Riemann-Hilbert problem for Gromov-Witten invariants

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K05193
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: MILANOV Todor 東京大学, カブリ数物連携宇宙研究機構, 准教授 (80596841)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: period integrals / quantum cohomology / vertex operators / mirror symmetry

Outline of Final Research Achievements

The results of this project are in the settings of the theory of semi-simple Frobenius manifolds. The main examples of such manifolds come from quantum cohomology and singularity theory. Motivated by Kyoji Saito's theory of primitive forms, we have introduced the notion of period vectors for any semi-simple Frobenius manifold. Using the period vectors and following ideas of Givental and Milanov we introduce vertex operators. The main result of this proposal is a connection formula for the operator product expansion (OPE) of the vertex operators, i.e., we found a general rule that allows us to analytically continue the OPE from one singularity to another one. The second main achievement of this proposal is a K-theoretic interpretation of the period integrals for simple singularities corresponding to vanishing cycles. Both results were applied to the problem of constructing integrable hierarchies of differential equations in the form of Hirota quadratic equations.

Free Research Field

complex geometry and differential equations

Academic Significance and Societal Importance of the Research Achievements

The project gives new methods to construct differential equations with possible applications to physics, engineering, and cosmology. Highly specialized results in complex geometry are becomming more accesible to young researchers.