The Period Map of a Two-Dimensional Semi-Simple Frobenius Manifold

2021 Fiscal Year Annual Research Report

• Project/Area Number: 20J20053
• Research Institution: The University of Tokyo
• Principal Investigator: ZHA Chenghan 東京大学, 数理科学研究科, 特別研究員(DC1)
• Project Period (FY): 2020-04-24 – 2023-03-31
• Keywords: singularity theory / invertible polynomial / relative K-theory / matrix factorizations / integrable hierarchies

Outline of Annual Research Achievements

The research is related to singularity theory, mirror symmetry, equivariant K-theory, integrable system and so on. In singularity theory, primitive form was invented by K. Saito and one can get Frobenius manifold from it.

Givental-Milanov used such integrals to construct integrable hierarchies for the applications to enumerative geometry but they did not find explicit formulas for the integrals. Alexandrov-Milanov needed explicit formulas in order to construct a matrix model for D type singularity. For ADE singularity, Fan-Jarvis-Ruan proved that the generating function of Fan-Jarvis-Ruan-Witten (FJRW) invariants of can be identified with a tau-function of a specific Kac-Wakimoto hierarchy.

The other goal is to figure out equivariant relative K-theory and furthermore to show the mirror isomorphism between them. K-theoretic side is motivated by the work of Iritani, Chiodo-Iritani-Ruan, and Chiodo-Nagel. For an invertible polynomial, a conjecture by Ebeling-Takahashi motivates a series of works. There is an equivalence of triangulated categories between triangulated category of graded matrix factorizations and derived directed Fukaya category. Futaki-Ueda proved homological mirror symmetry for Lefschetz fibrations obtained as Sebastiani-Thom sums of polynomials of types A or D. It would be interesting to compare the homological mirror map with the map mirror.

Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

In the case of invertible polynomial of chain type, Hirano-Ouchi first proved the existence of a full exceptional collection and they further explicitly constructed a full strong exceptional collection. Aramaki-Takahashi also constructed a full exceptional collection and computed the corresponding Euler matrix. Their construction helped us to find the basis for equivariant relative topological K-theory.

Strategy for Future Research Activity

We would like to generalize our result to the case of invertible polynomials, specifically, invertible polynomials of chain type. One can use the result by Otani-Takahashi. In their work, they constructed a basis for general chain type invertible polynomials inductively. While for the other side, i.e., equivariant relative topological K-theory interpretation of the mirror of the singularity, we expect that the technique of spectral sequence is needed.