[Paper Review] Random Lattice Gauge Theories and Differential Forms

Fernando L. Teixeira|arXiv (Cornell University)|Apr 11, 2013

Electromagnetic Simulation and Numerical Methods132 references3 citations

TL;DR

This paper presents a framework for formulating random lattice gauge theories using the exterior calculus of differential forms, employing discrete analogues of the exterior derivative and Hodge star operator to separate metric-free combinatorial structure from metric-dependent physics. The approach ensures consistency on irregular lattices by leveraging Whitney forms and primal-dual cell complexes, enabling stable, geometrically flexible simulations of field theories such as electrodynamics with exact conservation laws and reduced numerical artifacts.

ABSTRACT

We provide a brief overview on the application of the exterior calculus of differential forms to the ab initio formulation of field theories on random simplicial lattices. In this framework, discrete analogues of the exterior derivative and the Hodge star operator are employed for the factorization of discrete field equations into a purely combinatorial (metric-free) part and a metric-dependent part. The Hodge star duality (isomorphism) is invoked to motivate the use of primal and dual lattices (a dual cell complex). The natural role of Whitney forms in the construction of discrete Hodge star operators is stressed.

Motivation & Objective

• To develop a consistent ab initio formulation of field theories on irregular (random) simplicial lattices using the language of differential forms.
• To resolve numerical instabilities, convergence issues, and spurious modes in lattice simulations caused by inconsistent discretization of differential calculus on skewed or hybrid meshes.
• To establish a geometrically consistent discrete calculus by employing Whitney forms and dual lattices, ensuring exactness of the de Rham complex and conservation laws.
• To unify disparate numerical methods—such as finite elements, mimetic finite differences, and the cell method—under a common framework rooted in exterior calculus.
• To provide a tutorial-level, accessible foundation for applying discrete exterior calculus to non-perturbative lattice field theory, particularly in electrodynamics and gauge theories.

Proposed method

• Represent fields as discrete differential forms (cochains) on a primal simplicial lattice, with degrees of freedom defined via integration over p-chains.
• Define discrete exterior derivative as a coboundary operator acting on cochains, preserving the exactness property of the de Rham complex.
• Construct the discrete Hodge star operator using Whitney forms to map p-cochains to (n−p)-cochains, encoding metric information in a geometrically consistent way.
• Employ a dual cell complex (dual lattice) to model the Hodge star duality, enabling consistent pairing between primal and dual elements.
• Ensure compatibility between operators by enforcing commuting diagram structures (e.g., Tonti diagrams), which preserve fundamental identities like d² = 0.
• Use Whitney forms as basis functions for finite element-type discretization, enabling conforming approximations to Sobolev spaces such as H(curl) and H(div).

Experimental results

Research questions

• RQ1 How can field theories be consistently discretized on irregular, random simplicial lattices without introducing numerical instabilities or spurious modes?
• RQ2 What role do Whitney forms play in constructing stable, geometrically consistent discrete Hodge star operators on unstructured meshes?
• RQ3 How does the use of primal and dual lattices, combined with discrete differential forms, preserve the exactness of the de Rham complex in non-orthogonal or skewed lattices?
• RQ4 In what way does the exterior calculus framework unify finite element, mimetic finite difference, and cell method approaches in lattice field theory?
• RQ5 Can the ab initio formulation of electrodynamics on random lattices achieve exact conservation laws and correct dispersion properties without relying on finite difference approximations?

Key findings

• The use of discrete differential forms and Whitney-based Hodge star operators enables a consistent, metric-dependent discretization of field equations on irregular lattices, reducing numerical artifacts.
• Discrete exterior calculus ensures that the fundamental identity d² = 0 is preserved, which underpins the exactness of the de Rham complex and prevents spurious modes.
• The framework naturally incorporates conservation laws (e.g., charge conservation) through the exactness of the discrete complex and the use of dual lattices.
• Whitney forms provide a geometrically consistent basis for constructing conforming finite element spaces, such as Nedelec and Raviart-Thomas elements, ensuring stability in curl- and div-conforming problems.
• The method achieves better convergence and reduced dispersion errors on non-orthogonal or random lattices compared to standard finite difference schemes.
• The approach unifies multiple numerical methods—finite elements, mimetic finite differences, and the cell method—by embedding them within a common algebraic-topological framework based on cochains and dual complexes.