⚛️ high-energy theory

Conserved charges and $L_\infty$ algebras

This paper presents a novel formula for computing conserved charges in arbitrary Lagrangian field theories using only $L_\infty$ algebra data, thereby enabling calculations in nonlocal models like string field theory where conventional derivative-based methods fail, while also successfully recovering known results for general relativity and Yang-Mills theory.

Original authors: Vinícius Bernardes, Theodore Erler, Atakan Hilmi Fırat

Published 2026-06-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Vinícius Bernardes, Theodore Erler, Atakan Hilmi Fırat

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Finding the "Hidden Wallets" of the Universe

Imagine the universe is a giant, complex machine. In physics, when this machine runs, certain things stay the same no matter how the machine moves. These are called conserved quantities (like energy, momentum, or electric charge).

Usually, to find these "hidden wallets" of the universe, physicists use a specific recipe: they look at the Lagrangian. Think of the Lagrangian as the machine's instruction manual. It tells you how every part moves based on its position and how fast it's changing (its derivatives).

The Problem:
For simple machines (like a ball rolling down a hill), this recipe works perfectly. But for the most complex machines in physics—like String Theory—the instruction manual is broken. It involves infinite derivatives and "non-local" interactions (where a part of the machine affects another part instantly, regardless of distance). In these cases, the traditional recipe fails. You can't easily define "position" and "momentum," so you can't find the conserved charges.

The Solution:
This paper introduces a new, universal recipe. Instead of looking at the messy instruction manual (the Lagrangian), the authors look at the machine's algebraic skeleton. They use a mathematical structure called an $L_\infty$ algebra.

Think of an $L_\infty$ algebra not as a set of rules for movement, but as a recipe for relationships. It describes how different parts of the system interact with each other, regardless of whether those interactions happen locally or across the whole universe.

The Key Ingredients

To understand their new formula, we need to understand three main concepts the authors use:

1. The $L_\infty$ Algebra (The Relationship Map)

Imagine a social network. In a normal physics theory, you only care about who is standing next to whom. In this new approach, you care about the entire web of connections. The $L_\infty$ algebra is a map of all possible interactions, from simple pairs to complex groups of particles interacting at once.

Why it matters: This map exists even for the weird, non-local theories where the old "instruction manual" doesn't work.

2. The Sigmoid (The Time-Slice Switch)

The authors introduce a special tool called a sigmoid.

The Analogy: Imagine you are watching a movie. You want to know the total energy of the movie right now. But the movie is a continuous flow. The sigmoid is like a dimmer switch that slowly turns the movie "on" in the past and "off" in the future.
The Function: It acts as a filter. It isolates a specific moment in time (a "slice") so you can measure the charge without the measurement getting blurred by the infinite past or future. It turns a messy, continuous flow into a clean, measurable snapshot.

3. The Conserved Charge (The Result)

The paper provides a specific formula to calculate the "charge" (like energy or momentum) using only the relationship map ( $L_\infty$ ) and the time-slice switch (sigmoid).

The Magic: The formula works by looking at how the symmetry of the system (how it looks the same if you shift it in time or space) interacts with the sigmoid. If the symmetry and the sigmoid "fight" each other (they don't commute), that friction creates the conserved charge.

What They Actually Did (The Examples)

The authors didn't just write a theory; they tested it on three different "machines" to prove it works:

The Simple Ball (Scalar Field):
They applied their formula to a standard, simple physics problem (a scalar field in flat space).
- Result: It correctly calculated the Stress-Energy Tensor (the standard way physicists measure energy and momentum). This proved their new, complex formula agrees with the old, trusted methods for simple cases.
The Ghostly String (p-adic String Theory):
This is a "non-local" theory where particles interact across time and space in a weird way. The old method of finding energy breaks down here.
- Result: They used their formula to find the Hamiltonian (the total energy) of this ghostly string.
- Verification: They compared their result to a previous calculation done by other scientists using a very different, complicated method. They matched perfectly. This is a huge win because it shows their method works for the weird, non-local stuff where the old rules fail.
The Edge of the World (Surface Charges):
In theories like General Relativity (gravity) and Yang-Mills (electromagnetism), charges often live on the boundaries (the edges of the universe or a black hole's surface).
- Result: They showed their formula naturally handles these boundaries.
- Gravity: They derived the Brown-York stress tensor, which describes the energy of a gravitational system at its boundary.
- Yang-Mills: They found the "Killing charge," which includes contributions from the boundary.
- Significance: They did this without needing to add special "boundary terms" to the action manually. The formula seems to automatically know how to handle the edges of the universe.

The Bottom Line

This paper is a toolkit upgrade.

Old Toolkit: Requires a clean, local instruction manual (Lagrangian). Fails for String Theory and complex boundaries.
New Toolkit: Uses the algebraic skeleton ( $L_\infty$ ) and a time-slice switch (sigmoid).
The Claim: You can now calculate conserved charges (like energy) for any field theory, even the weird, non-local ones like String Theory, without needing to know the derivative structure of the Lagrangian.

The authors are essentially saying: "We found a way to weigh the universe's energy even when the scale itself is broken, by looking at the relationships between the parts rather than the parts themselves."

Technical Summary: Conserved Charges and $L_\infty$ Algebras

Problem Statement
The paper addresses the challenge of defining and computing conserved charges in arbitrary Lagrangian field theories, particularly those that are nonlocal or possess higher-derivative interactions where conventional canonical methods fail. In standard field theory, conserved charges (such as energy or momentum) are typically derived via Noether's theorem, relying on the derivative structure of the Lagrangian to define canonical coordinates and momenta. However, in theories like String Field Theory (SFT), interactions involve infinite derivatives, rendering the standard phase space construction ill-defined. Furthermore, the treatment of spatial boundaries in these nonlocal contexts remains problematic, as conventional variational principles often require specific boundary terms (e.g., Gibbons-Hawking-York) that are difficult to formulate for nonlocal actions. The authors aim to construct a universal formula for conserved charges that relies solely on the algebraic data of the theory, bypassing the need for a local Lagrangian density or a well-posed initial value problem.

Methodology
The authors utilize the framework of cyclic $L_\infty$ algebras to reformulate field theory. In this approach, the action is defined by a set of higher-order products ( $L_n$ ) and a Batalin-Vilkovisky (BV) inner product ( $\omega$ ), rather than a local Lagrangian density.

Formalism Setup: The action is expressed as $S = -\frac{1}{2}\omega(\Phi, Q\Phi) - \sum \frac{1}{(n+1)!}\omega(\Phi, L_n(\Phi, \dots, \Phi))$ , where $\Phi$ is the dynamical field and $Q$ is the kinetic operator. The equations of motion are given by the Euler-Lagrange state $q_\Phi = 0$ .
Symplectic Structure: Building on previous work [1], the authors employ a symplectic form on the phase space defined by $\Omega = \frac{1}{2}\omega(\delta\Phi, [Q_\Phi, \sigma]\delta\Phi)$ , where $\sigma$ is a "sigmoid" operator. The sigmoid acts as a time-slicing tool, transitioning from 0 to 1, effectively localizing the symplectic structure to finite times without requiring a specific Cauchy surface.
Derivation of Charges: A symmetry of the action is represented by a vector field $\xi$ with a generating parameter $\xi_\Phi$ and a generating operator $\Xi_\Phi$ . The authors derive the conserved charge $F_\xi$ by identifying the symmetry as a Hamiltonian vector field on the phase space. They utilize a "tau regularization" technique to handle temporal boundary terms, ensuring the cyclicity of operators holds during intermediate steps.
Coalgebra Perspective: The paper also presents the results using the coalgebra description of $L_\infty$ algebras, where the action and symmetries are encoded as coderivations on a tensor algebra. This provides an algebraic proof of the charge formula and clarifies the role of the sigmoid in preventing the commutation of symmetry and action products.

Key Contributions and Results

Universal Charge Formula: The primary result is a general formula for the conserved charge associated with a symmetry $\xi$ :
$F_\xi = \sum_{n=0}^\infty \sum_{k=0}^n \frac{1}{(n+1)!} \binom{n}{k} \omega \left( \Phi, \Xi_{k+1}(\sigma L_{n-k}(\Phi, \dots), \Phi, \dots) - L_{k+1}(\sigma \Xi_{n-k}(\Phi, \dots), \Phi, \dots) \right)$
This expression depends only on the $L_\infty$ data ( $L_n$ , $\Xi_n$ , $\omega$ ) and the sigmoid $\sigma$ , independent of the derivative structure of the Lagrangian.
Total Derivative Form: The authors derive a computationally convenient "total derivative" form:
$F_\xi = -\omega(\sigma \xi_\Phi, q_\Phi)$
When expanded, this yields a boundary term involving the derivative of the sigmoid, $\nabla_\mu \sigma$ , which localizes the charge to a finite time slice.
Nonlocal Application (p-adic String Theory): The formalism is successfully applied to p-adic string theory, a prototypical nonlocal model. The authors compute the Hamiltonian and demonstrate that it matches the energy functional derived by Moeller and Zwiebach [15] via a derivative expansion, validating the method for nonlocal theories.
Surface Charges and Boundaries: The paper investigates the behavior of charges in the presence of spatial boundaries.
- Yang-Mills: The derived surface charge matches the known expression for non-abelian gauge charge, naturally incorporating boundary contributions without ad-hoc boundary terms in the action.
- General Relativity: The formula correctly reproduces the surface charge of General Relativity in terms of the Brown-York stress tensor. The authors show that the necessary boundary contributions arise automatically from the formalism, even without explicitly adding the Gibbons-Hawking-York term to the action.
Killing Charges: In the presence of spatial boundaries, the Killing charge of Yang-Mills theory is shown to include a nontrivial boundary contribution arising from the gauge field, distinct from the standard stress tensor contribution.

Properties of the Charge
The derived charge satisfies several critical properties:

Gauge Invariance: It is a well-defined function on the physical phase space (gauge invariant on-shell).
Linearity: It is linear with respect to the symmetry generator.
Conservation: It is independent of the specific choice of the sigmoid, provided the boundary conditions (0 at $t \to -\infty$ , 1 at $t \to \infty$ ) are preserved.
On-Shell Nature: The charge is an on-shell observable; it does not define a gauge-invariant deformation of the action off-shell.

Significance and Claims
The paper claims that this approach extends Noether's theorem to a broader class of theories, specifically those where the canonical construction of phase space is obstructed by nonlocality or infinite derivatives. The authors emphasize that the method requires minimal input from the variational principle regarding boundary terms, as the correct boundary contributions for surface charges appear naturally within the $L_\infty$ framework.

The ultimate goal stated is to provide a fully nonperturbative (in $\alpha'$ ) framework for computing phase space observables in String Field Theory, such as energy and black hole entropy. While the current paper focuses on conventional and p-adic models to illustrate the formalism, the authors position this work as a foundational step toward applying these tools to full string field theory, where traditional variational principles are difficult to formulate. The work is presented as a necessary precursor to constructing Poisson brackets and a full Hamiltonian mechanics for these complex systems.