**Proceedings of Symposia in Pure Mathematics**

Volume: 96;
2017;
297 pp;
Hardcover

MSC: Primary 14; 51; 53; 81;

**Print ISBN: 978-1-4704-2951-5
Product Code: PSPUM/96**

List Price: $126.00

AMS Member Price: $100.80

MAA Member Price: $113.40

**Electronic ISBN: 978-1-4704-4276-7
Product Code: PSPUM/96.E**

List Price: $126.00

AMS Member Price: $100.80

MAA Member Price: $113.40

# String-Math 2015

Share this page *Edited by *
*Si Li; Bong H. Lian; Wei Song; Shing-Tung Yau*

A co-publication of the AMS and International Press of Boston

This volume contains the proceedings of the conference
String-Math 2015, which was held from December 31, 2015–January 4,
2016, at Tsinghua Sanya International Mathematics Forum in Sanya,
China. Two of the main themes of this volume are frontier research on
Calabi-Yau manifolds and mirror symmetry and the development of
non-perturbative methods in supersymmetric gauge theories. The
articles present state-of-the-art developments in these topics.

String theory is a broad subject, which has profound connections
with broad branches of modern mathematics. In the last decades, the
prosperous interaction built upon the joint efforts from both
mathematicians and physicists has given rise to marvelous deep results
in supersymmetric gauge theory, topological string, M-theory and
duality on the physics side, as well as in algebraic geometry,
differential geometry, algebraic topology, representation theory and
number theory on the mathematics side.

#### Readership

Advanced graduate students, post-docs, and post Ph.D. mathematicians and mathematical physicists interested in string theory.

# Table of Contents

## String-Math 2015

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- Superstring compactifications to all orders in πΌβ 18
- Supersymmetric partition functions on Riemann surfaces 1320
- 1. Introduction 1320
- 2. 3d theories on Ξ£_{π}ΓπΒΉ 1522
- 3. 2d theories on Ξ£_{π} 2633
- 4. 4d theories on Ξ£_{π}ΓπΒ² 2835
- 5. Examples 2936
- 6. Large π limit and black hole entropy 3946
- Acknowledgements 4249
- Appendix A. Notation, Lagrangians and supersymmetry variations 4249
- References 4451

- On the mathematics and physics of Mixed Spin P-fields 4754
- 1. Introduction 4754
- 2. Mirror Symmetry and Gromov-Witten Invariants of Quintics 4855
- 3. Wittenβs Gauged Linear Sigma Model (GLSM) 5158
- 4. Hyperplane Property, Ghost, and P-field 5259
- 5. Fields Valued in Two GIT Quotients 5663
- 6. Affine LG Phase and Spin Structure 5764
- 7. The Puzzle to Link Invariants in Opposite Phases 6067
- 8. Master Space 6168
- 9. Mixed Spin Fields: Quantization of the Master Space 6168
- 10. Vanishing and Polynomial Relations 6673
- 11. Comparison with Physical Theories 6875
- Acknowledgments 6976
- References 7077

- Homological mirror functors via Maurer-Cartan formalism 7582
- 1. Introduction 7582
- 2. Basic example of (\C,πΒΉ) and π(π₯)=π₯ 7784
- 3. Maurer-Cartan formalism and two different potentials 8188
- 4. Construction of the canonical \AI-functor 8491
- 5. Elliptic curve example 8592
- 6. Non-commutative mirrors and deformation quantization 8794
- 7. Complete intersection mirror of the torus 8996
- References 9198

- Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds 93100
- SL(2,β) Chern-Simons theory and four-dimensional quantum geometry 133140
- Quantum cohomology under birational maps and transitions 149156
- 0. Introduction 149156
- 1. Quantum cohomology 152159
- 2. Review on quantum Lefschetz 153160
- 3. Quantum LerayβHirsch 154161
- 4. Application I: Ordinary flops 156163
- 5. Application II: Blow-ups along complete intersection centers 159166
- 6. Application III: Simple flips 160167
- 7. Conifold transitions of CalabiβYau 3-folds 163170
- References 167174

- πΏΒ²-kernels of Dirac-type operators on monopole moduli spaces 169176
- π ππ/βππ correspondence: Instantons at crossroads and gauge origami 183190
- 1. Introduction 185192
- 2. Gauge and string theory motivations 187194
- 3. Spiked instantons 193200
- 4. The symmetries of spiked instantons 198205
- 5. Ordinary instantons 203210
- 5.1. ADHM construction and its πππππππππ‘ 203210
- 5.2. Ordinary instantons from spiked instantons 204211
- 5.3. Uhlenbeck spaces 204211
- 5.4. One-instanton example 205212
- 5.5. The canonical complex \CalS 205212
- 5.6. π-spaces 206213
- 5.7. Stratification and correspondences 206213
- 5.8. πΏ-spaces 207214
- 5.9. The symmetries of the ADHM space 208215
- 5.10. π versus ππ 208215
- 5.11. Tangent space 208215
- 5.12. Fixed locus 209216
- 5.13. Tangent space at the fixed point 211218
- 5.14. Canonical complex at the fixed point 212219
- 5.15. Smaller tori 213220
- 5.16. Fixed points of smaller tori 214221
- 5.17. Compactness of the fixed point set 215222
- 5.18. Ordinary instantons as the fixed set 218225

- 6. Crossed and folded instantons 218225
- 7. Reconstructing spiked instantons 227234
- 8. The compactness theorem 232239
- 9. Integration over the spiked instantons 234241
- 10. Quiver crossed instantons 236243
- 11. Spiked instantons on orbifolds and defects 242249
- 12. Conclusions and future directions 242249
- Acknowledgements 243250
- References 243250

- Balanced embedding of degenerating Abelian varieties 247254
- The modularity/automorphy of CalabiβYau varieties of CM type 265272
- Back Cover Back Cover1306