© 2018 Mark D Schubel DISCRETIZATION OF DIFFERENTIAL GEOMETRY FOR COMPUTATIONAL GAUGE THEORY BY MARK D SCHUBEL DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2018 Urbana, Illinois Doctoral Committee: Professor Eduardo Fradkin, Chair Associate Professor Anil N. Hirani, Director of Research Professor Peter Abbamonte Professor Daniel Berwick-Evans Professor Sheldon Katz Professor Adia X El-Khadra Abstract 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 푘-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 ana- log 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 elec- trodynamics 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 re- cover 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 ℝ3, 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. ii Acknowledgments First to my advisor Anil N. Hirani, thank you for your mentorship and for teaching me what can be learned by continuously questioning what you think you know. Daniel Berwick-Evans, thank you for helping shape and focus the topics for this thesis and especially providing guidance about where to find interesting physics applications of this discretization framework. Thank you Kaushik Kalyanaraman both for our discussions, which helped me better understand DEC and FEEC and provided pathways for further exploration, as well as our friendship which made my time at Illinois much more enjoyable. To my funding agencies, the Physics department at the University of Illinois, the NSF thank you for having financially supported my Ph.D. work. And thanks to the Institute of Mathematics and its Applications for funding my travel to attend several conferences, they were truly illuminating. To my parents, Gary and Linda, thank you for first encouraging me to go into physics and for your support through all of these years of life and education. And my siblings John, Katelyn and Kimberly, thank you for our adventures growing up and for teaching me what school could not. And lastly, to my wife Mary who helped me through the entire process and listened to me ramble on about this work an innumerable number of times, thank you! iii Table of Contents Chapter 1. Introduction .................................. 1 Chapter 2. Preliminaries .................................. 4 2.1. Cellular and Simplicial Complexes ............................. 4 2.1.1. Primal and Dual Complexes and Orientation .................. 5 2.1.2. Cochains and Differential Forms ......................... 6 2.1.3. Wedge Product ................................... 7 2.1.4. Hodge Stars and Discrete Inner Product ..................... 9 2.2. Vector Bundles ........................................ 11 2.3. Discrete Vector Bundles ................................... 14 2.4. Observables and the Space of Observables ......................... 16 2.4.1. Holonomy ...................................... 16 2.5. Connections and Curvature ................................. 18 2.6. Characteristic Classes .................................... 19 2.7. Discrete BF Models ..................................... 20 Chapter 3. Discrete Exterior Calculus with Boundary . 21 3.1. Exterior Derivative ...................................... 21 3.2. Codifferential and Adjointness ............................... 24 3.3. Primal-Dual Wedge Product & Discrete Exterior Derivative . 25 Chapter 4. Noether’s Theorem for Discrete Field Theories . 30 4.1. Fully-Discrete Field Theories ................................ 30 4.1.1. Euler-Lagrange Equations ............................. 31 4.1.2. Noether’s First Theorem .............................. 34 4.2. Semi-Discrete Field Theories ................................ 38 4.2.1. Euler-Lagrange Equations ............................. 40 4.2.2. Noether’s First Theorem .............................. 43 4.2.3. Hamiltonian Formulation ............................. 50 Chapter 5. Discrete Gauge Theory and Yang-Mills . 53 5.1. Connections and Parallel Transport ............................ 53 iv 5.1.1. Discrete Covariant Derivative: Existing Definition . 56 5.1.2. Discrete Covariant Derivative: An Extension to Higher Cochains . 57 5.2. Dual Discrete Covariant Exterior Derivative ........................ 62 5.3. Adjointness of the Covariant Exterior Derivative ..................... 62 5.3.1. Euler-Lagrange Equations ............................. 64 5.4. Discrete Yang-Mills ..................................... 65 5.5. Example: Complex 푈(1) Field Theory ........................... 66 5.6. Numerical Experiments on Scalar Field Theory ...................... 69 Chapter 6. Discrete Abelian BF Theory ......................... 71 6.1. Semi-Discrete Action .................................... 71 6.1.1. Euler-Lagrange Equations ............................. 72 6.1.2. Gauge Invariance .................................. 73 6.1.3. Quantization and Commutation Relations .................... 76 6.1.4. Wilson Loops .................................... 78 6.1.5. Consistency ..................................... 79 6.2. Fully Discrete Action .................................... 80 6.2.1. Euler-Lagrange Equations ............................. 81 6.2.2. Gauge Invariance .................................. 82 6.2.3. Charge Conservation ................................ 82 Chapter 7. Characteristic Classes and Topological Charge . 84 7.1. Bundle Operations ...................................... 84 7.2. First Chern Class ...................................... 85 7.3. Numerical Results ...................................... 87 7.4. Remaining Chern Classes .................................. 89 Chapter 8. Reduction of Structure Group ........................ 92 8.1. Determining Triviality .................................... 92 8.2. Determining Maximal Trivial Sub-bundle ......................... 93 8.3. Determining Block Structure ................................ 95 Chapter 9. Generalizing the Cheeger-Buser Inequalities to One-Laplacians on Hockey-Puck Domains ............................ 96 9.1. Introduction ......................................... 96 9.2. Numerical Experiments ................................... 97 9.2.1. Hollow Hockey Puck ................................ 97 9.2.2. Solid Hockey Puck ................................. 98 9.3. An Upper Bound .......................................100 Chapter 10.Conclusions and Future Aims . 102 v 10.1.Conclusions ..........................................102 10.2.Future Aims .........................................103 Bibliography ..........................................104 Appendix A. Proof of Upper Bound of One Form Laplacian Eigenvalue . 108 vi Chapter 1. Introduction Differential geometry is a valuable language for describing physical systems ranging from classical mechanics to modern field theories. A principal advantage is the ability to express quantities and operators in a coordinate-independent language, which eliminates the need for carefully constructed coordinate systems. In the discrete context this means that a system of discrete