How Supersymmetry Splits When Time Stretches or Shrinks to Nothing

Physicists have long known that the strange, ultra-fast world of relativity and the familiar, slow-motion world of everyday physics are connected by mathematical limits. Speed up the speed of light toward infinity and Einstein’s equations melt into Newton’s; shrink it toward zero and an even more exotic regime emerges, one where nothing can propagate through space at all. A new theoretical study published in The European Physical Journal C by Osman Ergec of Istanbul Technical University examines what happens to one of the most delicate structures in modern physics, supersymmetry, when it is pushed through these two opposite limits. The answer, it turns out, is that a single supersymmetric theory does not simply deform. It fractures, cleanly and consistently, into independent pieces, each of which remains a perfectly valid theory in its own right.

The two limits in question carry the names of the pioneers of non-Einsteinian physics. When the parameter c, which plays the role of the speed of light, is sent to infinity, the resulting framework is called Galilean, after Galileo’s principle that the laws of physics look the same to all observers moving at constant velocity. When c is sent to zero instead, the framework is called Carrollian, after Lewis Carroll, a whimsical nod to a world in which time exists but motion through space effectively freezes. Both frameworks are obtained by what mathematicians call a contraction of the relativistic symmetry algebra: one rescales the time direction by a factor involving c and then takes the singular limit, watching which terms survive and which vanish.

Supersymmetry adds a second layer of structure on top of spacetime symmetry. It pairs bosons, the particles that carry forces, with fermions, the particles that make up matter, and it does so through transformation rules that mix the two. In a fully relativistic theory, these transformations close on each other: applying two supersymmetry transformations in succession produces an ordinary spacetime translation, and the whole set of fields, known as a multiplet, transforms into itself in a rigidly prescribed way. The central question Ergec addresses is what happens to this intricate multiplet structure when the parent theory is contracted. Does the supersymmetry survive intact, break apart, or vanish entirely?

The answer is subtle and elegant. The contraction maps the algebraic relations of the relativistic theory into a new sector without discarding any charges, and the same operation can be applied directly to the multiplet of fields without spoiling its closure. But when one examines the Lagrangian, the function that encodes the dynamics of the theory, something remarkable happens. Under the non-relativistic scaling, the Lagrangian decomposes into distinct pieces, each multiplying a different power of the scaling parameter. In the schematic form used in the paper, the relativistic Lagrangian splits as a sum of two sectors, one weighted by a power of c and one unweighted, and in the limit each sector becomes an independently consistent theory.

Crucially, each sector is invariant under its own symmetry. One sector remains invariant under the complete contracted supersymmetry multiplet, while the other is invariant only under a submultiplet, a truncated subset of the transformation rules that nevertheless closes on itself and leaves its sector unchanged. Which sector carries the full multiplet and which carries the truncated one depends on the specific theory and the chosen scaling, and in some cases more than two sectors can appear. The contraction, in other words, acts like a prism: it takes one unified supersymmetric theory and refracts it into several self-contained theories, each with its own symmetry structure.

Ergec demonstrates this mechanism with explicit three-dimensional examples, working with both scalar and vector field theories in the Carrollian and Galilean settings. The scalar theories contain two real scalar fields, two Majorana fermions, and two auxiliary fields, the extra fields that allow supersymmetry to be realized without imposing equations of motion on the fields, the so-called off-shell formulation. The vector theories contain a gauge field, a scalar, two spinors, and an auxiliary scalar, with the gauge field conveniently rewritten in terms of its dual vector, a construction that simplifies the bookkeeping of temporal and spatial components. In every case, a carefully chosen scaling of the fields, the supersymmetry parameters, and the time derivative produces the same characteristic decomposition.

The decomposition also carries a physical interpretation connected to the electric and magnetic structure of non-relativistic theories. In recent years, physicists have systematically formulated electric and magnetic contractions of Lorentz-invariant theories in the Hamiltonian framework, where they arise from different consistent rescalings of the canonical variables. Electric sectors are dominated by timelike contributions, terms involving time derivatives, while magnetic sectors are dominated by spacelike contributions, terms involving spatial derivatives. In the examples studied, the sectors weighted by the highest power of the scaling parameter turn out to be magnetic, built entirely from spatial derivatives, while the unweighted sectors are electric, built from time derivatives. The complementary sectors contain both types of contributions and are therefore mixed, defying a clean electric or magnetic label.

One of the technical highlights of the work concerns the Galilean vector theory, where the temporal component of the gauge field and the auxiliary scalar are recombined into two linear combinations before the contraction is taken. After the contraction and the truncation to the submultiplet, the constraint that the dual vector be divergence-free reduces to a simple condition on one of these combinations, namely that its time derivative vanishes. This condition turns out to be essential: without it, the off-shell invariance of the truncated sector would fail. The example illustrates how constraints inherited from the relativistic parent theory can quietly reorganize themselves into exactly the conditions needed to keep the descendant theories consistent.

Why does this matter beyond the formalism? Non-relativistic field theories have attracted considerable attention over the last decade for reasons that range from condensed matter physics, where Newton-Cartan geometry underlies effective descriptions of the quantum Hall effect, to high-energy theory, where Carrollian physics appears in the study of flat-space holography, celestial mechanics of scattering amplitudes, and even connections between Carroll symmetry and dark energy and inflation. Supersymmetric versions of these theories are needed to extend such applications, and the sector decomposition identified here provides a systematic recipe: start from a known relativistic supersymmetric theory, choose a scaling, and read off the independent non-relativistic theories that emerge, each with its symmetry guaranteed rather than guessed.

The broader lesson is that limits in physics are not always losses. When a parameter is pushed to infinity or zero, one might expect structure to be destroyed, with symmetries breaking and fields decoupling. Here the opposite occurs in a precise sense: nothing is truncated at the level of the algebra, and the apparent fragmentation of the Lagrangian is actually a sorting operation, distributing the original theory’s content into sectors that are individually richer in interpretation than the undivided whole. Each sector knows exactly which supersymmetries it obeys, whether it is electric or magnetic in character, and how its fields must be constrained. For theorists constructing the next generation of non-relativistic supersymmetric models, that kind of map, drawn directly from well-understood relativistic parents, is a tool of considerable practical value, and it suggests that the Galilean and Carrollian corners of the theoretical landscape are far more organized than their whimsical names might imply.

Subject of Research: Non-relativistic contractions of supersymmetric field theories in Galilean and Carrollian limits

Article Title: Aspects of non-relativistic supersymmetric theories

Article References: Ergec, O. (2026). Aspects of non-relativistic supersymmetric theories. The European Physical Journal C, 86(9), Article 1046. https://doi.org/10.1140/epjc/s10052-026-16245-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16245-3

Keywords: supersymmetry, non-relativistic field theory, Galilean contraction, Carrollian symmetry, Lagrangian decomposition, electric and magnetic sectors, off-shell multiplets, theoretical physics, symmetry algebra contraction, three-dimensional field theory, Aspects, non-relativistic