[論文レビュー] Hydrodynamics as cospans of field theories into the BF theory

Simon Jonsson, Hyungrok Kim|arXiv (Cornell University)|Mar 9, 2026

Fluid dynamics and aerodynamics studies被引用数 0

ひとこと要約

The paper formalizes hydrodynamics as a cospan of differential graded manifolds connected via BF theory, linking microscopic and hydrodynamic descriptions through conserved currents.

ABSTRACT

Hydrodynamics is based on conservation laws of currents: one starts from the conserved currents of the theory describing the microscopic dynamics, and provides an alternative parameterisation of these currents in terms of hydrodynamic variables (density, pressure, velocity, etc.). This paradigm has recently been extended to incorporate higher-form symmetries. The conservation law of the $p$-form conserved currents can be regarded as the equations of motion of a $BF$ theory that treats the currents as fundamental fields. We argue that the hydrodynamic approximation to a microscopic theory can be regarded as a cospan of differential graded manifolds $X_\mathrm{micro} o X_{BF}\leftarrow X_\mathrm{hydro}$, where $X_\mathrm{micro}$ and $X_\mathrm{hydro}$ describe the microscopic and hydrodynamic theories, respectively, and $X_{BF}$ describes the $BF$ theory of conserved currents.

研究の動機と目的

• Motivate a unified framework for hydrodynamics via generalized (including higher-form) symmetries.
• Show how conserved currents can be treated as fundamental BF theory fields.
• Provide a cospan diagram linking microscopic theory, BF theory, and hydrodynamic theory within differential graded geometry.

提案手法

• Describe differential graded manifolds and the Batalin–Vilkovisky (BV) formalism as the language for field theories.
• Introduce BF theory with multiple p-form currents valued in bundles and discuss gauge structure.
• Define maps from microscopic theory to BF theory and from hydrodynamics to BF theory as a cospan.
• Explain how current conservation equations arise in both BF theory and hydrodynamics.
• Extend the construction to higher-form symmetries and nontrivial bundles.
• Relate standard hydrodynamic equations to the BV/dg-manifold framework.

実験結果

リサーチクエスチョン

• RQ1 How can hydrodynamics be formalized as a cospan of differential graded manifolds involving BF theory?
• RQ2 How do conserved Noether currents for higher-form symmetries map between microscopic, BF, and hydrodynamic descriptions?
• RQ3 What role do nontrivial vector bundles and connections play in the BF description of currents?
• RQ4 Can the BV formalism accommodate the matching between microscopic dynamics and hydrodynamic variables via current fields?
• RQ5 What are the limitations of neglecting thermodynamics and entropy in this cospan framework?

主な発見

• A cospan structure X_micro → X_BF ← X_hydro is proposed to connect microscopic dynamics, BF theory, and hydrodynamics.
• Conserved currents J^(p) of higher-form symmetries are treated as fundamental fields in BF theory, with dJ^(p)=0 as equations of motion.
• The BF action S_BF = ∫ Λ ∧ dJ generalizes to multiple p-form currents valued in bundles, possibly lacking gauge symmetry when connections are non-flat.
• Hydrodynamic parameterizations of currents J^(p) lead to the other leg of the cospan through dJ^(p)[ρ,u,...]=0.
• The framework is compatible with the BV formalism and differential graded geometry, providing a unified language for matching microscopic and hydrodynamic descriptions.