**Proceedings of Symposia in Pure Mathematics**

Volume: 103;
2021;
480 pp;
Softcover

MSC: Primary 14; 16; 20; 52; 53; 81;

**Print ISBN: 978-1-4704-5592-7
Product Code: PSPUM/103.2**

List Price: $137.00

AMS Member Price: $109.60

MAA Member Price: $123.30

**Electronic ISBN: 978-1-4704-6435-6
Product Code: PSPUM/103.2.E**

List Price: $137.00

AMS Member Price: $109.60

MAA Member Price: $123.30

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# Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry

Share this page *Edited by *
*Sergey Novikov; Igor Krichever; Oleg Ogievetsky; Senya Shlosman*

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.

The 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's early interests.

#### Readership

Graduate students and researchers interested in mathematical physics, enumerative and differential geometry, and geometric representation theory.

# Table of Contents

## Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry

- Cover Cover11
- Title page i4
- Contents vii8
- Preface ix10
- Selected Papers of Boris Dubrovin xiii14
- Geometry from Donaldson-Thomas invariants 124
- 1. Introduction 124
- 2. Frobenius structures and pencils of connections 831
- 3. Stokes data and isomonodromy 1235
- 4. Joyce structures 1740
- 5. BPS structures 2447
- 6. Inverting the Stokes map 2851
- 7. Linear data from Joyce structures 3558
- 8. The finite uncoupled case 4164
- 9. The case of Calabi-Yau threefolds without compact divisors 4871
- 10. The case of the A₂ quiver 5679
- Acknowledgements 6487
- References 6487

- Fricke identities, Frobenius 𝑘-characters and Markov equation 6790
- On Harish-Chandra theory of global nonsymmetric functions 79102
- Symmetries of genus zero modular operad 101124
- 1. Introduction and summary 101124
- 2. Cofinite combinatorics and operadic symmetries 102125
- 3. Stable curves of genus zero and their graphs 103126
- 4. Stratifications of \overline{𝑀}_{0,𝑛} 104127
- 5. Thin categories and groupoids 105128
- 6. Graphic skeleta of the symmetries of genus zero modular operad 106129
- References 108131

- On Schur problem and Kostka numbers 111134
- An analytic version of the Langlands correspondence for complex curves 137160
- \centerline{Part I} 143166
- \centerline{Part II} 166189
- 9. Darboux operators 166189
- 10. Eigenfunctions and monodromy for Darboux operators 170193
- 11. Essentially self-adjoint algebras of unbounded operators 175198
- 12. The main theorem 180203
- 13. Generalized Sobolev and Schwartz spaces attached to the operator 𝐿. 181204
- 14. Proof of Theorem 12.1 191214
- 15. The operator 𝐷=𝐿^{†}𝐿 and a second proof of Theorem 12.1 193216
- 16. A third proof of Theorem 12.1 195218
- 17. The spectral description of the global Sobolev and Schwartz spaces attached to 𝐿 197220
- 18. Remarks 197220
- 19. Degenerate cases 198221
- Acknowledgements 200223
- References 200223

- A Note On Some Minimally Supersymmetric Models In Two Dimensions 203226
- A moment map interpretation of the Ricci form, Kähler–Einstein structures, and Teichmüller spaces 223246
- Global gauge conditions in the Batalin–Vilkovisky formalism 257280
- Convention 258281
- 1. Odd symplectic structures 258281
- 2. Odd symplectic supermanifolds 261284
- 3. Simplicial supermanifolds 264287
- 4. Families of Lagrangian submanifolds 266289
- 5. Lagrangian submanifolds of simplicial odd symplectic supermanifolds 268291
- 6. Application to the superparticle 271294
- 7. Global symmetries 277300
- References 279302

- Interpolated Apéry numbers, quasiperiods of modular forms, and motivic gamma functions 281304
- Introduction: the Apéry numbers and their interpolations 281304
- 1. Hypergeometric interpolation, periods, and quasiperiods 283306
- 2. Proof of Theorems 1 and 2 285308
- 3. A second Apéry-like sequence and the proof of Theorem 3 286309
- 4. Periods and quasiperiods of modular forms 288311
- 5. Geometric interpretation: fibered motives 291314
- 6. Asymptotics 295318
- 7. The kappa series and the Gamma Conjecture 297320
- Acknowledgments 300323
- References 300323

- Quantum flips I: Local model 303326
- 0. Introduction 303326
- 1. From Picard–Fuchs to small 𝒟^{𝓏}-modules 307330
- 2. The GW system for 𝑄₀𝐻 312335
- 3. Decomposition of 𝑄₀𝐻 via block-diagonalization 316339
- 4. Existence of Ψ̂ as an 𝐹-embedding of 𝑄𝐻 323346
- 5. Exact determination of the Dubrovin connection 332355
- 6. An example on simple (2, 1) flip for fourfolds 337360
- Acknowledgements 351374
- References 352375

- Aspects of 𝑝-adic geometry related to entanglement entropy 353376
- Grothendieck-Teichmüller group, operads and graph complexes: a survey 383406
- 1. Introduction 383406
- 2. Completions of groups, Lie algebras and algebras 384407
- 3. Monoidal categories and monoidal functors 392415
- 4. Operads 400423
- 5. Operad of parenthesized braids and \wh{𝐺𝑇} 408431
- 6. Infinitesimal braids, 𝐺𝑅𝑇₁ and associators 421444
- 7. Grothendieck-Teichmüller group, graph complexes and T.Willwacher theorems 428451
- 8. Some applications of the theory of Drinfeld’s associators, 𝐺𝑅𝑇₁ and graph complexes 436459
- References 442465

- Platonic Compounds of Cylinders 447470
- 1. Introduction 447470
- 2. Interpretation of the configuration 𝑂₆: 𝔸₄-symmetric configurations 450473
- 3. Generalizations of the configuration 𝑂₆: 𝛿-rotation process 453476
- Neighboring tangent lines 453476
- 4. Platonic compounds: global picture 459482
- 5. Minima 466489
- 6. Parallel cylinders 472495
- 7. Concluding remarks 475498
- Acknowledgments 479502
- References 479502

- Back Cover Back Cover1506