Lie Theory and Its Applications in Physics: Sofia, Bulgaria, June 2021

Preface
Contents
List of Registered LT-14 Participants
*-1pc  Plenary Talks
Multiplicity in Restricting Minimal Representations
	1 Statement of Main Results
	2 Background and Motivation
		2.1 Bounded Multiplicity Pairs (G,K') with K' Compact
		2.2 Bounded/Finite Multiplicity Pairs (G,G')
		2.3 Uniform Estimates for a Family of Small Representations
	3 Coisotropic Action on Coadjoint Orbits
		3.1 Generalities: Coisotropic Actions on Coadjoint Orbits
		3.2 Real Minimal Nilpotent Orbits
		3.3 Complex Minimal Nilpotent Orbit
		3.4 Proof of Theorems in Sect.1
	References
From the String Landscape to the Mathematical Landscape: A Machine-Learning Outlook
	1 The String Landscape
		1.1 Calabi–Yau Manifolds: From Geometry to Physics
		1.2 Machine-Learning Algebraic Geometry
	2 The Landscape of Mathematics
		2.1 Methodology
		2.2 Across Disciplines
	References
Octonionic Clifford Algebra for the Internal Space of the Standard Model
	1 Introduction
		1.1 Octonions as a Composition Algebra. the Cayley-Dickson Construction
		1.2 Jordan Algebras; Grand Unified Theories; Clifford Algebras
	2 Triality Realization of Spin (8); Cell-6
		2.1 The Action of Octonions on Themselves. Spin (8) as a Subgroup of SO(8) timesSO(8) timesSO(8)
		2.2 Cell-6 as a Generating Algebra of mathbbO and of so(mathbbO)
	3 Cell10 = Cell4  otimes"0362otimes  Cell6 as Internal Space Algebra
		3.1 Equivalence Class of Lorentz Like Clifford Algebras
		3.2 Realization in Terms of Fermi Oscillators
		3.3 Expressing the Cell6 Pseudoscalar in Terms of (anti)particle Projectors
	4 Particle Subspace and the Higgs Field
		4.1 Particle Projection and Chirality
		4.2 The Higgs as a Scalar Part of a Superconnection
		4.3 Higgs Potential and Mass Formulas
	5 Outlook
		5.1 Coming to Cell10
		5.2 Two Ways to Avoid Fermion Doubling
		5.3 A Challenge
	References
The Jacobi Sigma Model
	1 Introduction
	2 The Model
		2.1 The Jacobi Sigma Model
	3 Conclusions
	References
Levi-Civita Connections on Braided Algebras
	1 Introduction
	2 Triangular Hopf Algebra Representations
	3 Differential and Cartan Calculus
	4 Braided Differential Geometry
		4.1 Connections
		4.2 Curvature
		4.3 Torsion
		4.4 Dual Connections, Cartan Structure Equations and Bianchi Identities
	5 Braided Riemannian Geometry
	References
Notes on AdS4 Holography and Higher-Derivative Supergravity
	1 Introduction
	2 Minimal Gauged Supergravity
	3 On-Shell Action
	4 Black Hole Thermodynamics
	5 Field Theory and Holography
	6 Outlook and Further Developments
	References
Homothetic Rota–Baxter Systems and Dyckm-Algebras
	1 Introduction
	2 Rota–Baxter Systems, Homothetisms and Dyckm-Algebras
		2.1 Rota–Baxter Systems and Generalized Rota–Baxter Operators
		2.2 Homothetisms and Homothetic Rota–Baxter Systems
		2.3 Dyckm-Algebras
	3 Dyckm-Algebras from Homothetic Rota–Baxter Systems
	References
Quantum Dynamics Far from Equilibrium: A Case Study in the Spherical Model
	1 What is a Quantum Langevin Equation?
	2 What is the Quantum Spherical Model?
	3 How to Formulate Quantum Dynamics of the Spherical Model?
	4 What is Physical Ageing?
	5 The Spherical Constraint
		5.1 Markovian Case
		5.2 Non-Markovian Case
	6 Quench into the Disordered Phase
	7 Quench onto Criticality or Into the Ordered Phase
	8 Conclusions
	References
On First Extensions in mathcalS-Subcategories of mathcalO
	1 Introduction and Description of the Results
	2 Preliminaries on Category mathcalO
		2.1 Category mathcalO
		2.2 Graded Category mathcalO
		2.3 Combinatorics of Category mathcalO0mathbbZ
		2.4 Kazhdan–Lusztig Orders and Cells
	3 First Extension from a Simple to a Verma Module in Category mathcalO
		3.1 First Extension to a Verma from the Anti-dominant Simple
		3.2 First Extension to a Verma Module from Other Simple Modules and Inclusion of Verma Modules
		3.3 Cokernel of Inclusion Between Verma Modules in Type A
		3.4 First Extension to a Verma from Other Simples in Type A
		3.5 Extensions in Singular Blocks in Type A
		3.6 The Graded Picture in Type A
		3.7 First Extension to a Verma from Other Simples in Other Types
	4 mathcalS-Subcategories in mathcalO
		4.1 Definition
		4.2 Origins and Motivation
		4.3 Stratified Structure
	5 First Extension from a Simple to a Proper Standard Module in mathcalSmathfrakp
		5.1 First Extension from the Antidominant Simple
		5.2 Inclusions Between Proper Standard Modules
		5.3 Cokernel of Inclusion of Proper Standard Modules
		5.4 Ungraded Statements in Type A
		5.5 Graded Statement in Type A
		5.6 First Extension from Other Simples to Proper Standard Modules in Other Types
	6 First Extension from a Simple to a Standard Module in mathcalSmathfrakp
		6.1 Elementary General Observations
		6.2 Reduction to Category mathcalO
		6.3 The Case of Standard Modules which Can be Obtained Using Projective Functors
		6.4 A Type A Formula
	7 Examples
		7.1 sl3-Example
		7.2 sl4-Example
	References
Higher Dimensional CFTs as 2D Conformally-Equivariant Topological Field Theories
	1 Introduction
	2 CFT4/TFT2 Construction of the Free Scalar Field
	3 Perturbative CFTs
	4 d = 4 -ε and Diagram Algebras
	5 Summary and Outlook
	References
Reducing the N = 1, 10-d, E8 Gauge Theory over a Modified Flag Manifold
	1 Introduction
	2 Dimensional Reduction of E8 over SU(3)/U(1)2
	3 Breaking by Wilson Flux Mechanism
	4 Specification of Parameters and GUT Breaking
		4.1 Choice of Radii
		4.2 The Breaking of SU(3)3
		4.3 Lepton Yukawa Couplings and µ Terms
	5 Phenomenological Analysis
		5.1 Gauge Unification
		5.2 1-Loop Results
	6 Conclusions
	References
*-1pc  String Theories, (Super-)Gravity, Cosmology
Ramond States of the D1-D5 CFT Away from the Free Orbifold Point
	1 Introduction
	2 Ramond Ground States and Their Four-Point Functions with the Deformation Operator
	3 Away from the Free Orbifold Point
	References
Primordial Black Hole Generation in a Two-Field Inflationary Model
	1 Introduction
	2 Two-Field Inflationary Models
		2.1 Important Characteristics
		2.2 Rotationally Invariant Field Spaces
	3 A Class of Exact Solutions
	4 Modified Solution and PBH Generation
	References
Late Time Cosmic Acceleration with Uncorrelated Baryon Acoustic Oscillations
	1 Introduction
	2 Review of the Theory
	3 Methodology
	4 The MCMC Results
	5 Conclusion
	References
On the Hidden Symmetries of D=11 Supergravity
	1 Introduction
	2 Lie Superalgebras and Maurer-Cartan Equations
	3 D=11 Supergravity and Its Free Differential Algebra
	4 Hidden Superalgebra Underlying D=11 Supergravity
		4.1 Role of the Nilpotent Fermionic Generator Q'
	5 Relation Between the DF-algebra and mathfrakosp(1|32)
	References
Defects at the Intersection: The Supergroup Side
	1 Intersecting Defects and Supermatrix-Like Models
	2 Towards Non-unitary Open/closed Duality
		2.1 Inclusion of Matter and Supergroup Higgsings
	3 Summary and Discussion
	References
A New S-matrix Formula and Extension of the State Space in Open String Field Theory
	1 Open String Field Theory
	2 On-shell Amplitudes from Classical Solutions
		2.1 KBc Subalgebra
		2.2 An S-matrix Formula
		2.3 Feynman Rules with an Unconventional Propagator
	3 1/K May Be a Key to Understanding the Theory
		3.1 Extension of the State Space
		3.2 Analogy with a Universal Cover of a Manifold
		3.3 Classical Solutions
	References
Dual Dilaton with mathcalR and mathcalQ Fluxes
	1 Introduction
	2 Dual Dilaton
		2.1 The Courant Algebroid and Its Connection
		2.2 The Curvature Scalar
	3 Conclusion
	References
*-1pc  Representation Theory
On 1-Dimensional Modules over the Super-Yangian of the Superalgebra Q(1)
	1 Introduction
	2 Preliminaries
	3 The Super Yangian of Q(1)
	4 1-Dimensional YQ(1)-Modules
	5 The Category YQ(1)–Mod
		5.1 The Subcategory (YQ(1))χ=0–Mod
	6 The Finite W-Algebra for Q(n)
	7 Wn Is a Quotient of YQ(1)
	8 1-Dimensional Wn-Modules
	9 The Category Wn–Mod
		9.1 The Subcategory (Wn)χ=0–Mod
	References
A Klein Operator for Paraparticles
	References
Principal and Complementary Series Representations at the Late-Time Boundary of de Sitter
	1 Introduction
	2 The de Sitter Geometry, The de Sitter Group and the Late-Time Boundary
	3 The Late-Time Operators
	4 The Late-Time Two-Point Functions
	References
Bulk Reconstruction from a Scalar CFT at the Boundary by the Smearing with the Flow Equation
	1 Introduction
	2 Bulk Reconstruction by the Flow
		2.1 From Conformal Symmetry to Bulk Symmetry
		2.2 Some properties
	3 Symmetries and Bulk Geometry
		3.1 Constraints to a Generic Correlation Function
		3.2 Metric Field
		3.3 VEV of the Metric Field
		3.4 Scalar Excited State Contribution
	4 Summary and Discussion
	References
Building Momentum Kernel from Shapovalov Form
	1 Introduction
	2 Conventions
	3 The Relation Between Momentum Kernel and Shapovalov Form
	4 Explaining Feynman Propagators as Shapovalov Dual Vectors
	5 Conclusion and Discussion
	References
Action of w0 on VL for Orthogonal and Exceptional Groups
	1 Basic Notations and Statement of Problem
	2 Background and Motivation
	3 Statement of Main Results
	References
Pairs of Spectral Projections of Spin Operators
	1 Introduction
	2 Main Results
		2.1 Slepian Spectral Concentration
	3 Concluding Remarks
		3.1 More Cases
		3.2 Numerical Phenomena of Spectral Projections of Spin Operators
	References
*-1pc  Integrable Systems
Algebraic Engineering and Integrable Hierarchies
	1 Introduction
	2 Algebraic Engineering of Supersymmetric Gauge Theories
	3 Integrable Hierarchies and Topological Strings
		3.1 Integrable Hierarchies
		3.2 Refined Topological Strings
	4 Discussion
	References
Nested Bethe Ansatz for RTT–Algebra mathcalAn
	1 Introduction
	2 The RTT–Algebra mathcalAn
	3 General Form of Common Eigenvectors of H(+)(x) and H(-)(x)
	4 Bethe Vectors and Bethe Condition
	5 Conclusion
	References
Lie Reductions and Exact Solutions of Generalized Kawahara Equations
	1 Introduction
	2 Lie Reductions and Exact Solutions
	References
Several Exactly Solvable Quantum Mechanical Systems and the SWKB Quantization Condition
	1 Introduction
	2 Exactly Solvable Quantum Mechanics
		2.1 SUSY QM and Shape Invariance
		2.2 Exactly Solvable Quantum Mechanical Systems
	3 SWKB Quantization Condition
		3.1 Multi-indexed Systems and SWKB Condition
		3.2 Krein–Adler Systems and SWKB Condition
	4 Conclusion
	References
Automorphic Symmetries and  AdSn  Integrable Deformations
	1 The Method: Automorphic Symmetry
	2  mathcalB  in  AdS  Integrability
	3  AdS2  and  AdS3  Integrable Backgrounds
	4 Conclusions and Remarks
	References
*-1pc  Applications to Quantum Theory
The Conformal-Symmetry–Color-Neutrality Connection in Strong Interaction
	1 The Puzzle of Color Confinement
	2 Spectroscopic Evidence for Conformal Symmetry of QCD
	3 The Geometric Foundations of the CS–CN Connection
	4 The Jordan Algebra View on Conformal Symmetry
	5 Summary
	References
sell(2) Gaudin Model with General Boundary Terms
	1 Introduction
	2 The sell(2) Linear Bracket
	3 Implementation of the Algebraic Bethe Ansatz
	4 Conclusion
	References
Entanglement of Mixed States in Kähler Quantization
	1 Preliminaries
	2 Main Result
	References
The Chirality-Flow Formalism for Standard Model Calculations
	1 Introduction
	2 Towards Chirality Flow
	3 Examples
	4 Massive Chirality Flow
	5 Conclusion
	References
Spacetime Stochasticity and Second Order Geometry
	1 Introduction
	2 Dynamics on Manifolds
	3 Second Order Geometry
	4 Lie Derivatives and Killing Vectors
	5 Stochastic Dynamics on Manifolds
	6 Conclusions and Outlook
	References
*-1pc  Special Mathematical Results
Velocity Reciprocity in Flat and Curved Space-Time
	1 Introduction
	2 Examples in Flat Space-Times
		2.1 Some Examples for which Velocity Reciprocity Holds
		2.2 Some Examples for which Velocity Reciprocity Does Not Hold
	3 Examples in Curved Space-Time
	4 Conclusions
	References
Meta-Schrödinger Transformations
	1 Introduction
	2 Construction of the Meta-Schrödinger Algebra
		2.1 The General Case: α=0
		2.2 Infinite-Dimensional Extension
		2.3 The Special Case α=0
		2.4 Representations Without Time-Translation-Invariance
	3 Covariant Two-Point Functions
		3.1 Stationary Case
		3.2 Ageing Case
	References
The Quantum Mirror to the Quartic del Pezzo Surface
	1 The Mirror to (dP4,D)
	2 The Quantum Mirror to (dP4,D)
	References
Bidirectional Processes—In Category Theory, Physics, Engineering, …
	1 Introduction
	2 String Diagrams, (co)ends, Optics
	3 Quantum Mechanics
	4 Graphical Linear Algebra
	5 Scattering and Transmission
	6 Discussion
	References
*-1pc  Gauge Theories and Applications
Nonholomorphic Superpotentials in Heterotic Landau-Ginzburg Models
	1 Introduction
	2 Action
	3 Supersymmetry Invariance for Holomorphic Superpotential
	4 Supersymmetry Invariance for Nonholomorphic Superpotential
	References
Automorphic Forms and Fermion Masses
	1 A Fresh Look into an Old Matter
	References
Wilson Lines and Their Laurent Positivity
	1 Wilson Loops
	2 Cluster Varieties Related to the Moduli Space LocG,
	3 Wilson Lines
	4 Multiplicativity of Wilson Lines
	5 Positivity of Wilson Lines
	References
Gauging Higher-Spin-Like Symmetries Using the Moyal Product
	1 Introduction
	2 Moyal-Higher-Spin Theory
		2.1 Gauging Free Field Symmetries
		2.2 Mimicking the Yang-Mills Construction
		2.3 Spacetime Content
		2.4 Coupling to Matter
		2.5 Geometric Interpretation
	3 Summary
	References
Integration of Double Field Theory Algebroids and Pre-rackoid in Doubled Geometry
	1 Introduction
	2 Leibniz, Courant and Vaisman Algebroids
	3 Racks and Rackoids
	4 Pre-rackoids and Doubled Cotangent Paths
	5 Summary
	References
Doubled Aspects of Algebroids and Gauge Symmetry in Double Field Theory
	1 Introduction
	2 Lie Bialgebroid
	3 Doubled Structure of Vaisman Algebroid
	4 The Vaisman Algebroid on a Para-Hermitian Geometry
	5 Conclusion and Discussion
	References
Lie Algebroids and Weight Systems
	1 Introduction
	2 Weight Systems
		2.1 What Is a Weight System
		2.2 Motivation: 3-Manifold Invariants from Weight Systems
		2.3 Weight Systems from Classical Lie Algebras
	3 The Rozansky-Witten Weight System
		3.1 Hyperkahler Manifolds
		3.2 RW-Weight Systems from Hyperkahler Manifolds
	4 The Kontsevich Weight System
		4.1 The Lie Algebra of Formal Hamiltonian Vector Fields
		4.2 Foliations and the Kontsevich Weight System
	5 Lie Algebroids and Voglaire-Xu Weight System
	6 Conclusions and Future Directions
	References
*-1pc  Structures on Lie Groups  and Lie Algebras
Visible Actions of Certain Affine Transformation Groups of a Siegel Domain of the Second Kind
	1 Introduction
	2 Coisotropic Action, Polar Action, and Visible Action
	3 Strongly Visible Action
	References
Quantum Particle on Lattices in Weyl Alcoves
	1 Position and Momentum Bases
	2 Hamiltonians and Energy Spectra
	3 Dual-Weight Models of A3
	References
Abelian J-Invariant Ideals on Nilpotent Lie Algebras
	1 Introduction
	2 Complex Structures on Nilpotent Lie Algebras
	3 Proof of Proposition 2
	References
The Dihedral Dunkl–Dirac Symmetry Algebra with Negative Clifford Signature
	1 Introduction
	2 The Dihedral Dunkl–Dirac Symmetry Algebra
	3 Sketch of the Finite-Dimensional Representations Construction
	References
Lie Structure on Hopf Algebra Cohomology
	1 Introduction
	2 Gerstenhaber Bracket on Cohomologies of a Taft Algebra
		2.1 Lie Structure on Hochschild Cohomology of A=k[x]/(xn)
		2.2 Lie Structure on Hochschild Cohomology of a Taft Algebra
		2.3 Lie Structure on Hopf Algebra Cohomology of a Taft Algebra
	3 Gerstenhaber Bracket for Hopf Algebras
	References
Filtration Associated to an Abelian Inner Ideal and the Speciality of the Subquotient of a Lie Algebra
	1 Preliminaries
	2 Filtration Associated to an Abelian Inner Ideal
	3 The Speciality of the Subquotient
	References
Nilpotent Inner Derivations in Prime Superalgebras
	1 Preliminaries
	2 Ad-nilpotent Elements of R and Skew(R,*)
	3 Examples
		3.1 Examples of Even Ad-nilpotent Elements of R and of Skew(R,*)
		3.2 Examples of Odd Ad-nilpotent Elements of R and of Skew(R,*)
	References