Authors: Marcelo Maciel Amaral
Journal: Gauge Freedom Field Notes (Preprint)
Abstract
We derive the classical equations of motion for nonlinear sigma models where the base space is a statistical manifold equipped with the Fisher–Rao metric. For canonical distributions (Bernoulli, Gaussian, Categorical, …) we compute the Laplace–Beltrami operators explicitly and show the resulting base geometries are constant-curvature (spherical, flat, or hyperbolic). This sets up a program where information geometry directly governs field dynamics.
Significance
- It ties field theory to statistical distinguishability, replacing an arbitrary base metric with the Fisher–Rao structure.
- Offers a route to data-informed effective theories: measured distributions can determine the base geometry and hence the dynamics.
- Bridges information geometry with sigma models, CFT/string intuitions, and potential AI/learning applications.
Key Findings
- A clean derivation of Euler–Lagrange equations for sigma models with Fisher–Rao base.
- Closed-form Laplace–Beltrami operators for standard families (Bernoulli, Gaussian, Categorical).
- Curvature classification of those bases (S² / ℝⁿ / Hⁿ analogues depending on family parameters).
- A roadmap to gauge-coupling base isometries and exploring curvature-squared terms.
Future Outlook
- Higher-loop structure and curvature-squared corrections on Fisher bases.
- Gauging base isometries and coupling to target-space gauge fields.
- Data-driven pipelines where empirical distributions fix the base geometry.
- Aperiodic order and tiling spaces applications.
How to cite
Marcelo Maciel Amaral (2025). Nonlinear Sigma Models on Statistical Manifolds: Equations of Motion. Gauge Freedom Field Notes. https://doi.org/10.65323/gf-lab.2025.001