A gauge theory for the 3+1 dimensional incompressible Euler equations

TL;DR

This paper reformulates the 3D incompressible Euler equations as an abelian gauge theory with topological terms, revealing new insights into fluid invariants and boundary effects.

Contribution

It introduces a novel gauge theory framework for the Euler equations, incorporating topological BF terms and dual descriptions involving 3-form and 2-form fields.

Findings

Fluid helicity is dual to a 3-form field strength.

The theory includes boundary edge modes potentially relevant for fluid flows.

Two formulations of the theory are presented, one with vorticity as a magnetic field, another using Clebsch scalars.

Abstract

We show that the incompressible Euler equations in three spatial dimensions can be expressed in terms of an abelian gauge theory with a topological BF term. A crucial part of the theory is a 3-form field strength, which is dual to a material invariant local helicity in the fluid. In one version of the theory, there is an additional 2-form field strength, with the magnetic field corresponding to fluid vorticity and the electric field identified with the cross-product of the velocity and the vorticity. In the second version, the 2-form field strength is instead expressed in terms of Clebsch scalars. We discuss the theory in the presence of the boundary and argue that edge modes may be present in the dual description of fluid flows with a boundary.