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2D discrete Yang-Mills equations on the torus

Volodymyr Sushch

TL;DR

This work develops a geometry-preserving discretization of the two-dimensional Yang-Mills equations on a torus via discrete exterior calculus. It defines a discrete covariant derivative $d^c_A$, its adjoint $\delta_A^c$, and a curvature $F=d^cA+A\cup A$, formulating the discrete YM equation $d_A^c\ast F=0$ along with the Bianchi identity and a discrete Laplacian $\Delta_A^c$. The authors specialize to a combinatorial torus ($N=M=2$), derive explicit component-wise difference equations for the YM system, and provide a comprehensive matrix representation using operators $D$, $S$, $D_1$, $D_2$, and $I_A$. This yields a concrete, implementable framework for structure-preserving numerical simulations of YM in 2D toroidal geometries. The approach demonstrates how discrete exterior calculus can faithfully capture gauge-theoretic structure in a lattice-like setting while enabling efficient, algebraic solution methods.

Abstract

In this paper, we introduce a discretization scheme for the Yang-Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang-Mills equations are presented in the form of both a system of difference equations and a matrix form.

2D discrete Yang-Mills equations on the torus

TL;DR

This work develops a geometry-preserving discretization of the two-dimensional Yang-Mills equations on a torus via discrete exterior calculus. It defines a discrete covariant derivative , its adjoint , and a curvature , formulating the discrete YM equation along with the Bianchi identity and a discrete Laplacian . The authors specialize to a combinatorial torus (), derive explicit component-wise difference equations for the YM system, and provide a comprehensive matrix representation using operators , , , , and . This yields a concrete, implementable framework for structure-preserving numerical simulations of YM in 2D toroidal geometries. The approach demonstrates how discrete exterior calculus can faithfully capture gauge-theoretic structure in a lattice-like setting while enabling efficient, algebraic solution methods.

Abstract

In this paper, we introduce a discretization scheme for the Yang-Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang-Mills equations are presented in the form of both a system of difference equations and a matrix form.
Paper Structure (5 sections, 9 theorems, 102 equations)

This paper contains 5 sections, 9 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Background on a discrete model
  3. Discrete Yang-Mills equations
  4. Discrete Yang-Mills equations on a combinatorial torus
  5. Conclusions

Key Result

Proposition 2.1

Let $\Phi\in K^r(2)$ and $\Omega\in K^{r+1}(2)$, $r=0,1$. Then we have where and $\ast^{-1}$ is the inverse of $\ast$.

Theorems & Definitions (18)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • ...and 8 more