Dialogues Between Physics and Mathematics: C. N. Yang at 100

Preface
Acknowledgements
Contents
1 Frank Yang at Stony Brook and the Beginning of Supergravity
1.1 Prologue
1.2 Frank Before Coming to Stony Brook
1.3 Early Years of Frank at Stony Brook
1.4 Supersymmetry and Quantum Gravity Before 1976
1.5 Some Recollections of the Path to Supergravity at Stony Brook
1.6 Supergravity Lives!
1.7 Epilogue
1.8 Later Years of Frank at Stony Brook
References
2 A Stacky Approach to Crystals
2.1 Introduction
2.1.1 A Theorem of Bhatt–Morrow–Scholze
2.1.2 A Generalization
2.1.3 Isocrystals
2.1.3.1 What We Mean by an Isocrystal
2.1.3.2 The Result on Isocrystals
2.1.3.3 ``Banachian Games'' and `3́9`42`"̇613A``45`47`"603ABunQ
2.2 Crystals and Crystalline Cohomology
2.2.1 A Class of Schemes
2.2.2 Some Simplicial Formal Schemes
2.2.2.1 The Simplicial Scheme P•
2.2.2.2 The Simplicial Formal Scheme F•
2.2.2.3 The Simplicial Formal Scheme A•
2.2.3 Notation and Terminology Related to Quasi-Coherent Sheaves
2.2.3.1 p-adic Formal Schemes and Stacks
2.2.3.2 The Notation QCoh (Y)
2.2.3.3 Zp-Flatness
2.2.3.4 Finite Generation
2.2.3.5 Cohomology
2.2.3.6 Equivariant Objects
2.2.3.7 Objects of QCoh (Y ) as Sheaves
2.2.3.8 Proof of (2.2)
2.2.4 Formulation of the Results
2.2.4.1 Convention
2.2.5 Proof of Theorem 2.1(i)
2.2.5.1 The Simplicial Formal Scheme X•
2.2.5.2 End of the Proof
2.2.6 Proof of Theorem 2.1(ii)
2.2.6.1 General Remark
2.2.6.2 The Functor in One Direction
2.2.6.3 Factorizing the Functor (2.8)
2.2.7 Proof of Theorem 2.1(iii)
2.2.7.1 General Remark
2.2.7.2 The Map in One Direction
2.2.7.3 End of the Proof
2.2.8 H0`3́9`42`"̇613A``45`47`"603Acris(X,O) and the Ring of Constants
2.2.8.1 The Ring of Constants
2.3 Isocrystals
2.3.1 A Class of Schemes
2.3.1.1 The Ring of Constants
2.3.2 Coherent Crystals and Isocrystals
2.3.3 Local Projectivity
2.3.4 Proof of Proposition 2.4
2.3.4.1 Strategy
2.3.5 Isocrystals as Vector Bundles
2.3.5.1 The Category `3́9`42`"̇613A``45`47`"603ABunQ(Y )
2.3.5.2 Flat Descent for `3́9`42`"̇613A``45`47`"603ABunQ(Y )
2.3.5.3 Equivariant Objects of `3́9`42`"̇613A``45`47`"603ABunQ(Y)
2.3.6 Banachian Games
2.3.6.1 One of the Goals
2.3.6.2 Proof of Theorem 2.2
2.3.7 Proof of Propositions 2.5 and 2.6
Appendix
The Isomorphism Between W(X`3́9`42`"̇613A``45`47`"603Aperf)/G and the Prismatization of X
The Goal
Prismatization of Semiperfect Fp-Schemes
Perfect Case
General Case
The Morphism W(X`3́9`42`"̇613A``45`47`"603Aperf)→`3́9`42`"̇613A``45`47`"603AWCartX
The Čech Nerve of (2.30)
References
3 The Potts Model, the Jones Polynomial and Link Homology
3.1 Introduction
3.2 Bracket Polynomial and Jones Polynomial
3.3 Khovanov Homology and the Cube Category
3.4 The Dichromatic Polynomial and the Potts Model
3.5 Khovanov Homology
3.6 Homology and the Potts Model
3.7 The Potts Model and Stosic's Categorification of the Dichromatic Polynomial
3.8 Imaginary Temperature, Real Time and Quantum Statistics
References
4 The Penrose–Onsager–Yang Approach to Superconductivity and Superfluidity
4.1 Quantum Condensation: The Onsager-Penrose-Yang Approach
4.2 Some Considerations and Questions Raised by the Content of Sect.4.1
4.3 What Is Special About Quantum Condensates?
4.4 Why Is Nature So Fond of ``Simple'' Quantum Condensation? Why Is ``Fragmentation'' So Rare?
4.5 When Does Fragmentation Occur?
4.6 Alternative Approaches to Quantum Condensation: Some Problems
4.6.1 ODLRO
4.6.2 Anomalous Averages
4.6.3 ``Spontaneously Broken U(1) Symmetry''
References
5 Quantum Operads
5.1 Introduction and Brief Survey
5.2 Quantum Structures in Symmetric Monoidal Categories
5.2.1 Monoidal (=Tensor) Categories V (Sm16, Sec. 2.2, 2.3)
5.2.2 Symmetric Monoidal Categories
5.2.3 Magmas, Comagmas, Bimagmas, Associativity and Commutativity for (co, bi)magmas in Symmetric Monoidal Categories (Sm16, Sec. 2.4)
5.2.4 Monoids, Comonoids, Bimonoids, and Hopf Algebras in Symmetric MonoIdal Categories (Sm16, Def. 2.7)
5.2.5 Quantum Quasigroups (Sm16, Sec. 3.1)
5.2.6 Quantum Loops
5.2.7 Functoriality (Sm16, Prop. 3.4)
5.2.8 Magmas etc. in the Categories of Sets with Direct Product
5.3 Monoidal Categories of Operads
5.3.1 Graphs and Their Categories
5.3.2 Operads and Categories of Operads (See BoMa08, Sec. 1.6, p. 262)
5.3.3 Operads and Collections as Symmetric Monoidal Categories
5.3.4 Operads as Monoids
5.3.4.1 Freely Generated Operads
5.3.5 Comonoids in Operadic Setup
5.3.6 The Magmatic Operad (See ChCorGi19)
5.3.7 Quasigroup Monomials and Planar Trees
5.4 Moufang Loops and Operads
5.4.1 Moufang Monomials and Their Encoding by Labeled Graphs
5.4.2 Passage to Moufang Operad: Basic Identity
5.4.3 Moufang Collections (See BoMa08, Sec. 1.5, pp. 259–261)
5.4.4 Latin Square Designs and Their Encoding by Graphs
5.4.4.1 Simplest Examples
5.4.5 From Loops to Latin Square Designs
5.5 Operadic Structures on Quantum States
5.5.1 Operads of Classical and Quantum Probabilities
5.5.1.1 Averages as an Algebra Over the Operad P
5.5.1.2 A∞-Operad and Entropy
5.5.2 Classical Probabilities from Quantum States
5.5.3 Non-unital Operads
5.5.4 The QP-Operad of Quantum States
5.5.5 The Q-Operad of Quantum States
5.5.6 Trees of Projective Quantum Measurements
5.5.7 Entropy Functionals
5.5.8 A∞-Operad of Quantum Channels
5.6 Operads and Almost-Symplectic Quantum Codes
5.6.1 Rational and Binary Little Square Operads
5.6.1.1 Binary Little Square Operad
5.6.1.2 Strict Binary Little Squares
5.6.2 Binary Little Square Operads and Almost Symplectic Spaces
5.6.3 Colored p-ary Little Squares
5.6.3.1 Operads and Almost-Symplectic Structures Over Fp
5.6.4 Operad Partial-Action on Quantum Codes
References
6 Quantum Computational Complexity with Photons and Linear Optics
6.1 Introduction
6.2 The Mathematics: Permanent and Hafnian
6.2.1 Permanent
6.2.2 Hafnian
6.3 The Model: Boson Sampling
6.4 Single-Photon Boson Sampling Experiments
6.5 Quantum Computational Advantage with Jiuzhang
6.6 Applications
References
7 Quantized Twistors, G2*, and the Split Octonions
7.1 A Key Motivation for the Formulation of Twistor Theory
7.2 The 2-Spinor Formalism
7.3 Projective Twistor Space
7.4 Twistor Kinematics
7.5 Quantized Twistor Theory and Masless Fields
7.6 Split Octonions and G2*
References
8 Kronecker Anomalies and Gravitational Striction
8.1 Introduction and General Discussion
8.2 Kronecker Anomaly in Thermal Harmonic Oscillator
Appendix
Kronecker Anomaly in Electromagnetic Theory
Mathematical Aspects of Kronecker Anomaly
Two Dimensional Scalar Electrodynamics on a Torus
Kronecker Anomaly in Spaces with Constant Curvature
Absence of Kronecker Anomalies in Even Dimensional de Sitter Spaces
De Sitter Lacuna
References
9 Projecting Local and Global Symmetries to the Planck Scale
9.1 Introduction
9.2 Quantum Mechanics
9.3 Classical Models Underlying Quantum Mechanics
9.4 The Standard Model
References
10 Gauge Symmetry in Shape Dynamics
10.1 Gauge Structure: Fundamental, Emergent, Productive
10.2 Dynamical Equation for Deformable Bodies
10.2.1 Referenced Angular Momentum
10.2.2 Inertia Tensor and Angular Motion
10.2.3 Gauge Symmetry and Gauge Field
10.2.4 Dynamical Equation
10.2.5 Three Dimensional Notation
10.2.6 Specializations
10.2.7 Angular Momentum and Energy
10.3 Extensions
10.3.1 Blobs, Media, and Swarms
10.3.2 Molecules and Nuclei
Appendix
Direct Calculation
References
11 Why Does Quantum Field Theory in Curved Spacetime Make Sense? And What Happens to the Algebra of Observables in the Thermodynamic Limit?
11.1 Introduction
11.2 Quantum Field Theory in Curved Spacetime
11.2.1 The Problem
11.2.2 Practicing with a Spin System
11.2.3 A System of Harmonic Oscillators
11.2.4 Back to Field Theory
11.2.5 What Is Quantum Field Theory in an Open Universe?
11.2.6 Non-Free Theories
11.3 Quantum Statistical Mechanics and the Thermodynamic Limit
11.3.1 The Thermofield Double
11.3.2 Surprises in the Thermodynamic Limit
11.3.3 Examples from Spin Systems
11.3.4 Relation to Quantum Field Theory
11.3.5 The Hagedorn Temperature
11.3.6 Density Matrices and Entropy
11.4 The Large N Limit and the Thermofield Double
References
12 Quantum Anomalous Hall Effect
13 Magic Superconducting States in Cuprates
References