Algebraic Geometry Seminar
Quantum $K$-Theory and $q$-Difference Toda Hamiltonians
2026/09/09 (Wed.) 15:00~16:30
- Date : 2026/09/09 (Wed.) 15:00~16:30
- Location : Seminar Room 638, Institute of Mathematics (NTU Campus)
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Speaker : Takeshi Ikeda (Waseda University)
- Organizer : Adeel Khan and Y.P. Lee (AS)
- Abstract : Let $G$ be a semisimple algebraic group and $G/B$ its flag variety. For $G=SL_n$, Givental and Lee showed that the quantum $K$-theoretic $J$-function is an eigenfunction of a $q$-difference Toda Hamiltonian. They also proposed a construction of commuting Hamiltonians for general $G$. It has since been pointed out that this construction requires modification in non-simply-laced types.
The general problem is to embed the Weyl-invariant representation ring into an algebra of $q$-difference operators so that the $J$-function is a joint eigenfunction, and to determine the resulting Hamiltonians explicitly. In this talk, I will discuss the current status of this problem and present an explicit formula for the Hamiltonian associated with a minuscule fundamental weight. This is joint work with Koushik Brahma, Yicen Huang, Takafumi Kouno, and Kohei Yamaguchi.
