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Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation

Abstract

We describe a systematic mathematical approach to the geometric discretization of gauge field theories that is based on Dirac and multi-Dirac mechanics and geometry, which provide a unified mathematical framework for describing Lagrangian and Hamiltonian mechanics and field theories, as well as degenerate, interconnected, and nonholonomic systems. Variational integrators yield geometric structure-preserving numerical methods that automatically preserve the symplectic form and momentum maps, and exhibit excellent long-time energy stability. The construction of momentum-preserving variational integrators relies on the use of group-equivariant function spaces, and we describe a general construction for functions taking values in symmetric spaces. This is motivated by the geometric discretization of general relativity, which is a second-order covariant gauge field theory on the symmetric space of Lorentzian metrics.

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Acknowledgements

The author would like to thank Evan Gawlik for helpful discussions and comments. The author has been supported in part by the National Science Foundation under Grants DMS-1010687, CMMI-1029445, DMS-1065972, CMMI-1334759, DMS-1411792, DMS-1345013, DMS-1813635.

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Correspondence to Melvin Leok.

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Communicated by Wolfgang Dahmen.

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Leok, M. Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation. Found Comput Math 19, 965–989 (2019). https://doi.org/10.1007/s10208-019-09420-4

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  • DOI: https://doi.org/10.1007/s10208-019-09420-4

Keywords

  • Geometric numerical integration
  • Variational integrators
  • Symplectic integrators
  • Hamiltonian field theories
  • Manifold-valued data
  • Gauge field theories
  • Numerical relativity

Mathematics Subject Classification

  • 37M15
  • 53C35
  • 65D05
  • 65M70
  • 65P10
  • 70H25