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Lecture Notes for Felix Klein Lectures Preprint typeset in JHEP style - HYPER VERSION Lecture Notes for Felix Klein Lectures Gregory W. Moore Abstract: These are lecture notes for a series of talks at the Hausdorff Mathematical Institute, Bonn, Oct. 1-11, 2012 December 5, 2012 Contents 1. Prologue 7 2. Lecture 1, Monday Oct. 1: Introduction: (2,0) Theory and Physical Mathematics 7 2.1 Quantum Field Theory 8 2.1.1 Extended QFT, defects, and bordism categories 9 2.1.2 Traditional Wilsonian Viewpoint 12 2.2 Compactification, Low Energy Limit, and Reduction 13 2.3 Relations between theories 15 3. Background material on superconformal and super-Poincar´ealgebras 18 3.1 Why study these? 18 3.2 Poincar´eand conformal symmetry 18 3.3 Super-Poincar´ealgebras 19 3.4 Superconformal algebras 20 3.5 Six-dimensional superconformal algebras 21 3.5.1 Some group theory 21 3.5.2 The (2, 0) superconformal algebra SC(M1,5|32) 23 3.6 d = 6 (2, 0) super-Poincar´e SP(M1,5|16) and the central charge extensions 24 3.7 Compactification and preserved supersymmetries 25 3.7.1 Emergent symmetries 26 3.7.2 Partial Topological Twisting 26 3.7.3 Embedded Four-dimensional = 2 algebras and defects 28 N 3.8 Unitary Representations of the (2, 0) algebra 29 3.9 List of topological twists of the (2, 0) algebra 29 4. Four-dimensional BPS States and (primitive) Wall-Crossing 30 4.1 The d = 4, = 2 super-Poincar´ealgebra SP(M1,3|8). 30 N 4.2 BPS particle representations of four-dimensional = 2 superpoincar´eal- N gebras 31 4.2.1 Particle representations 31 4.2.2 The half-hypermultiplet 33 4.2.3 Long representations 34 4.2.4 Short representations of = 2 36 N 4.3 Field Representations 36 4.4 Families of Quantum Vacua 37 4.5 Families of Hilbert Spaces 39 4.6 The BPS Index and the Protected Spin Character 40 4.7 An Open Problem: Compute the BPS Spectrum 42 – 1 – 4.8 Positivity Conjectures and geometric quantities 42 4.9 The wall-crossing phenomenon 44 4.10 The charge lattice Γ 45 4.10.1 Dirac quantization of dyonic charge in 4d Maxwell theory 45 4.10.2 General Picture: Global Symmetry Charges 46 4.11 Primitive Wall Crossing Formula 47 4.11.1 Denef’s boundstate radius formula 47 4.11.2 A simple physical derivation of the primitive wcf 48 5. The abelian tensormultiplet 49 5.1 Quick Reminder on the Hodge 49 ∗ 5.2 The (2, 0) tensormultiplet 50 5.3 A lightning review of generalized Maxwell theory 51 5.4 Differential cohomology 52 5.4.1 Examples 53 5.4.2 Ring Structure and Pairing 54 5.5 Choice of Dirac Quantization: The torus theories 56 5.6 A Reminder on Heisenberg Groups 57 5.7 Quantization of the Torus Theories 58 5.7.1 Partition function 58 5.7.2 Hilbert space 59 5.7.3 Vertex operators 60 5.8 Chiral theories 61 5.8.1 Classical chiral theory 61 5.8.2 Challenges for Quantum Theory 61 5.8.3 Approach via chiral factorization 62 5.9 Torus Chern-Simons Theories 63 5.10 Generalizing the notion of “field theory” 65 5.10.1 Anomalous Field Theories 65 5.10.2 n-dimensional field theory valued in an (n+1)-dimensional field theory 66 5.11 The issue of an action 66 5.12 Compactification to 5 dimensions: 5D Abelian SYM 68 5.13 Compactification to four dimensions 69 5.13.1 Self-dual abelian gauge theory in 4d 69 5.13.2 Lagrangian decomposition of a symplectic vector space with com- patible complex structure 70 5.13.3 Lagrangian formulation 71 5.13.4 Duality Transformations 72 5.14 Compactification to Three Dimensions 72 5.15 Compactification to Two Dimensions 73 – 2 – 6. Physical Heuristics: The Interacting (2, 0) theories 73 6.1 Due warning to the reader 73 6.2 Lightning review: Type II strings and D-branes 74 6.3 Physical Construction 1: Type IIB strings on a singular K3 surface 75 6.3.1 Hyperkahler resolution of surface singularities 75 6.3.2 IIA string theory on a hyperk¨ahler resolution of singularities 75 6.3.3 IIB string theory on a hyperk¨ahler resolution of singularities 75 6.4 Physical Construction 2: Parallel M5-branes 76 6.4.1 Lightning review of M-theory 76 6.4.2 Stacks of M5-branes 77 6.5 Duality relation between the two pictures 78 6.6 Ground rules for working with interacting (2, 0) theories 78 6.7 Remarks on and justification of the axioms 80 6.7.1 Basic Data 80 6.7.2 Axiom 1: Domain and codomain of definition 80 6.7.3 Axiom 2: Hilbert space as a representation of the superconformal algebra 82 6.7.4 Axiom 3: Relation to Super-Yang-Mills 83 6.7.5 Axiom 4: Coulomb branch in Minkowski space 84 6.7.6 Axiom 5: Low energy dynamics on the Coulomb branch 84 6.7.7 Axiom 6: String Excitations 84 6.7.8 Axiom 7: Surface Defects 84 6.7.9 Axiom 8: Half-BPS Codimension Two Defects 85 6.8 Information from the AdS/CFT correspondence 88 6.9 Relation to N=4 SYM 88 6.10 Attempts at a more rigorous construction 88 6.11 The “lattice gauge theory” or “deconstruction” approach 89 6.11.1 An elementary computation 89 6.11.2 The quiver gauge theory 90 6.12 Field theory vs. little string theory 91 7. Theories of class S 91 7.1 Definition 91 7.1.1 Data 91 7.1.2 Definition 92 7.2 Decoupling the Weyl modes: Gaiotto gluing, modular groupoid and S-duality 92 7.2.1 Gaiotto decomposition and S-duality 93 7.2.2 A conformal field theory valued in four-dimensional theories 94 7.3 The Higgs and Coulomb branches 94 7.4 Relation to the Hitchin system 94 7.5 Recovery of the Seiberg-Witten Paradigm 94 7.6 BPS states in class S 94 7.7 The Witten construction 95 – 3 – 7.8 Mirror picture 95 7.9 Some novel isomorphisms of Hitchin systems 96 8. Line defects, framed BPS states and wall-crossing 96 8.1 Definition of susy line defects 96 8.2 Framed BPS states 96 8.3 Wall-crossing of framed BPS states 96 8.4 Derivation of the motivic KSWCF 97 8.5 Special Cases of the KSWCF 97 8.6 The spectrum generator 98 8.7 Line defects in theories of class S 99 9. Darboux Functions and Hyperk¨ahler metrics 99 9.1 Line defect vevs 99 9.2 The Darboux expansion 99 9.3 The TBA construction of Darboux functions 100 9.4 3D Compactification and HK geometry 100 9.5 Twistors and Hitchin’s theorem 100 9.6 Twistor sections for 100 M 9.7 Relation to Fock-Goncharov coordinates for the A1 case 100 9.7.1 KS transformations from morphisms of decorated ideal triangulations 101 10. Coupled 2d-4d systems 101 10.1 Defining a class of susy surface defects 101 10.1.1 IR effective theory 101 10.2 Important Example: The canonical surface defects Sz 101 10.3 New BPS degeneracies 101 10.4 Supersymmetric Interfaces 101 10.4.1 Class S supersymmetric interfaces 102 10.5 2d/4d WCF 102 10.6 Reduction to 3D/1D systems: hyperk¨ahler geometry 102 10.6.1 Example: HH Line bundle over (generalized) PTN 102 10.6.2 The case of class S 102 10.7 Example: The CP 1 model coupled to the d = 4 = 2 SU(2) theory 102 N 11. Lecture 2, Tuesday Oct.2 : Theories of class S and string webs 102 11.1 Motivation 102 11.2 Recap for the impatient reader who has skipped Sections 2-9 103 11.3 WKB paths and string webs 108 11.3.1 WKB paths 109 11.3.2 Local Behavior 109 11.3.3 Global behavior 111 11.3.4 Definition of string webs 112 11.3.5 Counting string webs 114 – 4 – 12. Lecture 3, Thursday Oct. 4: Surface Defects and Spectral Networks 115 12.1 The canonical surface defect 115 12.2 Solitons on the canonical surface defect Sz 117 12.3 Physical Definition of a spectral network 119 12.4 Supersymmetric Interfaces and framed BPS states 121 12.5 The Formal Parallel Transport Theorem 123 13. Lecture 4, Friday October 5: Morphisms of Spectral Networks and the 2d-4d Wall-Crossing Formula 128 13.1 Morphisms of 128 Wϑ 13.2 How the formal parallel transport changes with ϑ 131 13.3 Examples 134 13.4 2d-4d WCF 135 14. Summary of material from week 1 136 15. Lecture 5, Monday, October 8: Wall-Crossing Formulae 137 15.1 Formal statement of the 2d4d WCF 137 15.1.1 The spectrum generator 139 15.2 Walls of Marginal Stability and the Four types of 2d4d wall crossing 140 15.3 Special Cases of the 4d KSWCF 142 15.3.1 Example 1: Pentagon identity 142 15.3.2 ADN theories 143 15.3.3 The Juggle/Vectormultiplet 145 15.3.4 2d4d Wall crossing for the SU(2) model 146 15.3.5 Wild wall crossing 147 15.4 The “motivic” generalization 149 16. Lecture 6: Tuesday, Oct. 9: Coordinates for Moduli Spaces of Flat Connections 153 16.1 Review: Moduli spaces of complex flat gauge fields and moduli spaces of local systems 153 16.1.1 Relation to Hitchin systems 154 16.2 Higgs bundles and the abelianization map 155 16.3 Generalized Spectral Networks 156 16.4 Nonabelianization map 157 16.5 Mapping moduli spaces 160 16.6 Darboux functions 162 16.7 The spectrum generator 163 17.
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