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The Non-Relativistic Superparticle in a Curved Background

Eric Bergshoeff, Joaquim Gomis, Marija Kovacevic, Lorena Parra, Jan Rosseel, Thomas Zojer

TL;DR

This work develops explicit actions for non-relativistic superparticles propagating in curved three-dimensional Newton–Cartan backgrounds, including Galilean, Newton–Cartan, and Newton–Hooke geometries with and without a supersymmetric cosmological constant. The construction proceeds by starting from a flat background, then promoting Galilean symmetries to a Galilean supergravity background, and finally gauging to a Newton–Cartan supergravity background, with supersymmetry encoded by a 3D N=2 Bargmann superalgebra and its Newton–Hooke contraction. Key contributions include the bosonic and fermionic actions for Galilean, Curved Galilean, NC, NH, Curved NH, and NC NH superparticles (to quartic fermions in the NC NH case), together with their background transformation rules and the realization of κ-symmetry as a Stückelberg gauge symmetry that can be fixed to simplify the dynamics. The results provide a concrete framework for non-relativistic holography and set the stage for higher-dimensional generalizations and extensions to non-relativistic strings or branes, with potential connections to a non-relativistic superspace formulation.

Abstract

Using a component formulation, we construct the supersymmetric action for a superparticle in a three-dimensional Newton-Cartan supergravity background and clarify its symmetries. Our construction proceeds by first constructing the superparticle in a flat background. Next, by boosting up the background symmetries, we replace in a first step the flat background by a Galilean supergravity background. In a second step the Galilean supergravity background is replaced by a Newton-Cartan supergravity background. We extend our results by adding a supersymmetric cosmological constant and compare the non-relativistic superparticle with the relativistic kappa-symmetric three-dimensional superparticle.

The Non-Relativistic Superparticle in a Curved Background

TL;DR

This work develops explicit actions for non-relativistic superparticles propagating in curved three-dimensional Newton–Cartan backgrounds, including Galilean, Newton–Cartan, and Newton–Hooke geometries with and without a supersymmetric cosmological constant. The construction proceeds by starting from a flat background, then promoting Galilean symmetries to a Galilean supergravity background, and finally gauging to a Newton–Cartan supergravity background, with supersymmetry encoded by a 3D N=2 Bargmann superalgebra and its Newton–Hooke contraction. Key contributions include the bosonic and fermionic actions for Galilean, Curved Galilean, NC, NH, Curved NH, and NC NH superparticles (to quartic fermions in the NC NH case), together with their background transformation rules and the realization of κ-symmetry as a Stückelberg gauge symmetry that can be fixed to simplify the dynamics. The results provide a concrete framework for non-relativistic holography and set the stage for higher-dimensional generalizations and extensions to non-relativistic strings or branes, with potential connections to a non-relativistic superspace formulation.

Abstract

Using a component formulation, we construct the supersymmetric action for a superparticle in a three-dimensional Newton-Cartan supergravity background and clarify its symmetries. Our construction proceeds by first constructing the superparticle in a flat background. Next, by boosting up the background symmetries, we replace in a first step the flat background by a Galilean supergravity background. In a second step the Galilean supergravity background is replaced by a Newton-Cartan supergravity background. We extend our results by adding a supersymmetric cosmological constant and compare the non-relativistic superparticle with the relativistic kappa-symmetric three-dimensional superparticle.

Paper Structure

This paper contains 21 sections, 96 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The Non-relativistic Bosonic Particle
  3. The Galilean Particle
  4. The Curved Galilean Particle
  5. The Newton--Cartan Particle
  6. Adding a Cosmological Constant (bosonic)
  7. The Non-relativistic Superparticle
  8. The Galilean Superparticle
  9. The Curved Galilean Superparticle
  10. The Newton--Cartan Superparticle
  11. Adding a Cosmological Constant
  12. The Newton--Hooke Superparticle
  13. The Curved Newton--Hooke Superparticle
  14. The Newton--Cartan Newton--Hooke Superparticle
  15. Kappa-symmetry
  16. ...and 6 more sections

Figures (1)

  • Figure 1: This figure displays the different backgrounds used in this paper and the sections where they are discussed. For each background, the first section discusses the bosonic case while the second section treats the super-sym-metric case. The arrows indicate the direction of gauge-fixing. From left to right we turn on a cosmological constant $\Lambda$.