Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 029, 60 pages      arXiv:2304.04365      https://doi.org/10.3842/SIGMA.2024.029

Reflection Vectors and Quantum Cohomology of Blowups

Todor Milanov and Xiaokun Xia
Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

Received May 30, 2023, in final form March 14, 2024; Published online April 05, 2024

Abstract
Let $X$ be a smooth projective variety with a semisimple quantum cohomology. It is known that the blowup $\operatorname{Bl}_{\rm pt}(X)$ of $X$ at one point also has semisimple quantum cohomology. In particular, the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ is a reflectiongroup. We found explicit formulas for certain generators of the monodromy group of the quantum cohomology of $\operatorname{Bl}_{\rm pt}(X)$ depending only on the geometry of the exceptional divisor.

Key words: Frobenius structures; Gromov-Witten invariants; quantum cohomology.

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