**Contemporary Mathematics**

Volume: 763;
2021;
203 pp;
Softcover

MSC: Primary 14;
Secondary 53; 32

**Print ISBN: 978-1-4704-5700-6
Product Code: CONM/763**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-6419-6
Product Code: CONM/763.E**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

# Singularities, Mirror Symmetry, and the Gauged Linear Sigma Model

Share this page *Edited by *
*Tyler J. Jarvis; Nathan Priddis*

This volume contains the proceedings of the
workshop Crossing the Walls in Enumerative Geometry, held in May 2018
at Snowbird, Utah. It features a collection of both expository and
research articles about mirror symmetry, quantized singularity theory
(FJRW theory), and the gauged linear sigma model.

Most of the expository works are based on introductory lecture
series given at the workshop and provide an approachable introduction
for graduate students to some fundamental topics in mirror symmetry
and singularity theory, including quasimaps, localization, the gauged
linear sigma model (GLSM), virtual classes, cosection localization,
\(p\)-fields, and Saito's primitive forms. These articles help readers
bridge the gap from the standard graduate curriculum in algebraic
geometry to exciting cutting-edge research in the field.

The volume also contains several research articles by leading
researchers, showcasing new developments in the field.

#### Readership

Graduate students and research mathematicians interested in enumerative geometry, singularity theory, and the relations between algebraic geometry and mathematical physics.

# Table of Contents

## Singularities, Mirror Symmetry, and the Gauged Linear Sigma Model

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Quasimaps and some examples of stacks for everybody 112
- 1. Introduction 112
- 2. Motivation for the study of quasimaps 112
- 3. Quotients in algebraic geometry 314
- 4. Quasimaps 516
- 5. Stability and moduli spaces of quasimaps 617
- 6. Virtual cycles via perfect obstruction theories 920
- 7. Quasimap 𝐼-functions 1122
- 8. A formula for the 𝐼-function 1324
- References 1627

- Introduction to the gauged linear sigma model 1930
- Localization and mirror symmetry 4354
- 0. Introduction 4354
- 1. Equivariant cohomology and localization 4455
- 2. Fan-Jarvis-Ruan-Witten theory of the quintic threefold 4960
- 3. Proof of the Fan-Jarvis-Ruan-Witten mirror theorem 5364
- 4. Gromov-Witten theory of the quintic threefold 5768
- 5. Proof of the Gromov-Witten mirror theorem 6172
- Acknowledgments 6475
- References 6475

- A brief introduction to cosection localization and 𝑃-fields 6778
- Virtual classes for hypersurfaces via two-periodic complexes 89100
- Localized Chern characters for 2-periodic complexes and virtual cycles 117128
- Singularity theory and mirror symmetry 131142
- Counting points with Berglund–Hübsch–Krawitz mirror symmetry 155166
- Variations on the theme of quantum Lefschetz 171182
- Type II extremal transitions in Gromov-Witten theory 183194
- A lecture on holomorphic anomaly equations and extended holomorphic anomaly equations 191202
- Back Cover Back Cover1218