A simplicial gauge theory

Christiansen, Snorre H.; Halvorsen, Tore Gunnar

doi:10.1063/1.3692167

Cited by 17 publications

(31 citation statements)

References 34 publications

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“…While the regularity of Regge elements seems adapted to general relativity theory, higher regularity, as achieved here, could be important to other PDEs in Riemannian geometry, such as those treated in [13,23,42]. Lattice Gauge Theory (LGT) [82], as extended to a finite element context [31], was also at the back of our minds during this work. In LGT, one defines discrete connections and curvature, as well as a discrete Yang-Mills functional, but it is less clear what the discrete covariant exterior derivative and Bianchi identity should be.…”

Section: Finite Element Elasticity Complexesmentioning

confidence: 80%

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Finite Element Systems for Vector Bundles: Elasticity and Curvature

Christiansen

2022

Found Comput Math

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We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes.

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Section: Finite Element Elasticity Complexesmentioning

confidence: 80%

“…LGT was initially defined for cubical complexes [82]. An analogue for simplicial complexes was developed in [31]. There, a discrete vector bundle corresponds to a choice of vector space attached to vertices only, whereas we here associate a vector space to each cell, of every dimension, in T .…”

Section: Discrete Vector Bundles With Connectionmentioning

confidence: 99%

Finite Element Systems for Vector Bundles: Elasticity and Curvature

Christiansen

2022

Found Comput Math

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“…A study of classical lattice gauge theory on simplicial lattices that allows for a Noether's theorem which is closely related to what we present in this section is presented in [14].…”

Section: Local Lagrangian Field Theory On Discrete Spacetimementioning

confidence: 99%

Multisymplectic effective general boundary field theory

Arjang¹,

Zapata²

2014

Class. Quantum Grav.

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The transfer matrix in lattice field theory connects the covariant and the initial data frameworks; in spin foam models, it can be written as a composition of elementary cellular amplitudes/propagators. We present a framework for discrete spacetime classical field theory in which solutions to the field equations over elementary spacetime cells may be amalgamated if they satisfy simple gluing conditions matching the composition rules of cellular amplitudes in spin foam models. Furthermore, the formalism is endowed with a multisymplectic structure responsible for local conservation laws.Some models within our framework are effective theories modeling a system at a given scale. Our framework allows us to study coarse graining and the continuum limit. *

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“…More importantly, discrete transcriptions of the Noether's theorem can be constructed for Lagrangian symmetries on a lattice [13], [111], to yield exact conservation laws of (properly defined) quantities such as discrete energy and discrete momentum [3].…”

Section: Appendix D: Classification Of Inconsistencies In Naïve Discrmentioning

confidence: 99%