**Proceedings of Symposia in Pure Mathematics**

Volume: 100;
2018;
549 pp;
Hardcover

MSC: Primary 14; 53; 81;

**Print ISBN: 978-1-4704-3541-7
Product Code: PSPUM/100**

List Price: $133.00

AMS Member Price: $106.40

MAA Member Price: $119.70

**Electronic ISBN: 978-1-4704-4992-6
Product Code: PSPUM/100.E**

List Price: $133.00

AMS Member Price: $106.40

MAA Member Price: $119.70

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

Share this page *Edited by *
*Chiu-Chu Melissa Liu; Motohico Mulase*

This volume contains the proceedings of the 2016 AMS von Neumann
Symposium on Topological Recursion and its Influence in Analysis,
Geometry, and Topology, which was held from July 4–8, 2016, at the
Hilton Charlotte University Place, Charlotte, North Carolina.

The papers contained in the volume present a snapshot of rapid and
rich developments in the emerging research field known as topological
recursion. It has its origin around 2004 in random matrix theory and
also in Mirzakhani's work on the volume of moduli spaces of hyperbolic
surfaces.

Topological recursion has played a fundamental role in connecting
seemingly unrelated areas of mathematics such as matrix models,
enumeration of Hurwitz numbers and Grothendieck's dessins d'enfants,
Gromov-Witten invariants, the A-polynomials and colored polynomial
invariants of knots, WKB analysis, and quantization of Hitchin moduli
spaces. In addition to establishing these topics, the volume includes
survey papers on the most recent key accomplishments: discovery of the
unexpected relation to semi-simple cohomological field theories and a
solution to the remodeling conjecture. It also provides a glimpse into
the future research direction; for example, connections with the Airy
structures, modular functors, Hurwitz-Frobenius manifolds, and
ELSV-type formulas.

#### Readership

Graduate students and researchers interested in topological recursion and its applications in various areas of mathematics.

# Table of Contents

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

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- 1. The 2016 AMS von Neumann Symposium vii8
- 2. What is topological recursion? ix10
- 3. The origin x11
- 4. Topological recursion and semi-simple cohomological field theories xii13
- 5. Topological recursion and toric mirror symmetry xii13
- 6. A knot theory twist and quantum curves xiv15
- 7. Relation to the Hitchin theory of Higgs bundles, and quantum curves as opers xv16
- 8. New developments xvi17
- Bibliography xviii19

- Modular functors, cohomological field theories, and topological recursion 124
- 1. Introduction 124
- 2. Construction of vector bundles from a modular functor 427
- 3. Cohomological field theories 1437
- 4. Topological recursion 2245
- 5. Example: Modular functors associated to finite groups 3457
- 6. Example: WZW model for compact Lie groups 3861
- 7. Discussion about global spectral curves 4871
- Appendix A. Extra properties of the 𝑆-matrix 5275
- Acknowledgments 5376
- References 5477

- On the Gopakumar–Ooguri–Vafa correspondence for Clifford–Klein 3-manifolds 5982
- Bouchard-Klemm-Marino-Pasquetti conjecture for ℂ³ 83106
- The hybrid Landau–Ginzburg models of Calabi–Yau complete intersections 103126
- Singular vector structure of quantum curves 119142
- Towards the topological recursion for double Hurwitz numbers 151174
- Quantization of spectral curves for meromorphic Higgs bundles through topological recursion 179202
- 1. Introduction 180203
- 2. A walk-through of the simplest example 191214
- 3. Quantum curves for Higgs bundles 196219
- 4. Geometry of spectral curves in the compactified cotangent bundle 201224
- 5. The spectral curve as a divisor and its minimal resolution 209232
- 6. Construction of the quantum curve 214237
- 7. The classical differential equations as quantum curves 221244
- Acknowledgments 226249
- References 226249

- Topological recursion and Givental’s formalism: Spectral curves for Gromov-Witten theories 231254
- Primary invariants of Hurwitz Frobenius manifolds 297320
- Hopf algebras and topological recursion 333356
- Graph sums in the remodeling conjecture 359382
- 1. Introduction 359382
- 2. Geometry and the A-model of a toric Calabi-Yau 3-orbifold 362385
- 3. Mirror curves and the Landau-Ginzburg mirror 370393
- 4. A quick review of the genus zero mirror theorem for toric orbifolds 375398
- 5. A-model quantization: The orbifold Givental formula 379402
- 6. B-model quantization: The topological recursion 388411
- 7. Comparing the graph sums: Proving the Remodeling Conjecture 394417
- Acknowledgments 398421
- References 398421

- Double quantization of Seiberg–Witten geometry and W-algebras 405428
- 1. Introduction and summary 405428
- 2. Seiberg–Witten spectral curve and 1st quantization 409432
- 3. Operator formalism and 2nd quantization 411434
- 4. From double quantization to Virasoro/W-algebra 412435
- 5. 𝑍-state 413436
- 6. Quiver W-algebra 417440
- 7. Affine quiver W-algebra 423446
- 8. Quiver elliptic W-algebra 425448
- Acknowledgments 428451
- References 428451

- Airy structures and symplectic geometry of topological recursion 433456
- 1. Introduction 433456
- 2. Airy structures 437460
- 3. Comparison with topological recursion 448471
- 4. Spectral curves 461484
- 5. Affine symplectic connection and local embedding of the moduli space of spectral curves 463486
- 6. Formal discs and universal Airy structure 468491
- 7. Hamiltonian reduction and Holomorphic Anomaly Equation 473496
- 8. Semi-affine Lagrangian embeddings 480503
- 9. Two more speculations 486509
- References 488511

- Periods of meromorphic quadratic differentials and Goldman bracket 491514
- 1. Introduction 491514
- 2. Canonical covering of a Riemann surface 495518
- 3. Second order equation with meromorphic potential on a Riemann surface 498521
- 4. Variational formulas 500523
- 5. Canonical symplectic structure on 𝑇*\Mcal_{𝑔,𝑛} via periods of 𝑄 502525
- 6. From canonical symplectic structure on 𝑇*\Mcal_{𝑔,𝑛} to Goldman bracket 507530
- 7. Riemann sphere with four marked points 512535
- Acknowledgments 514537
- References 514537

- On ELSV-type formulae, Hurwitz numbers and topological recursion 517540
- Quantum curves for simple Hurwitz numbers of an arbitrary base curve 533556
- 1. Introduction and the main results 533556
- 2. A cut-and-join equation for simple Hurwitz numbers 537560
- 3. The discrete Laplace transform 539562
- 4. A Schrödinger equation 539562
- 5. The heat equation and its consequences 542565
- 6. The quantum curve 544567
- 7. Semi-classical limit 545568
- References 547570

- Back Cover Back Cover1578