Lifshitz symmetry: Lie algebras, spacetimes and particles

Abstract

We study and classify Lie algebras, homogeneous spacetimes and coadjoint orbits ("particles") of Lie groups generated by spatial rotations, temporal and spatial translations and an additional scalar generator. As a first step we classify Lie algebras of this type in arbitrary dimension. Among them is the prototypical Lifshitz algebra, which motivates this work and the name "Lifshitz Lie algebras". We classify homogeneous spacetimes of Lifshitz Lie groups. Depending on the interpretation of the additional scalar generator, these spacetimes fall into three classes: (1) ($d+2$)-dimensional Lifshitz spacetimes which have one additional holographic direction; (2) ($d+1$)-dimensional Lifshitz--Weyl spacetimes which can be seen as the boundary geometry of the spacetimes in (1) and where the scalar generator is interpreted as an anisotropic dilation; and (3) ($d+1$)-dimensional aristotelian spacetimes with one scalar charge, including exotic fracton-like symmetries that generalise multipole algebras. We also classify the possible central extensions of Lifshitz Lie algebras and we discuss the homogeneous symplectic manifolds of Lifshitz Lie groups in terms of coadjoint orbits.

30 pages, 2 figures. (v2: added some references and acknowledgments, v3: various minor updates)

References in corpus (9)

• Effective Holographic Theories for low-temperature condensed matter systems
• Fracton Phases of Matter
• Fractons, dipole symmetries and curved spacetime
• Goldstones with Extended Shift Symmetries
• Lifshitz scaling effects on the holographic p-wave superconductors coupled to nonlinear electrodynamics
• Informational properties of holographic Lifshitz field theory
• Holomorphic Structure and Quantum Critical Points in Supersymmetric Lifshitz Field Theories
• Symplectic Actions and Central Extensions
• Dynamical realizations of the Lifshitz group

Cited by in corpus (9)

• Krylov Complexity in Lifshitz-type Scalar Field Theories
• A symmetry principle for gauge theories with fractons
• Entanglement in Lifshitz Fermion Theories
• From pp-Waves to Galilean Spacetimes
• Planons and their Carroll-Galilei symmetries
• 1/c deformations of AdS${}_3$ boundary conditions and the Dym hierarchy
• Possible ambient kinematics
• A non-lorentzian primer
• New dynamical realizations of the Lifshitz group