Hydrodynamics as cospans of field theories into the BF theory

Here is an explanation of the paper "Hydrodynamics as cospans of field theories into the BF theory," translated into everyday language with creative analogies.

The Big Picture: Connecting the Micro to the Macro

Imagine you are trying to understand a massive, chaotic crowd at a music festival.

The Microscopic View: You are a fly on the wall, watching every single person (the "microscopic degrees of freedom"). You see them shoving, dancing, tripping, and shouting. This is described by complex Quantum Field Theory (like the Standard Model of physics). It's incredibly detailed but impossible to track for the whole crowd.
The Hydrodynamic View: You step back and look at the crowd as a whole. You don't care about individual people; you care about the "flow." You see a wave of people moving left, a density of bodies in the center, and the pressure of the crowd. This is Hydrodynamics. It's a simplified, smooth description that works perfectly for large groups.

The Problem: How do we mathematically prove that the smooth "flow" (Hydrodynamics) actually comes from the chaotic "individuals" (Microscopic Theory)? Usually, we just assume they match. This paper proposes a rigorous, geometric way to connect them.

The Middleman: The "BF Theory"

The authors introduce a middleman to bridge the gap. They call it a BF Theory.

Think of the BF Theory as a universal translator or a rulebook of conservation.

In physics, certain things are never lost: energy, momentum, electric charge. These are called "conserved currents."
The BF Theory is a special, simple mathematical framework where the only rule is: "What goes in must stay closed." (Mathematically, the "currents" must be "closed forms").
It doesn't care what is flowing (whether it's water, electricity, or abstract quantum fields); it only cares that the flow is conserved.

The "Cospan": The Bridge

The core idea of the paper is a shape called a Cospan. Imagine a bridge with two roads leading to a central island.

Road A (Left): Starts at the Microscopic Theory (the complex quantum world).
Road B (Right): Starts at the Hydrodynamic Theory (the fluid flow world).
The Island (Center): The BF Theory (the conservation rulebook).

The paper argues that both the Microscopic world and the Hydrodynamic world can be mapped onto this central island.

Microscopic $\to$ BF: We take the complex quantum fields and calculate their currents. We show that these currents obey the "conservation rules" of the BF Theory.
Hydrodynamic $\to$ BF: We take the fluid variables (density, velocity) and calculate their currents. We show that these also obey the same "conservation rules."

Because both roads lead to the same island, we know they are describing the same underlying physics, just from different perspectives.

The Secret Sauce: "Differential Graded Manifolds"

You might wonder, "How do they actually draw these maps?"

The authors use a fancy mathematical tool called Differential Graded Geometry (and the Batalin-Vilkovisky formalism).

The Analogy: Imagine a video game character.
- The "Field": The character's position and appearance.
- The "Antifield": A ghostly shadow that tracks the character's potential moves or errors.
- The "Homological Vector Field" ( $Q$ ): This is the game engine's logic. It checks: "If the character moves here, does it break the laws of physics?" If the character is on the "correct" path (satisfying the equations of motion), the engine says "Good." If not, it flags an error.

In this paper, they treat the Microscopic theory, the Hydrodynamic theory, and the BF theory as three different "video games" (mathematical spaces). They show that there are valid "portals" (maps) that translate the characters and rules from the Micro game to the BF game, and from the Hydro game to the BF game.

Why "Higher-Form Symmetries"?

The paper also talks about "Higher-Form Symmetries."

Normal Symmetry: Think of a spinning top. It has a symmetry because it looks the same if you rotate it. The "current" is a simple flow (like a line).
Higher-Form Symmetry: Imagine a soap bubble. The symmetry isn't just about a point; it's about the surface of the bubble. The "current" is a sheet or a volume, not just a line.

The authors show that their "Cospan bridge" works even for these complex, multi-dimensional flows (like the magnetic fields in a plasma or the flow of branes in string theory). They prove that even for these weird, high-dimensional symmetries, the Hydrodynamic description is just a different way of parameterizing the same conservation laws found in the microscopic world.

The "Overdetermined" Problem

The paper also touches on a tricky situation called an Overdetermined System.

Analogy: Imagine you are trying to solve a puzzle. You have 10 clues (equations) but only 3 pieces (variables). Usually, this is impossible.
The Fix: In hydrodynamics, sometimes the rules are so strict that you have more conservation laws than variables. The authors show that in these cases, you can introduce "potentials" (like a hidden map) that automatically satisfy the rules, making the puzzle solvable again.

Summary

In a nutshell:
This paper builds a mathematical bridge between the chaotic world of tiny particles and the smooth world of fluids. It uses a "conservation rulebook" (BF Theory) as the central hub. By proving that both the microscopic world and the fluid world can be translated into this rulebook using a specific geometric language (Differential Graded Manifolds), the authors provide a rigorous, unified framework for understanding how the macroscopic world emerges from the microscopic one, even for the most complex types of symmetries.

It's like proving that the smooth flow of a river and the chaotic motion of individual water molecules are just two different languages describing the exact same set of conservation laws.

Here is a detailed technical summary of the paper "Hydrodynamics as cospans of field theories into the $BF$ theory" by David Simon Henrik Jonsson and Hyungrok Kim.

1. Problem Statement

The paper addresses the formalization of the relationship between microscopic quantum field theories (QFTs) and their macroscopic hydrodynamic limits.

The Gap: While hydrodynamics is understood as the long-wavelength, long-time limit of a microscopic theory governed by conservation laws, a rigorous mathematical framework connecting the two descriptions (especially in the context of modern generalized symmetries) has been lacking.
The Challenge: Microscopic theories are often described using the Batalin–Vilkovisky (BV) formalism (a symplectic differential graded geometry framework) to handle gauge symmetries and quantization. Hydrodynamics, conversely, is typically described by partial differential equations (PDEs) that may not always arise from an action principle (due to dissipation) and often involve higher-form symmetries.
The Goal: To unify these descriptions by treating the matching between microscopic and hydrodynamic variables as a cospan of differential graded manifolds, mediated by a topological $BF$ theory that treats conserved currents as fundamental fields.

2. Methodology

The authors employ differential graded (dg) geometry and the Batalin–Vilkovisky (BV) formalism to construct a categorical framework.

A. Theoretical Framework

Differential Graded Manifolds (dg-manifolds): The authors model physical theories as dg-manifolds $(X, Q)$ , where $X$ is a graded manifold (coordinates have integer degrees, with odd degrees being Grassmann-valued) and $Q$ is a homological vector field of degree 1 ( $[Q, Q]=0$ ) encoding the equations of motion.
BV Formalism: For theories with an action principle, the dg-manifold is equipped with a symplectic structure $\omega$ of degree $-1$ . The action $S$ generates the homological vector field via the Poisson bracket: $Q = \{S, \cdot\}$ .
The $BF$ Theory as the Mediator: The authors introduce an Abelian $BF$ theory where the fundamental fields are the conserved currents $J^{(p)}$ (p-forms) and Lagrange multipliers $\Lambda^{(d-p-1)}$ . The action is:
$S_{BF} = \int_M \Lambda^{(d-p-1)} \wedge dJ^{(p)}$
The equations of motion are simply $dJ^{(p)} = 0$ (conservation laws) and $d\Lambda = 0$ . This theory treats the currents as independent fields, decoupled from the microscopic origin.

B. The Cospan Construction

The core methodology is defining a cospan of dg-manifolds:
$X_{micro} \xrightarrow{f} X_{BF} \xleftarrow{g} X_{hydro}$

$X_{micro}$ : The dg-manifold of the microscopic QFT (fields $\phi$ , ghosts, antifields).
$X_{BF}$ : The dg-manifold of the $BF$ theory (currents $J$ , multipliers $\Lambda$ , and their antifields).
$X_{hydro}$ : The dg-manifold of the hydrodynamic theory (hydrodynamic variables like density $\rho$ , velocity $u$ , etc.).

The Morphisms:

Microscopic $\to$ $BF$ ( $f$ ): Maps microscopic fields $\phi$ to the conserved currents $J[\phi]$ . Crucially, this map handles the fact that conservation laws ( $dJ=0$ ) often hold only "on-shell" (when equations of motion are satisfied). In the BV language, $dJ$ is $Q$ -exact ( $dJ = Q(\dots)$ ), ensuring the map is a morphism of dg-manifolds.
Hydrodynamic $\to$ $BF$ ( $g$ ): Maps hydrodynamic variables to the currents $J[\rho, u, \dots]$ . Since hydrodynamic equations are often imposed directly without an action, $X_{hydro}$ is defined as a dg-manifold without a symplectic structure, where the homological vector field $Q_{hydro}$ encodes the conservation laws directly.

C. Handling Technicalities

Non-orientable Manifolds: The authors utilize twisted differential forms (valued in the orientation bundle $or_M$ ) to define currents on manifolds without a fixed orientation, ensuring the formalism is generally covariant.
Higher-Form Symmetries: The framework naturally generalizes to $p$ -form currents, where the conservation law is $dJ^{(p)} = 0$ . This covers standard vector currents ( $p=1$ ) and higher-form symmetries (e.g., magnetic flux conservation).
Overdetermined Systems: The paper addresses cases where the number of conservation equations exceeds the number of current components (common in higher-form symmetries), suggesting the use of potentials to solve the system.

3. Key Contributions

Categorical Formulation of Hydrodynamics: The paper provides the first rigorous formulation of hydrodynamics as a cospan in the category of differential graded manifolds. This places the matching of microscopic and macroscopic scales on the same footing as cobordisms in topological field theory.
Unification of Action and Non-Action Theories: It successfully bridges the gap between microscopic theories (which usually have actions and symplectic structures) and hydrodynamic theories (which may lack an action due to dissipation) by using the $BF$ theory as a common "target" that only requires the differential graded structure.
Generalization to Higher-Form Symmetries: The formalism explicitly incorporates continuous higher-form symmetries, treating their Noether currents as fundamental fields in a $BF$ theory. This offers a new perspective on magnetohydrodynamics (MHD) and other systems with extended symmetries.
Explicit Construction of Morphisms: The authors provide explicit coordinate maps for the cospans, demonstrating how microscopic fields map to currents and how hydrodynamic variables parameterize those currents.

4. Results and Examples

The paper validates the framework through three specific examples:

Relativistic Hydrodynamics (Scalar Field):
- Microscopic: A real scalar field $\phi$ .
- Hydrodynamic: Perfect fluid variables $(\rho, u^\mu)$ .
- Result: The map $X_{micro} \to X_{BF}$ recovers the stress-energy tensor $T_{\mu\nu}$ from $\phi$ . The map $X_{hydro} \to X_{BF}$ recovers the perfect fluid stress-energy tensor. The conservation $\nabla_\mu T^{\mu\nu}=0$ is the shared equation.
Relativistic Irrotational Fluid:
- Microscopic: A scalar field valued in a circle $S^1$ (periodic), leading to a conserved 1-form current $J = d\phi$ .
- Hydrodynamic: Includes both stress-energy conservation and the irrotationality constraint ( $dJ=0$ ).
- Result: The framework handles the overdetermined nature of the system (more equations than variables in high dimensions) and correctly maps the irrotational condition to the conservation of the 1-form current.
Magnetohydrodynamics (MHD) of a $p$ -form Gauge Field:
- Microscopic: A $p$ -form gauge field $A$ coupled to branes.
- Hydrodynamic: Generalized MHD where the field strength $F=dA$ is treated as a conserved current.
- Result: The authors derive a parameterization of the stress-energy tensor and the field strength $F$ in terms of hydrodynamic variables (density, velocity, and auxiliary forms $h, k$ ). This recovers known MHD equations and generalizes them to arbitrary $p$ -form symmetries.

5. Significance

Foundational Rigor: It moves hydrodynamics from a phenomenological set of equations to a structured geometric object within the BV formalism, allowing for potential quantization and deformation studies.
Symmetry Topological Field Theories (SymTFTs): The cospan structure mirrors the structure of SymTFTs, suggesting a deep connection between generalized symmetries in QFT and the hydrodynamic limit.
Future Applications: The framework opens the door to:
- Systematic derivation of hydrodynamic transport coefficients from microscopic data using dg-geometry.
- Studying anomalies and topological terms in hydrodynamics via the $BF$ mediator.
- Extending the formalism to include entropy production (currently neglected) and discrete symmetries (which require cohomology with discrete coefficients rather than de Rham forms).

In summary, the paper successfully reinterprets hydrodynamics not merely as an approximation, but as a specific geometric realization (a cospan) of the conservation laws inherent in a microscopic theory, mediated by a topological $BF$ theory. This provides a powerful new language for analyzing the interplay between micro and macro physics in the presence of generalized symmetries.