研究実績の概要 |
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.
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今後の研究の推進方策 |
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.
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