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Homotopy lattice gauge fields 1: The fields and their properties

Juan Orendain, Ivan Sanchez, José A. Zapata

Abstract

We introduce homotopy lattice gauge fields (HLGFs), a version of gauge fields over a discretized base, based on a notion of higher parallel transport that enriches the usual parallel transport along paths on a lattice to also consider higher dimensional paths. Higher dimensional data keeps information about the parallel transport along homotopies of curves. With this data, a HLGF on a base space of dimension two or three determines a principal bundle over the base manifold. This data is also responsible for our formulas for the topological charge on two-dimensional bases. Our framework is an application of a nonabelian algebraic topology framework developed to solve the local to global problem in higher dimensional homotopy. No previous knowledge of higher category theory is assumed. The second part will be devoted to the space of fields as an arena for doing Quantum Field Theory, and to give the first examples of how our framework refines standard lattice gauge theory.

Homotopy lattice gauge fields 1: The fields and their properties

Abstract

We introduce homotopy lattice gauge fields (HLGFs), a version of gauge fields over a discretized base, based on a notion of higher parallel transport that enriches the usual parallel transport along paths on a lattice to also consider higher dimensional paths. Higher dimensional data keeps information about the parallel transport along homotopies of curves. With this data, a HLGF on a base space of dimension two or three determines a principal bundle over the base manifold. This data is also responsible for our formulas for the topological charge on two-dimensional bases. Our framework is an application of a nonabelian algebraic topology framework developed to solve the local to global problem in higher dimensional homotopy. No previous knowledge of higher category theory is assumed. The second part will be devoted to the space of fields as an arena for doing Quantum Field Theory, and to give the first examples of how our framework refines standard lattice gauge theory.
Paper Structure (19 sections, 59 equations, 21 figures)

This paper contains 19 sections, 59 equations, 21 figures.

Table of Contents

  1. Introduction: The lattice homotopy cutoff, a better cutoff for gauge fields
  2. Preparing gauge fields in the continuum for the homotopy lattice cutoff
  3. Parallel transport
  4. Higher homotopy parallel transport in the continuum
  5. The homotopy lattice cutoff: Essential ingredients
  6. The homotopy lattice cutoff: A preliminary example
  7. HLGFs as higher parallel transport maps on the homotopy lattice
  8. Prelude to nonabelian algebraic topology
  9. Abstract HLGFs and the higher homotopy Atiyah groupoid
  10. HLGFs after a trivialization over vertices
  11. Gauge transformations
  12. The homotopy lattice cutoff
  13. A set of generators for HLGFs: ELGFs
  14. General higher gauge fields on the homotopy lattice
  15. The topological charge and other topological considerations
  16. ...and 4 more sections

Figures (21)

  • Figure 1:
  • Figure 2: The blue singular curves belong to $\tilde{P}(M, X_0)$, and the red does not.
  • Figure 3: A homotopy of singular curves is a singular 2-globe sharing endpoints. The gauge field transports homotopies of initial conditions over the source along a singular 2-globe to a homotopy of final conditions over the target.
  • Figure 4:
  • Figure 5:
  • ...and 16 more figures

Theorems & Definitions (1)

  • Remark 1