Topology and Quantum Theory in Interaction.

Cover -- Title page -- Contents -- Preface -- Introduction -- Geometry and physics: An overview -- 1. Dirac quantization -- 2. Missed opportunities -- 3. Yang-Mills theory and connections on fiber bundles -- 4. Unreasonable effectiveness -- 5. Anomalies and index theory -- 6. Donaldson invariants -- 7. Topological quantum field theory -- 8. Seiberg-Witten theory -- 9. Conclusions -- Acknowledgments -- References -- An introduction to spin systems for mathematicians -- Introduction -- 1. General principles of quantum mechanics: a reminder -- 2. A few words about field theories -- 3. From field theories to spin systems -- 4. An example: The Heisenberg spin chain and integrability -- 5. Questions to ask about systems, or, what is a phase of matter? -- 6. Stacking -- 7. Topological order and the toric code -- Acknowledgments -- References -- The Arf-Brown TQFT of pin⁻ surfaces -- 1. Introduction -- 2. Preliminaries -- 2.1. Clifford algebras, pin groups, and pin structures -- 2.2. Homotopy theory -- 3. The Arf-Brown invariant of a pin\textsuperscript{-} surface -- 3.1. Intersection-theoretic descriptions of the invariants -- 3.2. Index-theoretic description of the invariants -- 3.3. -theoretic descriptions of the invariants -- 4. Invertible TQFTs via stable homotopy theory -- 4.1. What is a (2D) invertible TQFT? -- 4.2. The homotopy hypothesis and stable 2-types -- 4.3. Classifying invertible TQFTs up to isomorphism -- 5. The Arf-Brown TQFT -- 6. The time-reversal-invariant Majorana chain -- 6.1. Symmetry-protected topological phases -- 6.2. Combinatorial spin and pin structures -- 6.3. Defining the Majorana chain -- 6.4. The low-energy TQFT -- Acknowledgments -- References -- A guide for computing stable homotopy groups -- 1. Introduction and organization -- 2. A working guide to spectra -- 3. The Steenrod algebra -- 4. The Adams spectral sequence.

5. Examples from the classification problems -- Acknowledgments -- References -- Flagged higher categories -- Introduction -- 1. Comparing Segal sheaves and flagged higher categories -- 2. Univalent-completion -- 3. Using the formula to prove the main result -- Acknowledgments -- References -- How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism -- 0. Introduction -- 1. Example: Wick's lemma -- 2. Example: Counting trivalent graphs -- 3. The general case: From homological algebra to diagrammatics -- 4. Motivation: Finite dimensional integrals -- Acknowledgments -- References -- Homotopy RG flow and the non-linear -model -- 1. Introduction -- 2. The classical -model -- 3. One-loop quantization -- 4. -function generalities -- 5. The one-loop -function for the -model -- Appendix A. Homotopy renormalization group flow -- Acknowledgments -- References -- The holomorphic bosonic string -- 1. Introduction -- 2. The classical holomorphic bosonic string -- 3. Deformations of the theory and string backgrounds -- 4. Quantizing the holomorphic bosonic string on a disk -- 5. OPE and the string vertex algebra -- 6. The holomorphic string on closed Riemann surfaces -- 7. Looking ahead: Curved targets -- Appendix A. Calculation of anomaly -- Acknowledgments -- References -- Back Cover.

This volume contains the proceedings of the NSF-CBMS Regional Conference on Topological and Geometric Methods in QFT, held from July 31-August 4, 2017, at Montana State University in Bozeman, Montana. In recent decades, there has been a movement to axiomatize quantum field theory into a mathematical structure. In a different direction, one can ask to test these axiom systems against physics. Can they be used to rederive known facts about quantum theories or, better yet, be the framework in which to solve open problems? Recently, Freed and Hopkins have provided a solution to a classification problem in condensed matter theory, which is ultimately based on the field theory axioms of Graeme Segal. Papers contained in this volume amplify various aspects of the Freed-Hopkins program, develop some category theory, which lies behind the cobordism hypothesis, the major structure theorem for topological field theories, and relate to Costello's approach to perturbative quantum field theory. Two papers on the latter use this framework to recover fundamental results about some physical theories: two-dimensional sigma-models and the bosonic string. Perhaps it is surprising that such sparse axiom systems encode enough structure to prove important results in physics. These successes can be taken as encouragement that the axiom systems are at least on the right track toward articulating what a quantum field theory is.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.