Topological Recursion and its Influence in Analysis, Geometry, and Topology

Cover
Title page
Contents
Preface
1. The 2016 AMS von Neumann Symposium
2. What is topological recursion?
3. The origin
4. Topological recursion and semi-simple cohomological field theories
5. Topological recursion and toric mirror symmetry
6. A knot theory twist and quantum curves
7. Relation to the Hitchin theory of Higgs bundles, and quantum curves as opers
8. New developments
Bibliography
Modular functors, cohomological field theories, and topological recursion
1. Introduction
2. Construction of vector bundles from a modular functor
3. Cohomological field theories
4. Topological recursion
5. Example: Modular functors associated to finite groups
6. Example: WZW model for compact Lie groups
7. Discussion about global spectral curves
Appendix A. Extra properties of the ?-matrix
Acknowledgments
References
On the Gopakumar–Ooguri–Vafa correspondence for Clifford–Klein 3-manifolds
1. Introduction
2. A 0-dimensional aperçu: Matrix models, the topological recursion and enumerative geometry
3. The GOV correspondence for ?³
4. The GOV correspondence for Clifford–Klein 3-manifolds
5. Conclusions
Acknowledgments
References
Bouchard-Klemm-Marino-Pasquetti conjecture for ℂ³
1. Introduction
2. Marino-Vafa formula and Symmetrized Cut-Join Equation
3. The BKMP Conjecture
4. Residue Calculus
5. Some Formal Analysis
6. The Left Hand Side
7. The Right Hand Side
References
The hybrid Landau–Ginzburg models of Calabi–Yau complete intersections
1. Introduction
2. Setup
3. Calabi–Yau side
4. Hybrid Landau–Ginzburg side
5. The correspondence
References
Singular vector structure of quantum curves
1. Introduction
2. Higher level (Virasoro) quantum curves
3. Higher level (super-Virasoro) quantum curves
4. Summary
Acknowledgments
References
Towards the topological recursion for double Hurwitz numbers
1. Introduction
2. Combinatorics of double Hurwitz numbers
3. Evidence for the conjecture
4. Towards a proof of the conjecture
Appendix A. Table of double Hurwitz numbers
Appendix B. Table of pruned double Hurwitz numbers
References
Quantization of spectral curves for meromorphic Higgs bundles through topological recursion
1. Introduction
2. A walk-through of the simplest example
3. Quantum curves for Higgs bundles
4. Geometry of spectral curves in the compactified cotangent bundle
5. The spectral curve as a divisor and its minimal resolution
6. Construction of the quantum curve
7. The classical differential equations as quantum curves
Acknowledgments
References
Topological recursion and Givental’s formalism: Spectral curves for Gromov-Witten theories
1. Introduction
2. Preliminaries
3. Local spectral curves for cohomological field theories
4. Dubrovin’s superpotential as a global spectral curve
Acknowledgments
References
Primary invariants of Hurwitz Frobenius manifolds
1. Introduction
2. Frobenius manifolds
3. Topological recursion and cohomological field theory
4. Hurwitz Frobenius manifolds
5. Topological recursion for compact spectral curves
6. Topological recursion for families of spectral curves
References
Hopf algebras and topological recursion
1. Introduction
2. The topological recursion of Eynard and Orantin
3. The Loday-Ronco Hopf algebra of planar binary trees
4. The solution of topological recursion
5. The antipode
6. Discussion and other topics
References
Graph sums in the remodeling conjecture
1. Introduction
2. Geometry and the A-model of a toric Calabi-Yau 3-orbifold
3. Mirror curves and the Landau-Ginzburg mirror
4. A quick review of the genus zero mirror theorem for toric orbifolds
5. A-model quantization: The orbifold Givental formula
6. B-model quantization: The topological recursion
7. Comparing the graph sums: Proving the Remodeling Conjecture
Acknowledgments
References
Double quantization of Seiberg–Witten geometry and W-algebras
1. Introduction and summary
2. Seiberg–Witten spectral curve and 1st quantization
3. Operator formalism and 2nd quantization
4. From double quantization to Virasoro/W-algebra
5. ?-state
6. Quiver W-algebra
7. Affine quiver W-algebra
8. Quiver elliptic W-algebra
Acknowledgments
References
Airy structures and symplectic geometry of topological recursion
1. Introduction
2. Airy structures
3. Comparison with topological recursion
4. Spectral curves
5. Affine symplectic connection and local embedding of the moduli space of spectral curves
6. Formal discs and universal Airy structure
7. Hamiltonian reduction and Holomorphic Anomaly Equation
8. Semi-affine Lagrangian embeddings
9. Two more speculations
References
Periods of meromorphic quadratic differentials and Goldman bracket
1. Introduction
2. Canonical covering of a Riemann surface
3. Second order equation with meromorphic potential on a Riemann surface
4. Variational formulas
5. Canonical symplectic structure on ?*\Mcal_{?,?} via periods of ?
6. From canonical symplectic structure on ?*\Mcal_{?,?} to Goldman bracket
7. Riemann sphere with four marked points
Acknowledgments
References
On ELSV-type formulae, Hurwitz numbers and topological recursion
1. Introduction
2. Chiodo classes
3. From the spectral curve to the Givental R-matrix
4. Equivalence statements: A new proof of the Johnson-Pandharipande-Tseng formula
References
Quantum curves for simple Hurwitz numbers of an arbitrary base curve
1. Introduction and the main results
2. A cut-and-join equation for simple Hurwitz numbers
3. The discrete Laplace transform
4. A Schrödinger equation
5. The heat equation and its consequences
6. The quantum curve
7. Semi-classical limit
References
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