⚛️ high-energy theory

10D Supergravity Numerical Data Sets for L & R Matrices

This paper reviews 10D superspace theories, derives the super-current and non-closure terms for on-shell $\mathcal{N}=1$ linearized supergravity, and provides complete numerical datasets for the $L_{\rm I}$ and $R_{\rm I}$ adjacency matrices as a preliminary step toward embedding them into off-shell formulations.

Original authors: Jacob Cigliano, Bergen Dahl, S. James Gates Jr

Published 2026-09-10

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CC BY 4.0

Original authors: Jacob Cigliano, Bergen Dahl, S. James Gates Jr

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

In the vast landscape of modern physics, there is a persistent quest to find a single, unified set of rules that governs everything from the smallest subatomic particles to the largest galaxies. For decades, scientists have explored a framework called supersymmetry, a concept that proposes every known particle has a heavier, invisible partner. This idea is particularly powerful in theories that attempt to describe gravity alongside other forces, such as string theory, which suggests that the fundamental building blocks of the universe are tiny, vibrating strings rather than point-like dots. However, a major hurdle has long stood in the way of fully understanding these theories: while physicists can describe how these systems behave when particles are moving and interacting in a specific, simplified way, they have struggled to describe the same systems when they are "off-shell." In the language of the field, being "on-shell" means the particles obey their standard equations of motion, like a car driving smoothly down a highway. Being "off-shell" is like the car being in a garage, where it is not moving but still exists as a complete object with all its parts intact. Finding a description that works in the garage is essential for a complete theory, yet for ten-dimensional supergravity—a version of gravity that includes supersymmetry and exists in a universe with ten dimensions—this off-shell description has remained elusive.

A team of researchers at the University of Maryland, led by Jacob Cigliano, Bergen Dahl, and S. James Gates, Jr., has taken a significant step toward solving this decades-old puzzle. They have performed a rigorous, detailed calculation of the mathematical relationships that govern a specific version of ten-dimensional supergravity. Their work focuses on a set of fields that include the graviton, the particle responsible for gravity, and its supersymmetric partner, the gravitino, along with other related fields like a scalar field and a two-form field. By working through the complex algebra of how these fields transform into one another under supersymmetry, the team has derived a precise set of numerical data. Specifically, they have constructed two large tables of numbers, known as adjacency matrices, which act as a map showing exactly how the different components of the theory connect and interact. These matrices are not just abstract numbers; they are the result of explicitly calculating how the theory behaves when reduced to a single point in time, stripping away spatial complexity to reveal the core structure of the relationships.

The researchers began by writing down the most general possible equations for the energy and motion of these fields, ensuring they respected the fundamental symmetries of the universe. They then calculated how these fields change when acted upon by a supersymmetry generator, a mathematical operation that swaps particles of different types. A critical part of their work involved checking whether these transformations "close," meaning that if you perform two supersymmetry operations in a row, you end up with a result that is consistent with the laws of physics, such as a simple shift in time or a gauge transformation. They found that for the bosonic fields, which include the graviton and other force-carrying particles, the transformations close perfectly. However, for the fermionic fields, which include the gravitino and the dilatino, the transformations do not close completely unless the particles are on their standard equations of motion. This "non-closure" is a known feature of this specific theory in its current form, and the team explicitly calculated the exact terms that remain when the algebra fails to close. These remaining terms are not errors; they are the specific mathematical signatures that tell physicists exactly what is missing to make the theory complete off-shell.

Having established the precise rules of the on-shell theory, the team then translated these rules into the language of graphs and matrices. They reduced the ten-dimensional fields to a single time dimension, a process that simplifies the problem by removing spatial dependencies while keeping the essential algebraic structure intact. In this reduced view, the bosonic fields amount to 82 independent components, and the fermionic fields amount to 176 independent components. The researchers then computed two massive matrices: one that describes how the 16 different supersymmetry charges transform the 82 bosonic components into the 176 fermionic components, and another that describes the reverse process. These matrices, which they have made available as a public data set, are the first time such a complete numerical representation has been derived directly from the on-shell description of ten-dimensional supergravity. They serve as a concrete, unambiguous record of the theory's structure, free from the ambiguity of symbolic notation.

The significance of this work lies in what it enables for the future. The researchers describe their matrices as a piece of a larger puzzle. In previous work, other scientists had proposed various candidate structures, called prepotentials, that could describe the off-shell version of this theory. These candidates were like blueprints for a building that had never been constructed. The new matrices generated by Cigliano, Dahl, and Gates represent the actual, measured dimensions of the foundation of the on-shell theory. By comparing their data against the proposed blueprints, physicists can now systematically test which candidate, if any, correctly contains the on-shell theory as a subset. The team acknowledges that they do not yet have the software tools to perform this final comparison, but they have provided the necessary raw material. Their work does not solve the off-shell problem immediately, but it provides the first rigorous, numerical bridge between the known on-shell reality and the unknown off-shell possibilities. By turning a complex web of theoretical relationships into a clear set of numbers, they have given the community a new tool to finally determine whether a complete, off-shell description of ten-dimensional supergravity exists, and if so, what it looks like.

Technical Summary: 10D Supergravity Numerical Data Sets for L & R Matrices

Problem Statement
The primary objective of this work is to address the challenge of constructing an off-shell formulation for ten-dimensional ( $D=10$ ), $N=1$ linearized supergravity. While the on-shell theory is well-established, the existence of a complete off-shell supergravity multiplet (one that closes the supersymmetry algebra without relying on equations of motion or auxiliary fields) remains an open problem. Recent developments in the 2020s have introduced "Adynkra" and "Adinkra" graphical methods to analyze supermultiplets, requiring specific numerical input in the form of adjacency matrices ( $L$ and $R$ ) derived from the on-shell theory. This paper aims to provide the necessary numerical data sets to bridge the gap between established on-shell supergravity results and these new off-shell search methodologies.

Methodology
The authors undertake a rigorous, explicit calculation within the framework of on-shell $N=1$ linearized supergravity in $D=10$ . The methodology proceeds through the following steps:

Formulation of the On-Shell Theory: The authors define the field content of the supermultiplet: the graviton ( $h_{\mu\nu}$ ), gravitino ( $\psi_{\mu a}$ ), two-form ( $B_{\mu\nu}$ ), dilaton ( $\phi$ ), and dilatino ( $\chi_a$ ). They construct the most general quadratic free-field Lagrangian consistent with Lorentz invariance, reality, and engineering dimensions.
Derivation of Transformation Laws: By imposing the closure of the supersymmetry algebra on-shell and normalizing the supercurrent, the authors uniquely determine the coefficients for the supersymmetry transformation laws ( $Q_a$ ) acting on these component fields.
Algebraic Closure and Non-Closure Analysis: The commutators of two supersymmetry transformations $\{Q_a, Q_c\}$ are explicitly computed. The authors demonstrate that while the bosonic sector closes into translations and gauge transformations, the fermionic sector yields "non-closure terms" proportional to the equations of motion (Rarita-Schwinger and Dirac equations). These terms are explicitly isolated and categorized.
Supercurrent Construction: Using the Noether procedure, the conserved supercurrent is derived, ensuring compatibility with the closure relations and canonical kinetic term normalizations.
0-Brane Reduction and Matrix Extraction: To generate the required data for Adynkra analysis, the theory is dimensionally reduced to one time dimension (0-brane reduction). Spatial dependencies are removed, and specific gauge choices are applied ( $h_{0\nu} = 0$ $h_{0 ν} = 0$ and $B_{0\nu} = 0$ $B_{0 ν} = 0$ ).
- The bosonic degrees of freedom are reduced to $n_b = 82$ independent components.
- The fermionic degrees of freedom are reduced to $n_f = 176$ independent components.
- The supersymmetry transformations are recast as linear maps between these bosonic and fermionic vectors.
Matrix Generation: The authors compute the explicit numerical forms of the $L_I$ (Left) and $R_I$ (Right) adjacency matrices, where $I$ runs from 1 to 16 (representing the 16 supercharges). These matrices satisfy the 1-dimensional "Garden algebra" relations, including terms encoding the non-closure of the on-shell algebra.

Key Contributions and Results

Explicit Coefficients: The paper provides the unique set of coefficients for the linearized supersymmetry transformations in $D=10$ , derived strictly from on-shell closure and supercurrent normalization.
Supercurrent Formula: A complete, explicit expression for the supercurrent $(J_\mu)_a$ is presented, including all interaction terms between the graviton, gravitino, two-form, and dilaton sectors.
Non-Closure Tensors: The specific tensor structures representing the failure of the algebra to close off-shell (the $E_{IJ}$ tensor) are explicitly calculated and displayed.
Numerical Data Sets: The primary result is the generation of the $L_I$ $L_{I}$ and $R_I$ $R_{I}$ matrices.
- $L_I$ matrices are of dimension $82 \times 176$ .
- $R_I$ matrices are of dimension $176 \times 82$ .
- These matrices are provided as numerical data sets accessible via a GitHub repository, rather than being printed in the text due to their size.
Algebraic Verification: The matrices are shown to satisfy the 0-brane Garden algebra: $L_I R_J + L_J R_I = 2\delta_{IJ} I_{82\times82}$ and $R_I L_J + R_J L_I = 2\delta_{IJ} I_{176\times176} + 2E_{IJ}$ , where $E_{IJ}$ encodes the on-shell non-closure.

Significance and Future Pathways
The authors position this work as a "preliminary step" and a "library" of data essential for advancing the search for off-shell supergravity.

Enabling Off-Shell Searches: The paper argues that these matrices serve as the necessary input to "scan" and embed the known on-shell $D=10$ supergravity into potential off-shell prepotential candidates.
Integration with Adynkra/Adynkra Methods: The results are intended to be used in conjunction with "Adynkra" technology (which maps superfield expansions to representation-theoretic data) and previous work on $D=10$ prepotentials.
The "Jigsaw Puzzle" Analogy: The authors describe the on-shell matrices derived here as the "right-hand side" (on-shell) piece of a puzzle, while previous work on prepotentials provides the "left-hand side" (off-shell candidate) pieces. The goal is to determine how the on-shell data fits into the off-shell candidates to solve the off-shell problem.
Limitations: The authors explicitly state that solving the full off-shell problem is currently impossible due to "insufficiency of the appropriate codes" to perform the necessary merging of these data sets. Thus, the paper claims significance not as a solution to the off-shell problem, but as the provision of the critical numerical infrastructure required to attempt that solution.

In summary, this paper does not propose new physical theories or experimental tests but provides a foundational computational resource: the explicit numerical adjacency matrices for $D=10$ linearized supergravity, derived from first principles, to facilitate the application of modern graphical algebraic methods to the problem of off-shell supersymmetry.

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