2025/2026
Moduli Spaces, Dubrovin-Frobenius Manifolds and Topological Recursion
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of all HSE University campuses
Language:
English
ECTS credits:
6
Contact hours:
60
Course Syllabus
Abstract
"Cohomological field theories were defined in the mid-90s by Kontsevich and Manin to describe the formal properties of the virtual fundamental class in Gromov-Witten theory. The Givental-Teleman classification of cohomological field theories states that any semisimple CohFT with a unit is uniquely determined by its genus 0, descendent-free part (which corresponds to a semisimple Dubrovin-Frobenius manifold) through the so-called Givental R-matrix.
The Chekhov-Eynard-Orantin topological recursion is a universal recursion that arises in various enumerative problems in combinatorics, algebraic geometry, and mathematical physics.
The course will focus on identifying the Givental-Teleman construction with topological recursion, which, in particular, provides an algebro-geometric interpretation of many enumerative combinatorics problems."
Learning Objectives
- Cohomological field theories were defined in the mid-90s by Kontsevich and Manin to describe the formal properties of the virtual fundamental class in Gromov – Witten theory. The Givental – Teleman classi- fication of cohomological field theories states that any semisimple CohFT with a unit is uniquely determined by its genus 0, descendent-free part (which corresponds to a semisimple Dubrovin – Frobenius manifold) through the so-called Givental R-matrix. The Chekhov – Eynard – Orantin topological recursion is a universal recursion that arises in various enumerative problems in combinatorics, algebraic geometry, and mathematical physics. The course focuses on identifying the Givental – Teleman construction with topological recursion, which, in particular, provides an algebro-geometric interpretation of many enumerative combinatorics problems.
Expected Learning Outcomes
- The student will study the basic concepts of the intersection theory on moduli spaces of algebraic curves
- The students will study the basic concepts of Cohomological Field Theories
- The students will study the basic concepts of Frobenius manifolds.
- The students will study the basic concepts of theory of Chekhov-Eynard-Orantin topological recursion
- The students will study the basic concepts of the Givental-Teleman classification of the semisimple Frobenious manifold
- The students will study the basic concepts of the theory of Hurwitz numbers
- The students will study the basic concepts of theory of the theory of Hurwitz numbers
Course Contents
- Integration over the moduli space of algebraic curves
- Cohomological field theories
- Dubrovin – Frobenius manifolds
- Topological recursion
- Identification of CohFT and TR
- Hurwitz numbers
- The ELSV formula
Interim Assessment
- 2025/2026 2nd module0.4𝐸 + 0.2(𝐻𝑊1 + 𝐻𝑊2 + 𝐻𝑊3), where E is the grade for the exam, and HW1,2,3 are the homework grades.
Bibliography
Recommended Core Bibliography
- Yuri I. Manin. (1999). Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces. AMS.
- Казарян, М. Э. Алгебраические кривые. По направлению к пространствам модулей : учебное пособие / М. Э. Казарян, С. К. Ландо, В. В. Прасолов. — Москва : МЦНМО, 2019. — 272 с. — ISBN 978-5-4439-3353-5. — Текст : электронный // Лань : электронно-библиотечная система. — URL: https://e.lanbook.com/book/267665 (дата обращения: 00.00.0000). — Режим доступа: для авториз. пользователей.
- Натанзон, С. М. Модули римановых поверхностей, вещественных алгебраических кривых и их супераналоги : сборник научных трудов / С. М. Натанзон. — Москва : МЦНМО, 2021. — 175 с. — ISBN 978-5-4439-2185-3. — Текст : электронный // Лань : электронно-библиотечная система. — URL: https://e.lanbook.com/book/267506 (дата обращения: 00.00.0000). — Режим доступа: для авториз. пользователей.
Recommended Additional Bibliography
- 267410 - ЛАНЬ - Введение в пучки, расслоения и классы Черна - Московский центр непрерывного математического образования - Натанзон С. М. - 2014 - 978-5-4439-2029-0 - https://e.lanbook.com/book/267410
- Kazarian, M., & Lando, S. (2015). Combinatorial solutions to integrable hierarchies. https://doi.org/10.1070/RM2015v070n03ABEH004952