Perfect fluid equations with nonrelativistic conformal supersymmetries

Abstract: Our recent result on the construction of perfect fluid equations with N=1,2 Schrödinger supersymmetry [Mod. Phys. Lett. A 41 (2026) 2550214] is extended to accommodate nonrelativistic conformal supersymmetries of other types. Two cases are considered in detail, which include the N=2 conformal Newton-Hooke superalgebra and N=1 l-conformal Galilei superalgebra with arbitrary half-integer parameter l. Supersymmetric fluid models are built within the Hamiltonian framework by introducing real (for N=1) or complex (for N=2) anticommuting field variables as superpartners for the density and velocity. For both the cases the full set of conserved charges associated with the superalgebras is constructed and the Lagrangian description is given. Subtleties with the construction of perfect fluid equations with N=2 l-conformal Galilei supersymmetry are discussed as well.

• The paper presents explicit Hamiltonian and Lagrangian formulations for perfect fluid equations extended by nonrelativistic conformal supersymmetries.
• It employs noncanonical Poisson brackets and a generalized Clebsch decomposition to achieve closure of the supersymmetry charges and full set of conserved quantities.
• The study identifies algebraic obstacles for realizing full N=2 l-conformal Galilei supersymmetry for l > 1, suggesting the need for additional bosonic generators.

Supersymmetric Extensions of Perfect Fluid Equations with Nonrelativistic Conformal Symmetries

Background and Motivation

The study of hydrodynamics under the lens of nonrelativistic conformal (super)symmetries has gained momentum due to its relevance across condensed matter physics, gravitation theory, and integrable systems. In particular, perfect fluid models with generalized symmetry structures offer a fertile ground for investigating strongly-coupled many-body systems, the fluid/gravity correspondence, and the nonrelativistic AdS/CFT framework. Nonrelativistic conformal algebras are characterized by anisotropic dilatations, with the dynamical critical exponent $l$ governing the scaling behavior. Supersymmetric extensions of these structures—especially the Schrödinger and $l$-conformal Galilei algebras and their Newton-Hooke relatives—serve as a bridge between fluid dynamics and supersymmetric field theories, including superstrings and supermembranes.

This paper ("Perfect fluid equations with nonrelativistic conformal supersymmetries" (2605.19356)) expands on previous constructions of perfect fluid dynamics with $\mathcal{N}=1,2$ Schrödinger supersymmetry, aiming to systematically cover $\mathcal{N}=2$ conformal Newton-Hooke and $\mathcal{N}=1$ $l$-conformal Galilei superalgebras, including arbitrary half-integer $l$. The paper scrutinizes the Hamiltonian and Lagrangian formulations, derives the corresponding supercharges and conserved quantities, and addresses subtleties in the attempted extension to $\mathcal{N}=2$ $l$-conformal Galilei supersymmetry.

Supersymmetric Hydrodynamics: Hamiltonian Formalism

The construction begins with the canonical perfect fluid equations (continuity and Euler equations) supplemented by polytropic equations of state, yielding models invariant under Schrödinger and $l$-conformal Galilei symmetries for special exponents. Within the Hamiltonian formalism, supersymmetrization proceeds by introducing fermionic partners for the density and velocity fields: real for $l$0, complex for $l$1. Noncanonical Poisson brackets are enforced to ensure supersymmetry closure, and supersymmetry charges are defined to generate the super-extended Hamiltonians.

For the Schrödinger case ($l$2), full representations for $l$3 and $l$4 supersymmetries are realized, including all symmetry-generating charges, e.g., time/space translations, boosts, dilatations, special conformal transformations, and superconformal partners. The super-extended Hamiltonians introduce explicit boson-fermion coupling terms and, at the $l$5 level, cubic fermionic contributions, encapsulating $l$6 R-symmetry.

Extensions to Conformal Newton-Hooke and $l$7-Conformal Galilei Algebras

The paper systematically extends these basic constructions to models with a nonzero cosmological constant, replacing the Schrödinger algebra with the conformal Newton-Hooke algebra. The presence of harmonic oscillator potentials compatible with $l$8 supersymmetry ensures closure of the superalgebra and preserves the full symmetry set, albeit with modified expressions for conserved charges and the necessity for additional fermionic terms.

For the $l$9-conformal Galilei superalgebras, the paper constructs supersymmetric perfect fluid models for arbitrary half-integer $\mathcal{N}=1,2$0—primarily at $\mathcal{N}=1,2$1. The higher-derivative structure of the Euler equation necessitates Ostrogradski-like auxiliary variables; the Poisson structure and supersymmetry charges are carefully adapted to accommodate the increased field content. All conserved charges reflecting the extended symmetry algebra are constructed, and the Hamiltonian/Lagrangian frameworks are explicitly worked out.

Analytical Results and Structural Claims

Strong results:

• The paper provides explicit Hamiltonian and Lagrangian formulations for perfect fluid equations with $\mathcal{N}=1,2$2 $\mathcal{N}=1,2$3-conformal Galilei supersymmetry for arbitrary half-integer $\mathcal{N}=1,2$4 and for $\mathcal{N}=1,2$5 conformal Newton-Hooke supersymmetry.
• Full sets of conserved charges are constructed for both the bosonic and fermionic sectors, forming representations of the respective superalgebras.

Crucial structural claim:

• It is shown that the standard superpartner construction fails to realize the full $\mathcal{N}=1,2$6 $\mathcal{N}=1,2$7-conformal Galilei superalgebra for $\mathcal{N}=1,2$8. While the supersymmetric extension of the Galilei subalgebra is feasible, conformal symmetry generators and the superconformal charges cannot be realized without additional bosonic degrees of freedom. This is traced to the necessity for extra bosonic generators in the algebra closure (as mandated by prior results for $\mathcal{N}=1,2$9, $\mathcal{N}=2$0).

Technical achievement:

• The Clebsch-type decomposition is generalized to provide Lagrangian formulations consistent with the Hamiltonian dynamics, successfully yielding all equations of motion and highlighting the role of fermionic variables as generalized Gaussian potentials in vorticity representations.

Implications and Future Directions

From a theoretical standpoint, the work advances the understanding of symmetry-enhanced hydrodynamics, offering constructions that could inform developments in nonrelativistic holography, integrable supersymmetric field theories, and effective descriptions of strongly-anisotropic systems. The inability to construct full $\mathcal{N}=2$1 $\mathcal{N}=2$2-conformal Galilei supersymmetric fluid models for $\mathcal{N}=2$3 underscores nontrivial algebraic obstructions, suggesting parallels with the structure of supersymmetric higher-derivative mechanics.

Practically, the models established herein provide templates for exploring exact solutions, the effects of supersymmetric couplings on transport properties, and potential realizations in condensed matter systems with higher-order symmetries. The paper signals several future directions, including the search for solutions via symmetry-based techniques, generalization to $\mathcal{N}=2$4 or Lifshitz-type symmetries, and the inclusion of extra bosonic fluid variables to circumvent the observed algebraic obstructions.

Conclusion

This work rigorously constructs supersymmetric perfect fluid models invariant under nonrelativistic conformal superalgebras, extending existing results beyond Schrödinger symmetry to encompass $\mathcal{N}=2$5 $\mathcal{N}=2$6-conformal Galilei and $\mathcal{N}=2$7 conformal Newton-Hooke superalgebras. While the approach successfully yields the Hamiltonian and Lagrangian structures, including the full complement of conserved charges, it identifies barriers to realizing $\mathcal{N}=2$8 $\mathcal{N}=2$9-conformal Galilei symmetry for $\mathcal{N}=1$0, opening a pathway for future research to incorporate additional degrees of freedom and broader symmetry structures. The presented models constitute a robust foundation for supersymmetric hydrodynamics and its applications in mathematical physics, holography, and integrable systems.