Carroll/fracton particles and their correspondence

TL;DR

By applying the coadjoint-orbit method to the Carroll and dipole algebras, the paper defines and classifies classical Carroll particles and fractons, establishing a precise Carroll/fracton correspondence. The correspondence maps Carroll energy $E$ to fracton charge $q$ and center-of-mass $\boldsymbol{k}$ to dipole moment $\boldsymbol{d}$, so immobile massive Carroll particles correspond to fracton monopoles while certain massless Carroll states become fracton dipoles. The authors provide explicit particle actions for all massive and massless branches, show a GL$(2,\mathbb{R})$ automorphism connecting different sectors, and extend the framework to curved spaces and field theories. These results offer a concrete classical foundation for fractons, clarifying their definition and mobility constraints and guiding future quantum and field-theoretic formulations.

Abstract

We exploit the close relationship between the Carroll and fracton/dipole algebras, together with the method of coadjoint orbits, to define and classify classical Carroll and fracton particles. This approach establishes a Carroll/fracton correspondence and provides an answer to the question "What is a fracton?". Under this correspondence, carrollian energy and center-of-mass correspond to the fracton electric charge and dipole moment, respectively. Then immobile massive Carroll particles correspond to the fracton monopoles, whereas certain mobile Carroll particles ("centrons") correspond to fracton elementary dipoles. We uncover various new massless carrollian/neutral fractonic particles, provide an action in each case and relate them via a $GL(2,\mathbb{R})$ symmetry. We also comment on the limit from Poincaré particles, the relation to (electric and magnetic) Carroll field theories, contrast Carroll boosts with dipole transformations and highlight a generalisation to curved space ((A)dS Carroll).

Figures (2)

• Figure 1: This figure is a sketch of the movement of fractons (and carrollions, dually) in space. When the dipole moment is conserved, monopoles are restricted to a point in space as pictured by the straight left line. On the other hand, the mobility of elementary dipoles is not restricted: dipole conservation $\dot {\bm{d}} ={\bm{0}}$ implies the dipole vector is inert (right). Dually, massive carrollions are stuck at a point since the center-of-mass charge is conserved as can be visualised by thinking of a massive Poincaré particle for which the light cone closes to a line. For the massless carrollian centrons again mobility is not restricted.
• Figure 2: This figure shows the coadjoint orbits of the Poincaré and Carroll group in momentum space $(E,\boldsymbol{p})$. The Poincaré orbits are given by two families of massive particles (green), two massless orbits (yellow), one family of tachyonic orbits (gray) and the vacuum (black dot). In the Carroll limit the green massive Poincaré orbits (hyperboloids) flatten out and lead to the planes (see Section \ref{['sec:massive-case']}). The lightcone would lead to a the $E=0$ plane, but since $E=\|{\bm{p}} \|$ this orbit actually vanishes in the limit. The sphere ($\bm{p}^{2} = const.$), represented as a circle in the $E=0$ plane on the right-hand side, can be seen to arise as a limit of the tachyonic orbits (see Section \ref{['sec:massless-case']} for details). The whole carrollian $E=0$ plane is foliated by such spheres. Let us emphasise that this figure only represents the $(E,\boldsymbol{p})$ part of the full dual space $({\bm{j}},\boldsymbol{v},\boldsymbol{p},E)$ and the complete structure of the orbits is more intricate and involves spin degrees of freedom.

Theorems & Definitions (4)

• Lemma 1
• proof
• Lemma 2
• proof