Fluid dynamics as intersection problem

TL;DR

This paper presents a novel geometric intersection-theoretic framework for fluid dynamics on infinite-dimensional symplectic manifolds, clarifying structures and extending to complex fluids and self-dual fields.

Contribution

It introduces a new geometric formulation of fluid dynamics using intersection theory, connecting it to topological field theories and extending to various complex fluid types.

Findings

Derives covariant hydrodynamics from geometric principles.

Clarifies the role of hydrodynamic invariants and velocities.

Extends formalism to multicomponent, charged, and superfluids, including anomalies.

Abstract

We formulate the equations of fluid dynamics as an intersection-theoretic problem on an infinite-dimensional symplectic manifold naturally associated with spacetime. This perspective separates the structures determined by the equation of state and the spacetime geometry from the differential-topological data of spacetime. It leads to a geometric derivation of the covariant formulation of hydrodynamics due to Lichnerowicz and Carter, clarifies the role of the canonical velocity and hydrodynamic invariants, including the asymptotic Hopf invariant and the Ertel invariant, and yields a generalized Kelvin circulation theorem. We also explain the relation between the canonical velocity, the four-velocity, and other choices of hydrodynamic frame. In addition, we identify a five-dimensional geometric origin of the formalism underlying covariant hydrodynamics. The formalism extends naturally to fluids with additional degrees of freedom, including multicomponent fluids, charged fluids, and superfluids, and incorporates the chiral anomaly and Onsager quantization. It also suggests a possible bridge between hydrodynamics, Poisson sigma models, and topological field theories. We further argue that the same intersection-theoretic viewpoint applies to self-dual fields, including chiral bosons in 1+1 dimensions, tensor fields of the (2,0) theory in 1+5 dimensions, and the self-dual four-form field of type-IIB supergravity in 1+9 dimensions.