# Mathematical Aspects of Quantization

Share this page *Edited by *
*Samuel Evens; Michael Gekhtman; Brian C. Hall; Xiaobo Liu; Claudia Polini*

This book is a collection of expository
articles from the Center of Mathematics at Notre Dame's 2011 program
on quantization.

Included are lecture notes from a summer school on quantization on
topics such as the Cherednik algebra, geometric quantization, detailed
proofs of Willwacher's results on the Kontsevich graph complex, and
group-valued moment maps.

This book also includes expository articles on quantization and
automorphic forms, renormalization, Berezin–Toeplitz quantization in
the complex setting, and the commutation of quantization with
reduction, as well as an original article on derived Poisson brackets.

The primary goal of this volume is to make topics in quantization
more accessible to graduate students and researchers.

# Table of Contents

## Mathematical Aspects of Quantization

- Preface ix10 free
- Dunkl operators and quasi-invariants of complex reflection groups 112 free
- Notes on algebraic operads, graph complexes, and Willwacher’s construction 2536
- 1. Introduction 2637
- 2. Trees 3142
- 3. Operads, pseudo-operads, and their dual versions 3445
- 4. Convolution Lie algebra 4657
- 5. To invert, or not to invert: that is the question 5162
- 6. Twisting of operads 5768
- 7. The operad 𝖦𝗋𝖺 and its link to the operad 𝖦𝖾𝗋 7990
- 8. The full graph complex 𝖿𝖦𝖢: the first steps 8293
- 9. Analyzing the dg operad 𝑇𝑤𝖦𝗋𝖺 8697
- 10. The full graph complex 𝖿𝖦𝖢 revisited 105116
- 11. Deformation complex of 𝖦𝖾𝗋 108119
- 12. Tamarkin’s rigidity in the stable setting 114125
- 13. Deformation complex of 𝖦𝖾𝗋 versus Kontsevich’s graph complex 127138
- Appendix A. Lemma on a quasi-isomorphism of filtered complexes 131142
- Appendix B. Harrison complex of the cocommutative coalgebra 𝑆(𝑉) 132143
- Appendix C. Filtered dg Lie algebras. The Goldman-Millson theorem 135146
- Appendix D. Solutions to selected exercises 139150
- References 144155

- Geometric quantization; a crash course 147158
- 1. An outline of the notes 148159
- 2. From Newton’s law of motion to geometric mechanics in one hour 149160
- 3. Prequantization 152163
- 4. Polarizations 156167
- 5. Prequantization of differential cocycles 161172
- Appendix A. Elements of category theory 165176
- Appendix B. Densities 168179
- References 173184

- Lectures on group-valued moment maps and Verlinde formulas 175186
- 1. Introduction 175186
- 2. Motivation: Moduli spaces of flat bundles 176187
- 3. Group-valued moment maps 178189
- 4. Quantization of Hamiltonian 𝐺-spaces 188199
- 5. The level 𝑘 fusion ring 192203
- 6. Pre-quantization of q-Hamiltonian spaces 196207
- 7. Twisted Spin_{𝑐}-structures on q-Hamiltonian spaces 198209
- 8. Quantization of q-Hamiltonian 𝐺-spaces 202213
- References 207218

- Quantization and automorphic forms 211222
- Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras 219230
- Renormalization by any means necessary 247258
- Berezin-Toeplitz quantization and star products for compact Kähler manifolds 257268
- 1. Introduction 257268
- 2. The geometric setup 260271
- 3. Berezin-Toeplitz operator quantization 261272
- 4. Deformation quantization –star products 265276
- 5. Global Toeplitz operators 269280
- 6. Coherent states and symbols 273284
- 7. The Berezin transform and Bergman kernels 276287
- 8. Berezin transform and star products 278289
- 9. Other constructions of star products –Graphs 282293
- 10. Excursion: The Kontsevich construction 286297
- 11. Some applications of the Berezin-Toeplitz operators 288299
- References 290301

- Commutation of geometric quantization and algebraic reduction 295306