Integrability, Quantization, and Geometry: I. Integrable Systems

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This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher.\n\nThe contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin\x27s early interests. Cover Title page Contents Preface Selected Papers of Boris Dubrovin Primitive Forms without Higher Residue Structure and Integrable Hierarchies (I) 1. Introduction 2. Filtered De Rham cohomology modules \HH_{𝐹} 3. Section and opposite filtration of \HH_{𝐹} 4. Formal analysis on deformation parameter space 5. Primitive forms without metric structure 6. Construction of formal primitive forms with or without metric structure 7. Flat structure without metric structure Acknowledgements References Solutions of 𝐵𝐶_{𝑛} Type of WDVV Equations 1. Introduction 2. Metric for a family of 𝐵𝐶_{𝑛} type configurations 3. Proof through restrictions 4. Application to supersymmetric mechanics References Topology of the Stokes phenomenon 1. Introduction 2. Summary of some data canonically determined by a connection 3. Linear algebra 4. Topological basics 5. Irregular classes and associated topological data 6. Stokes filtered local systems 7. Stokes graded local systems 8. Stokes local systems 9. Stokes local systems and Stokes graded local systems 10. Operations on Stokes filtered local systems 11. Stokes filtrations from Stokes gradings 12. Canonical splittings 13. Wild character varieties and moduli problems Appendix A. Analytic black boxes Acknowledgments References Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and \textcyr{B}-Theorem 1. Introduction 1.1. 1.2. 1.3. 1.4. 1.5. 2. Equivariant exceptional collections and bases 2.1. Basic notions. 2.2. Equivariant Grothendieck-Euler-Poincaré characteristic 2.3. Exceptional collections in 𝒟^{𝒷}_{𝒢}(𝒳) and their mutations 2.4. Dual exceptional collections and helices 2.5. Exceptional bases in equivariant 𝐾-theory 2.6. Dual exceptional bases 2.7. Serre functor and canonical operator 3. Equivariant derived category, exceptional collections and 𝐾-theory of ℙⁿ⁻¹ 3.1. Symmetric functions 3.2. Torus action 3.3. Derived category 3.4. Equivariant 𝐾-theory 3.5. Diophantine constraints on Gram matrices. 4. Equivariant cohomology of ℙⁿ⁻¹ 4.1. Equivariant cohomology 4.2. Extension of scalars 4.3. Poincaré pairing and 𝒟-matrix 4.4. Equivariant characteristic classes 5. Equivariant quantum cohomology of ℙⁿ⁻¹ 5.1. Equivariant Gromov-Witten invariants 5.2. Equivariant Gromov-Witten potential 5.3. Equivariant quantum cohomology 5.4. Quantum connection 5.5. Small equivariant quantum product for ℙⁿ⁻¹ 5.6. 𝑅-matrices and 𝑞𝐾𝑍 operators 5.7. Equivariant 𝑞𝐷𝐸 and 𝑞𝐾𝑍 difference equations 6. Equivariant 𝑞𝐷𝐸 of ℙⁿ⁻¹ and its topological-enumerative solution 6.1. Equivariant quantum differential equation 6.2. Levelt Solution 6.3. Topological-enumerative solution 6.4. Scalar equivariant quantum differential equation 7. Solutions of the equivariant 𝑞𝐷𝐸 and 𝑞𝐾𝑍 difference equations 7.1. 𝑞-Hypergeometric Solutions 7.2. Identification of solutions with 𝐾-theoretical classes 7.3. Module 𝒮_{𝓃} of solutions 7.4. Integral representations for solutions 7.5. Coxeter element, and elements 𝛾_{𝑛},𝛿_{𝑛,𝑜𝑑𝑑},𝛿_{𝑛,𝑒𝑣𝑒𝑛}∈ℬ_{𝓃} 7.6. Exceptional bases 𝑄_{𝑘},𝑄_{𝑘}’,𝑄_{𝑘}”,̃𝑄_{𝑘},̃𝑄_{𝑘}’,̃𝑄_{𝑘}” 7.7. Asymptotic expansion of bases 𝑄_{𝑘}’ and 𝑄_{𝑘}” in sectors 𝒱_{𝓀}’ and 𝒱_{𝓀}” 8. B-classes and B-Theorem 8.1. Morphism \textcyr{B} 8.2. \textcyr{B}-Theorem 9. Formal solutions of the system of 𝑞𝐷𝐸 and 𝑞𝐾𝑍 equations 9.1. Matrix form of 𝑞𝐷𝐸 and 𝑞𝐾𝑍 difference equations 9.2. Shearing transformation 9.3. The ℰ-matrix 9.4. Formal reduction of the system of 𝑞𝐷𝐸 and 𝑞𝐾𝑍 equations 9.5. Formal solutions of the system of 𝑞𝐷𝐸 and 𝑞𝐾𝑍 equations at 𝑞=∞ 10. Stokes bases of the system of 𝑞𝐷𝐸 and 𝑞𝐾𝑍 equations 10.1. Stokes rays, Stokes sectors 10.2. Stokes bases and Stokes matrices 10.3. Properties of Stokes matrices and lexicographical order 10.4. Stokes bases ̃𝑄_{𝑘}’ and ̃𝑄_{𝑘}” 10.5. Stokes bases as 𝕋-full exceptional collections 11. Stokes matrices as Gram matrices of exceptional collections 11.1. Musical notation for braids 11.2. An identity in ℬ_{𝓃} 11.3. Stokes matrices as Gram matrices 12. Specialization of the 𝑞𝐷𝐸 at roots of unity 12.1. Specialization of equivariant 𝐾-theory at roots of unity 12.2. Identities for Stirling numbers 12.3. Scalar equivariant quantum differential equation at roots of unity Appendix A. Formal reduction of the joint system Appendix B. Relation of 𝑞𝐷𝐸 to Dubrovin’s equation for 𝑄𝐻^{∙}(ℙⁿ⁻¹) References Meromorphic Connections over F-manifolds 1. Introduction 2. A review of F-manifolds 3. Frobenius manifolds and flat F-manifolds, with or without Euler fields 4. A dictionary of connections with different enrichments 5. From (𝑇)-structures to pure (𝑇𝐿)-structures 6. Freedom and constraints in the steps from F-manifolds to Frobenius manifolds 7. A conjecture on (𝑇𝐸)-structures over irreducible germs of generically semisimple F-manifolds with Euler fields 8. (𝑇𝐸)-structures over the 2-dimensional F-manifolds 𝐼₂(𝑚) References Canonical maps and integrability in 𝑇𝑇̄ deformed 2d CFTs 1. Introduction 2. Hamiltonian formulation of 2d field theory 3. 𝑇𝑇̄ deformation of 2d Hamiltonian systems 4. Integrability of the deformed 2d massless free field 5. Generalization to 2d CFTs and to (non-conformal) models with a potential Acknowledgements Appendix A. Solution for the light-cone chiral fields Appendix B. String energy in the static and light-cone gauges References Incarnations of XXX ̂𝔰𝔩_{𝔑} Bethe ansatz equations and integrable hierarchies 1. Introduction 2. Incarnations of the Bethe ansatz equations 3. Generation of solutions of Bethe ansatz equations 4. Generating linear problem 5. Spectral transforms for the rational RS system 6. Solution of the rational RS hierarchy 7. Spectral transform for 𝑁-periodic Bethe ansatz equations 8. Bethe ansatz equations and integrable hierarchies 9. Combinatorial data 10. Tau-functions and Baker-Akhieser functions 11. Appendix References The Kowalewski separability conditions 1. Introduction 2. The Kowalewski separability conditions. 3. The method of syzygies 4. The method of complete lifts. References On the Liouville Integrable Reduction of the Associativity Equations in the Case of Three Primary Fields 1. Introduction 2. The reduction theorem 3. The associativity equations 4. The reduction of the associativity equations 5. Liouville integrability of the reduction 6. Integrals of the associativity equations in the Lenard–Magri scheme 7. Appendix References Hurwitz numbers from matrix integrals over Gaussian measure 1. Introduction 2. Definitions and a review of known results 3. Integrals and Hurwitz numbers 4. Hurwitz numbers and quantum and classical integrable models Acknowledgements References Spin Calogero-Moser models on symmetric spaces Introduction 1. Two–sided Spin Calogero-Moser systems 2. Two–sided Calogero-Moser systems for symmetric pairs of Cartan type 3. Spin Calogero-Moser systems on 𝐺×𝐺 4. Two–sided spin Calogero-Moser model for rank one orbits for 𝑆𝐿_{𝑛} 5. Conclusion Appendix A. Cotangent bundle 𝑇*𝐺 as a symplectic manifold Appendix B. Poisson manifold 𝐾\𝑇*𝐺/𝐾 Appendix C. Poisson manifold 𝐺\𝑇*(𝐺×𝐺)/𝐺 Appendix D. Matrix element functions Acknowledgements References Quantum Toda Lattice: a Challenge for Representation Theory 1. Introduction 2. Quantum Toda Lattice and Representation Theory 3. Quantum Toda Lattice: the point of view of QISM 4. Whittaker vectors and spherical vectors in Gelfand–Zetlin representation References Finite-Gap Solutions of the Mikhalëv Equation 1. Introduction 2. Solutions related to the KdV hierarchy 3. Solutions corresponding to quadratic spectral problem Appendix A. Calculation of the parameters of the 3-elliptic two-gap solutions of Mikhalëv equations Appendix B. Calculation of the parameters of the 2-elliptic two-gap solutions of Mikhalëv equations Appendix C. Some theta-functional identities Concluding remarks References Flat coordinates on orbit spaces: from Novikov algebras to cyclic quotient singularities 1. Introduction 2. Algebraic preliminaries 3. The Gauss-Manin equations 4. A Frobenius Theorem for solvable vector fields 5. Solutions of the Gauss-Manin equations for Novikov algebra 6. Finite monodromy and polynomial solutions 7. Cyclic quotient singularities and orbit spaces 8. Novikov structures on the cotangent bundle 9. Conclusion Acknowledgements References Cubic Hodge integrals and integrable hierarchies of Volterra type 1. Introduction 2. Two-partition Hodge integrals 3. Lift to tau function 4. Perspectives in Lax formalism 5. Integrable structures in cubic Hodge integrals Acknowledgements References Gauged Witten Equation and Adiabatic Limit 1. Introduction 2. Gauged Linear Sigma Model Spaces 3. The GLSM Correlation Functions 4. The Adiabatic Limit 5. Relation with the Mirror Map References Back Cover