Discretization of differential geometry for computational gauge theory

computational analysis;discrete exterior calculus;discrete exterior calculus;DEC;spaces with boundary;discrete curvature;covariant derivative;discrete connection;characteristic classes;Chern classes;Yang-Mills;BF theory;discrete field theory

Schubel, Mark D.

关键词: computational analysis;    discrete exterior calculus;    discrete exterior calculus;    DEC;    spaces with boundary;    discrete curvature;    covariant derivative;    discrete connection;    characteristic classes;    Chern classes;    Yang-Mills;    BF theory;    discrete field theory;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/100963/SCHUBEL-DISSERTATION-2018.pdf?sequence=1&isAllowed=y

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来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis develops a framework for discretizing field theories that is independent of the chosen coordinates of the underlying geometry. This independence enables the framework to be more easily utilized in a variety of domains such as those with non-trivial geometry and topology. To do this, we build on discretizations of exterior calculus including Discrete Exterior Calculus and Finite Element Exterior Calculus. We apply these methods to discrete differential geometric objects by providing a new definition of the discrete exterior derivative on dual cochains, allowing us to incorporate more general boundary conditions and prove a discrete version of adjointness of the discrete exterior derivative and the codifferential. We also provide a definition of fundamental constructions of discrete vector bundles such as the Whitney sum, tensor bundle and pullback bundles, and a definition of a discrete covariant exterior derivative on general vector-valued $k$-cochains that extends to endomorphism-valued cochains while leading to discrete analogs of properties of endomorphism-valued forms. As part of our investigations of discrete vector bundles, we consider the problem of under what conditions the structure group of a discrete vector bundle can be simplified and give algorithms to perform the reduction when such a reduction is possible.We also develop discrete variational mechanics deriving the Euler-Lagrange equations for both fully-discrete (both space and time are discretized) as well as semi-discrete (space is discretized and time is left smooth) theories with and without gauge symmetries. We further derive a discrete analog of Noether's theorem and define discrete analogs of conserved current and charge densities. We apply our discretization scheme to classic examples including complex scalar field theory and electrodynamics as well as to non-Abelian Yang-Mills. Our last application is to Abelian Chern-Simons, where we consider fully- and semi-discrete discretizations utilizing both primal and dual complexes to provide simpler discrete descriptions of physical quantities and demonstrate our ability to recover other topological properties of smooth theories. In examining discretizations of topological charge, we extend a definition of the first Chern class to all vector bundles, and in addition we discuss possible discretization of the second Chern class. Finally, we consider a generalization of the Cheeger-Buser inequalities to a ``hockey puck shaped" domain in $\mathbb{R}^3$, showing how the eigenvalues of the one-form Laplacian change as the hockey puck shape approaches that of a solid torus. Our framework for discretizing field theories enables broader use of techniques in exterior calculus to improve numerical methods for solving physical and geometric systems.