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The Lab Notebook Literature References, Definitions, Theorems, Conventions et al. and Christian Sämann The Lab Notebook Literature references, Definitions, Theorems, Conventions Version April 22, 2019 These notes are a collection of literature references relevant to my research and useful as a reference for my PhD and summer project students. The level of explanation ranges from very basic to very advanced. Mostly, I’m bad at keeping notes, and this is material that I need repeatedly for references or didn’t fit in any of my papers. The style is certainly more colloquial than in proper papers, these are notes after all. Some paragraphs may be copied literally from some of my papers with other authors. There- fore, these notes are by no means meant to be original. They should not be cited nor should there be any credit attributed to them. “Everything is in flow,” and in particular these notes are. Statements in this document might be incomplete, contain sloppy formulations or even errors. Sometimes they may be very old and then reflect an outdated understanding of mine. Also, conventions and notations may jump between and even within sections. Therefore, use everything contained in here with great care and consult the original literature. Finally, some sections of these notes may require serious tiding. Contents 1 Introductory remarks7 2 Foundations 9 2.1 Set theory . .9 2.2 Homotopy type theory . .9 2.3 Category theory and higher category theory . 10 2.3.1 Ordinary category theory . 10 2.3.2 Higher category theory . 10 2.3.2.1 1-categories . 11 2.3.2.2 Categorification . 11 2.3.3 Other . 11 3 Algebra 13 3.1 Group like objects . 13 3.1.1 Group theory . 13 3.1.2 Groupoids . 13 3.1.3 Higher groupoids and groups . 13 3.1.3.1 2-groups . 13 3.1.3.2 Higher than 2 . 14 3.1.3.3 String and Fivebrane groups . 15 3.2 L1-algebroids and related . 21 3.2.1 Homotopy Algebras . 21 3.2.2 Differential graded algebras . 21 3.2.2.1 NQ-manifolds . 21 3.2.3 Lie 2-algebras . 21 3.2.3.1 Strict Lie 2-algebras . 22 3.2.3.2 Semistrict Lie 2-algebras . 22 3.2.4 L1-algebras . 23 3.2.4.1 Loop L1-algebras . 25 3.2.5 L1-algebroids . 25 3.2.5.1 Lie algebroids . 25 3.2.5.2 Courant algebroids . 25 3.2.5.3 Higher Lie n-algebroids . 26 3.2.6 2-Crossed modules of Lie algebras . 26 3.2.7 Other . 26 3.3 Representation theory . 27 3.4 Algebraic Topology . 27 4 Geometry 29 4.1 Generalities . 29 4.1.1 Manifolds . 29 4.1.1.1 Spheres . 29 4.1.1.2 Loop spaces . 30 4.1.2 Differential geometry . 31 4.2 Symplectic and Poisson geometry . 31 4.2.1 Basics . 31 4.2.2 Multisymplectic geometry . 31 4.2.3 Poisson and Nambu–Poisson geometry . 31 4.3 Complex geometry . 32 4.3.1 Calabi–Yau spaces . 32 4.3.2 Twistor geometry . 32 4.4 Graded geometry . 33 4.4.1 Graded vectorspaces . 33 4.4.2 N-manifolds . 33 4.4.3 Supermanifolds . 33 4.4.4 Generalized geometry . 33 4.5 Fiber bundles . 33 4.5.1 Tangent and cotagent bundles . 33 4.5.2 Jet bundles . 35 4.5.3 Principal bundles . 36 4.5.3.1 Ordinary principal bundles . 36 4.5.3.2 Higher principal bundles . 36 4.5.3.3 Connections and related . 37 4.5.3.4 Characateristic classes . 38 4.5.4 Other fibrations . 38 4.6 Stacks . 38 4.6.1 Lie groupoids and Stacks . 38 4.6.2 1-stacks . 39 4.7 Other areas . 39 5 Other areas 41 5.1 Arithmetic . 41 5.2 Analysis . 41 5.3 Supermathematics . 41 5.3.1 Supergeometry . 41 6 Classical and Quantum Mechanics 43 6.1 Classical Mechanics . 43 6.2 Quantum Mechanics . 43 6.2.1 Noncommutative geometry . 44 6.2.1.1 Fuzzy geometry . 44 6.2.1.2 Higher quantization . 44 7 Field Theory 47 7.1 Classical Field Theory . 47 7.1.1 Spin and all that . 47 7.1.2 Examples of field theories . 47 7.1.2.1 General gauge theories . 47 7.1.2.2 Yang–Mills theory . 47 7.1.2.3 Yang–Mills–Higgs theory . 47 7.1.2.4 Chern–Simons theory . 47 7.1.2.5 BF-type theories . 48 7.1.2.6 General higher gauge theories . 48 7.1.2.7 (1,0)- and (2,0)-theory . 48 7.1.2.8 General Relativity . 48 7.1.3 Kaluza–Klein reduction . 48 7.1.4 Classical Integrability . 49 7.1.4.1 Integrable field equations . 49 7.1.4.2 Penrose–Ward transform and related . 49 7.1.4.3 ADHM-construction . 50 7.1.4.4 ADHMN-construction . 50 7.2 Quantum Field Theory . ..
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