Differential Forms in Lattice Field Theories: An Overview

We provide an overview on the application of the exterior calculus of differential forms to the ab initio formulation of lattice field theories, with a focus on irregular or “random” lattices. 1. Introduction The need to formulate field theories on a lattice (mesh, grid) arises from two main reasons, which may occur simultaneously or not. First, the lattice provides a natural “regularization” of divergences in lieu of renormalization techniques [1]. Such regularization does not need to be viewed as an ad hoc step, but instead as a natural consequence of assuming the field theory to be, at some fundamental level, an effective (“low’’-energy) description [2]. Second, the lattice provides a direct route to compute, in a nonperturbative fashion, quantities of interest by numerical simulations. Nontrivial domains and complex boundary conditions can then be easily treated as well [3–6]. For these, the use of irregular (“random”) lattices are often of interest to gain geometrical flexibility. Irregular lattices are also of interest as a means to provide a potentially faster convergence to the continuum limit, near-isotropic lattice dispersion properties, and better “conservation” of some (e.g., long-range translational and rotational) symmetries [7, 8]. In some cases, irregular lattices are useful for universality tests as well [9, 10]. Lattice theories are typically developed by taking the counterpart continuum theory as starting point and then applying discretization techniques whereby derivatives are approximated by finite differences or some constraints are enforced on the functional space of admissible solutions to be spanned by a finite set of “basis” functions (e.g., “Galerkin methods” such as spectral elements and finite elements). These discretization strategies have proved very useful in many settings; however, they often produce difficulties in the case of irregular (“random”) lattices. Among such difficulties are (i) numerical instabilities in marching-on-time algorithms (regardless of the time integration method used), (ii) convergence problems in algorithms relying on iterative linear solvers, and (iii) spurious (“ghost”) modes and/or extraneous degrees of freedom. These problems often (but not always) appear associated with highly skewed or obtuse lattice elements, or at the boundary between heterogeneous (hybrid) lattices subcomponent, comprising overlapped domains or “mesh-stitching” interfaces, for example. Clearly, such difficulties put a constraint on the geometric flexibility that irregular lattices are intended for, and may require