We present a Cartan-like first-order action principle for electric Carrollian gravity. The action is invariant under the local homogeneous Carroll group, albeit in a different representation than the one obtained by gauging the Carroll algebra. Additionally, we show that this first-order action can be derived from a smooth Carrollian limit of the Einstein-Cartan action. The connection with the Hamiltonian and metric forms of the action for electric Carrollian gravity, as well as with previous works in the literature, is also discussed.

France

Simon Pekar

Chile

Alfredo Pérez

Poland

Austria

Patricio Salgado-Rebolledo

Cartan-like formulation of electric Carrollian gravity

We construct Carroll-invariant theories with fields propagating outside the Carroll light cone, i.e., at a speed strictly greater than zero (“Carroll swiftons”). We first consider models in flat Carroll spacetime in general dimensions, where we present scalar and vector Carroll swifton field theories. We then turn to the coupling to gravity and achieve in particular in two dimensions a Carroll-invariant scalar swifton by coupling it suitably to Carroll dilaton gravity. Its backreaction on the geometry generates dynamical torsion.

Florian Ecker

Daniel Grumiller

Belgium

Marc Henneaux

Carroll-invariant propagating fields

We show that the fractonic dipole-conserving algebra can be obtained as an Aristotelian (and pseudo-Carrollian) contraction of the Poincaré algebra in one dimension higher. Such contraction allows to obtain fracton electrodynamics from a relativistic higher-dimensional theory upon dimensional reduction. The contraction procedure produces several scenarios including the some of the theories already discussed in the literature. A curved space generalization is given, which is gauge invariant when the Riemann tensor of the background geometry is harmonic.

Francisco Peña-Benítez

Fracton gauge fields from higher-dimensional gravity

By incorporating two gauge connections, transgression forms provide a generalization of Chern-Simons actions that are genuinely gauge-invariant on bounded manifolds. In this work, we show that, when defined on a manifold with a boundary, the Hamiltonian formulation of a transgression field theory can be consistently carried out without the need to implement regularizing boundary terms at the level of first-class constraints. By considering boundary variations of the relevant functionals in the Poisson brackets, the surface integral in the very definition of a transgression action can be translated into boundary contributions in the generators of gauge transformations and diffeomorphisms. This prescription systematically leads to the corresponding surface charges of the theory, reducing to the general expression for conserved charges in (higher-dimensional) Chern-Simons theories when one of the gauge connections in the transgression form is set to zero.

Czechia

Pablo Pais

Aldo Vera

A note on the Hamiltonian structure of transgression forms

We show that infinite families of Galilean spin-3 symmetries in <math><mn>2</mn><mo>+</mo><mn>1</mn></math> dimensions, which include higher-spin extensions of the Bargmann, Newton-Hooke, nonrelativistic Maxwell, and nonrelativistic AdS-Lorentz algebras, can be obtained as Lie algebra expansions of two different spin-3 extensions of the Nappi-Witten symmetry. These higher-spin Nappi-Witten algebras, in turn, are obtained by means of Inönü-Wigner contractions applied to suitable direct product extensions of <math><mi>s</mi><mi>l</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>R</mi><mo>)</mo></math>. Conversely, we show that the same result can be obtained by considering contractions of expanded <math><mi>s</mi><mi>l</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>R</mi><mo>)</mo></math> algebras. The method can be used to define nonrelativistic higher-spin Chern-Simon gravity theories in <math><mn>2</mn><mo>+</mo><mn>1</mn></math> dimensions in a systematic way.

Ricardo Caroca

Diego M. Peñafiel

Nonrelativistic spin-3 symmetries in <math><mn>2</mn><mo>+</mo><mn>1</mn></math> dimensions from expanded and extended Nappi-Witten algebras

In this paper, three things are done. First, we study from an algebraic point of view the infinite-dimensional Bondi-Metzner-Sachs (BMS)-like extensions of the Carroll algebra relevant to the asymptotic structure of the electric and magnetic Carrollian limits of Einstein gravity. In the course of this study we exhibit by the “Carroll-Galileo duality” a new infinite-dimensional BMS-like extension of the Galilean algebra and of its centrally extended Bargmann algebra. Second, we consider the electric Carrollian limit of the pure Einstein theory and indicate that more flexible boundary conditions than the ones that follow from just taking the limit of the Einsteinian boundary conditions are actually consistent. These boundary conditions lead to a bigger asymptotic symmetry algebra that involves spatial supertranslations depending on three functions of the angles (instead of one). Third, we turn to the Carrollian limit of the coupled Einstein-Yang-Mills system. An infinite-dimensional color enhancement of the gauge algebra is found in the electric Carrollian limit of the Yang-Mills field, which allows angle-dependent Yang-Mills transformations at spatial infinity, not available in the Einstein-Yang-Mills case prior to taking the Carrollian electric limit. This enhancement does not occur in the magnetic limit.

Oscar Fuentealba

Jakob Salzer

Asymptotic structure of Carrollian limits of Einstein-Yang-Mills theory in four spacetime dimensions

We explicitly establish the equivalence between the magnetic Carrollian limit of Einstein gravity defined through the Hamiltonian formalism and the Carrollian theory of gravity defined through a gauging of the Carroll algebra along the lines of standard Poincaré (or (A)dS) gaugings.

Andrea Campoleoni

Magnetic Carrollian gravity from the Carroll algebra

We consider Carroll-invariant limits of Lorentz-invariant field theories. We show that just as in the case of electromagnetism, there are two inequivalent limits, one “electric” and the other “magnetic”. Each can be obtained from the corresponding Lorentz-invariant theory written in Hamiltonian form through the same “contraction” procedure of taking the ultrarelativistic limit c → 0 where c is the speed of light, but with two different consistent rescalings of the canonical variables. This procedure can be applied to general Lorentz-invariant theories (p-form gauge fields, higher spin free theories etc) and has the advantage of providing explicitly an action principle from which the electrically-contracted or magnetically-contracted dynamics follow (and not just the equations of motion). Even though not manifestly so, this Hamiltonian action principle is shown to be Carroll invariant. In the case of p-forms, we construct explicitly an equivalent manifestly Carroll-invariant action principle for each Carroll contraction. While the manifestly covariant variational description of the electric contraction is rather direct, the one for the magnetic contraction is more subtle and involves an additional pure gauge field, whose elimination modifies the Carroll transformations of the fields. We also treat gravity, which constitutes one of the main motivations of our study, and for which we provide the two different contractions in Hamiltonian form.

Carroll contractions of Lorentz-invariant theories

We show that Merons in D-dimensional Einstein–Massive–Yang–Mills theory can be mapped to solutions of the Einstein–Skyrme model. The identification of the solutions relies on the fact that, when considering the Meron ansatz for the gauge connection  $$A=\lambda U^{-1}dU$$  <math> <mrow> <mi>A</mi> <mo>=</mo> <mi>λ</mi> <msup> <mi>U</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>d</mi> <mi>U</mi> </mrow> </math>  , the massive Yang–Mills equations reduce to the Skyrme equations for the corresponding group element U. In the same way, the energy–momentum tensors of both theories can be identified and therefore lead to the same Einstein equations. Subsequently, we focus on the SU(2) case and show that introducing a mass for the Yang–Mills field restricts Merons to live on geometries given by the direct product of  $$S^3$$  <math> <msup> <mi>S</mi> <mn>3</mn> </msup> </math>   (or  $$S^2$$  <math> <msup> <mi>S</mi> <mn>2</mn> </msup> </math>  ) and Lorentzian manifolds with constant Ricci scalar. We construct explicit examples for  $$D=4$$  <math> <mrow> <mi>D</mi> <mo>=</mo> <mn>4</mn> </mrow> </math>   and  $$D=5$$  <math> <mrow> <mi>D</mi> <mo>=</mo> <mn>5</mn> </mrow> </math>  . Finally, we comment on possible generalisations.

Marcelo Ipinza

United Kingdom

Gravitating Meron-like topological solitons in massive Yang–Mills theory and the Einstein–Skyrme model

We construct the non-relativistic and Carrollian versions of Jackiw-Teitelboim gravity. In the second order formulation, there are no divergences in the non-relativistic and Carrollian limits. Instead, in the first order formalism, some divergences can be avoided by starting from a relativistic BF theory with (A)dS2 × ℝ gauge algebra. We show how to define the boundary duals of the gravity actions using the method of non-linear realisations and suitable Inverse Higgs constraints. In particular, the non-relativistic version of the Schwarzian action is constructed in this way. We derive the asymptotic symmetries of the theory, as well as the corresponding conserved charges and Newton-Cartan geometric structure. Finally, we show how the same construction applies to the Carrollian case.