2025/2026

Moduli Spaces, Dubrovin-Frobenius Manifolds and Topological Recursion

Category 'Best Course for Career Development'

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

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

Instructors: Boris Bychkov, Petr Dunin-Barkowski

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

Assessment Elements

• Homework
• In-class assignment

Interim Assessment

• 2025/2026 2nd module
  0.4𝐸 + 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

Authors

• Dunin-Barkovskii PETR IGOREVICH
• BYCHKOV BORIS SERGEEVICH