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Fluid dynamics as intersection problem

Nikita Nekrasov, Paul Wiegmann

TL;DR

The work develops a covariant, intersection-theory approach to relativistic hydrodynamics by recasting fluid dynamics as an intersection on an infinite-dimensional symplectic space tied to spacetime. It builds a four-dimensional covariant framework from two coisotropic/Lagrangian data sets, one encoding the equation of state and the other the geometric background, and shows how this naturally reduces to three-dimensional Eulerian descriptions via Novikov-type groups ${\mathcal{G}}^{(2)}_{B^3}$ and their duals. The paper then generalizes to anomalous fluids, magnetic couplings, and higher-dimensional formalisms, and reveals deep connections with topological field theories, Poisson sigma models, and topological strings, including links to the five- and six-dimensional theories and Langlands-type structures. The framework yields covariant expressions for Kelvin’s theorem, Landau velocity, and Onsager quantization, and suggests a unified viewpoint where EOS, geometry, and topology interlace to govern fluid dynamics and its quantum/thermodynamic extensions. Overall, it provides a versatile geometric language for ideal and anomalous fluids, with promising ties to gravity, string theory, and modern topological field theory.

Abstract

We formulate the covariant hydrodynamics equations describing the fluid dynamics as the problem of intersection theory on the infinite dimensional symplectic manifold associated with spacetime. This point of view separates the structures related to the equation of state, the geometry of spacetime, and structures related to the (differential) topology of spacetime. We point out a five-dimensional origin of the formalism of Lichnerowicz and Carter. Our formalism also incorporates the chiral anomaly and Onsager quantization. We clarify the relation between the canonical velocity and Landau $4$-velocity, the meaning of Kelvin's theorem. Finally, we discuss some connections to topological strings, Poisson sigma models, and topological field theories in various dimensions.

Fluid dynamics as intersection problem

TL;DR

The work develops a covariant, intersection-theory approach to relativistic hydrodynamics by recasting fluid dynamics as an intersection on an infinite-dimensional symplectic space tied to spacetime. It builds a four-dimensional covariant framework from two coisotropic/Lagrangian data sets, one encoding the equation of state and the other the geometric background, and shows how this naturally reduces to three-dimensional Eulerian descriptions via Novikov-type groups and their duals. The paper then generalizes to anomalous fluids, magnetic couplings, and higher-dimensional formalisms, and reveals deep connections with topological field theories, Poisson sigma models, and topological strings, including links to the five- and six-dimensional theories and Langlands-type structures. The framework yields covariant expressions for Kelvin’s theorem, Landau velocity, and Onsager quantization, and suggests a unified viewpoint where EOS, geometry, and topology interlace to govern fluid dynamics and its quantum/thermodynamic extensions. Overall, it provides a versatile geometric language for ideal and anomalous fluids, with promising ties to gravity, string theory, and modern topological field theory.

Abstract

We formulate the covariant hydrodynamics equations describing the fluid dynamics as the problem of intersection theory on the infinite dimensional symplectic manifold associated with spacetime. This point of view separates the structures related to the equation of state, the geometry of spacetime, and structures related to the (differential) topology of spacetime. We point out a five-dimensional origin of the formalism of Lichnerowicz and Carter. Our formalism also incorporates the chiral anomaly and Onsager quantization. We clarify the relation between the canonical velocity and Landau -velocity, the meaning of Kelvin's theorem. Finally, we discuss some connections to topological strings, Poisson sigma models, and topological field theories in various dimensions.
Paper Structure (59 sections, 280 equations, 2 figures)

This paper contains 59 sections, 280 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Euler-Poincare-Arnold equations
  4. Separating geometry from the equation of state
  5. Novikov groups
  6. A curious symmetry
  7. The inertia map and generalized angular velocity
  8. Classical mechanics
  9. Poisson and symplectic
  10. Variational principle
  11. Symplectic manifold out of Poisson pointwise
  12. Symplectic out of Poisson: the hard way, part I
  13. Symplectic out of Poisson: the hard way, part II
  14. The physical meaning of ${\mathcal{G}}^{(2)}_{B^3}$
  15. Symplectic manifold out of Poisson via pathways
  16. ...and 44 more sections

Figures (2)

  • Figure 1: The coisotropic ${\mathcal{C}}_{M^4}$ and Lagrangian ${\mathcal{L}}_{{{\varepsilon}}, {\bf g}}$ inside ${\mathscr{P}}_{M^4}$
  • Figure 2: ${\widetilde{\mathcal{L}}}_{S}$