Cover

Title page

Contents

Preface

Superstring compactifications to all orders in 𝛼’

1. Introduction

2. Type II string theory on 𝐺₂ manifolds

3. Interpretation in 3𝑑 and 4𝑑 field theory

4. Massive Kaluza-Klein modes in superspace

References

Supersymmetric partition functions on Riemann surfaces

1. Introduction

2. 3d theories on Σ_{𝑔}×𝑆¹

3. 2d theories on Σ_{𝑔}

4. 4d theories on Σ_{𝑔}×𝑇²

5. Examples

6. Large 𝑁 limit and black hole entropy

Acknowledgements

Appendix A. Notation, Lagrangians and supersymmetry variations

References

On the mathematics and physics of Mixed Spin P-fields

1. Introduction

2. Mirror Symmetry and Gromov-Witten Invariants of Quintics

3. Witten’s Gauged Linear Sigma Model (GLSM)

4. Hyperplane Property, Ghost, and P-field

5. Fields Valued in Two GIT Quotients

6. Affine LG Phase and Spin Structure

7. The Puzzle to Link Invariants in Opposite Phases

8. Master Space

9. Mixed Spin Fields: Quantization of the Master Space

10. Vanishing and Polynomial Relations

11. Comparison with Physical Theories

Acknowledgments

References

Homological mirror functors via Maurer-Cartan formalism

1. Introduction

2. Basic example of (\C,𝑆¹) and 𝑊(𝑥)=𝑥

3. Maurer-Cartan formalism and two different potentials

4. Construction of the canonical \AI-functor

5. Elliptic curve example

6. Non-commutative mirrors and deformation quantization

7. Complete intersection mirror of the torus

References

Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds

1. Introduction

2. Setup and preliminary results

3. Batyrev-Borisov mirror symmetry

4. Dolgachev-Nikulin mirror symmetry

5. Beyond Batyrev-Borisov mirror symmetry for threefolds

6. Non-commutative fibrations

References

SL(2,ℂ) Chern-Simons theory and four-dimensional quantum geometry

1. Introduction

2. Classical Correspondence

3. Quantum Correspondence

4. Wilson Graph Operator and Loop Quantum Gravity

Acknowledgements

References

Quantum cohomology under birational maps and transitions

0. Introduction

1. Quantum cohomology

2. Review on quantum Lefschetz

3. Quantum Leray–Hirsch

4. Application I: Ordinary flops

5. Application II: Blow-ups along complete intersection centers

6. Application III: Simple flips

7. Conifold transitions of Calabi–Yau 3-folds

References

𝐿²-kernels of Dirac-type operators on monopole moduli spaces

1. Introduction and Conclusion

2. (Singular) Monopole Moduli Space

3. Consequences of 𝑁=2 Supersymmetry

4. Mathematical Predictions

5. Two Examples

References

𝔅𝔓𝔖/ℭ𝔉𝔗 correspondence: Instantons at crossroads and gauge origami

1. Introduction

1.1. Organization of the paper

2. Gauge and string theory motivations

2.1. Generalized gauge theory

2.2. Gauge origami

2.3. Symmetries, twisting, equivariance

2.4. Gauge theories on stacks of D-branes

3. Spiked instantons

3.1. Generalized ADHM equations

3.2. Holomorphic equations

3.3. The moduli spaces \mM_{𝑘}*(⃗𝑛)

3.4. Stability

4. The symmetries of spiked instantons

4.1. Framing and spatial rotations

4.2. Subtori

4.3. Orbifolds, quivers, defects

4.4. Our goal: compactness theorem

5. Ordinary instantons

5.1. ADHM construction and its 𝑓𝑖𝑛𝑒𝑝𝑟𝑖𝑛𝑡

5.2. Ordinary instantons from spiked instantons

5.3. Uhlenbeck spaces

5.4. One-instanton example

5.5. The canonical complex \CalS

5.6. 𝑃-spaces

5.7. Stratification and correspondences

5.8. 𝐿-spaces

5.9. The symmetries of the ADHM space

5.10. 𝑈 versus 𝑃𝑈

5.11. Tangent space

5.12. Fixed locus

5.13. Tangent space at the fixed point

5.14. Canonical complex at the fixed point

5.15. Smaller tori

5.16. Fixed points of smaller tori

5.17. Compactness of the fixed point set

5.18. Ordinary instantons as the fixed set

6. Crossed and folded instantons

6.1. Crossed instantons

6.2. One-instanton crossed example

6.3. Folded instantons

6.4. One-instanton folded example

6.5. Fixed point sets: butterflies and zippers

7. Reconstructing spiked instantons

7.1. The local K-spaces

7.2. Toric spiked instantons

8. The compactness theorem

9. Integration over the spiked instantons

9.1. Cohomological field theory

9.2. Localization and analyticity

10. Quiver crossed instantons

10.1. Crossed quivers

10.2. Orbifolds and defects: ADE ×𝑈(1)× ADE

11. Spiked instantons on orbifolds and defects

12. Conclusions and future directions

Acknowledgements

References

Balanced embedding of degenerating Abelian varieties

1. Introduction

2. Setup and Main Theorems

3. Construction of Theta Functions

4. Proofs of Main Theorems

5. Appendix

Acknowledgements

References

The modularity/automorphy of Calabi–Yau varieties of CM type

1. Introduction

2. The (cohomological) 𝐿-series

3. The modularity results

4. Modularity/automorphy of K3 surfaces with non-symplectic automorphisms

5. Calabi–Yau threefolds of Borcea–Voisin type

6. Acknowledgments

References

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