions for an action principle, and presents a concrete action that retains finite and infinite Carrollian conformal invariance in d=4. Conserved charges associated with these symmetries are derived via Noether's theorem, rewritten in canonical variables, and shown to realize the infinite Carrollian conformal algebra without central extensions. The work also clarifies limitations of the magnetic sector for action formulations and lays out future directions toward Carrollian Yang–Mills, supersymmetric extensions, and flat-holography applications.

Abstract

We argue that generic field theories defined on null manifolds should have an emergent BMS or conformal Carrollian structure. We then focus on a simple interacting conformal Carrollian theory, viz. Carrollian scalar electrodynamics. We look at weak (on-shell) and strong invariance (off-shell) of its equations of motion under conformal Carrollian symmetries. Helmholtz conditions are necessary and sufficient conditions for a set of equations to arise from a Lagrangian. We investigate whether the equations of motion of Carrollian scalar electrodynamics satisfy these conditions. Then we proposed an action for the electric sector of the theory. This action is the first example for an interacting conformal Carrollian Field Theory. The proposed action respects the finite and infinite conformal Carrollian symmetries in d = 4. We calculate conserved charges corresponding to these finite and infinite symmetries and then rewrite the conserved charges in terms of the canonical variables. We finally compute the Poisson brackets for these charges and confirm that infinite Carrollian conformal algebra is satisfied at the level of charges.

Field Theories on Null Manifolds

TL;DR

The paper argues that field theories on null manifolds generically acquire BMS or conformal Carrollian structures and develops the first explicit interacting conformal Carrollian field theory: Carrollian scalar electrodynamics in the electric sector. It demonstrates strong (off-shell) and weak (on-shell) invariance under conformal Carrollian symmetries, verifies Helmholtz conditions for an action principle, and presents a concrete action that retains finite and infinite Carrollian conformal invariance in d=4. Conserved charges associated with these symmetries are derived via Noether's theorem, rewritten in canonical variables, and shown to realize the infinite Carrollian conformal algebra without central extensions. The work also clarifies limitations of the magnetic sector for action formulations and lays out future directions toward Carrollian Yang–Mills, supersymmetric extensions, and flat-holography applications.

Abstract

We argue that generic field theories defined on null manifolds should have an emergent BMS or conformal Carrollian structure. We then focus on a simple interacting conformal Carrollian theory, viz. Carrollian scalar electrodynamics. We look at weak (on-shell) and strong invariance (off-shell) of its equations of motion under conformal Carrollian symmetries. Helmholtz conditions are necessary and sufficient conditions for a set of equations to arise from a Lagrangian. We investigate whether the equations of motion of Carrollian scalar electrodynamics satisfy these conditions. Then we proposed an action for the electric sector of the theory. This action is the first example for an interacting conformal Carrollian Field Theory. The proposed action respects the finite and infinite conformal Carrollian symmetries in d = 4. We calculate conserved charges corresponding to these finite and infinite symmetries and then rewrite the conserved charges in terms of the canonical variables. We finally compute the Poisson brackets for these charges and confirm that infinite Carrollian conformal algebra is satisfied at the level of charges.

Paper Structure

This paper contains 33 sections, 113 equations.

Table of Contents

  1. Introduction
  2. Carrollian symmetries and conformal extensions
  3. Ultra-relativistic limit of conformal symmetry
  4. The $N$-Conformal Carroll Algebra
  5. Representation Theory of CCA
  6. Scale- Spin representation
  7. For scalars ($\varphi$):
  8. For gauge fields $(B,A_i)$:
  9. For scalars ($\varphi$):
  10. For gauge fields $(B,A_i)$:
  11. Interacting Carrollian Field Theory
  12. Carrollian electrodynamics: a quick recap
  13. Carrollian scalar electrodynamics
  14. Electric sector
  15. Symmetries of the electric sector
  16. ...and 18 more sections