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

SIGMA 21 (2025), 061, 32 pages      arXiv:2401.00500      https://doi.org/10.3842/SIGMA.2025.061

Deformation Quantization with Separation of Variables of $G_{2,4}(\mathbb{C})$

Taika Okuda a and Akifumi Sako b
a) Graduate School of Science, Department of Mathematics and Science Education, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
b) Faculty of Science Division II, Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received December 10, 2024, in final form July 14, 2025; Published online July 23, 2025

Abstract
We construct a deformation quantization with separation of variables of the Grassmannian $G_{2,4}(\mathbb{C})$. A star product on $G_{2,4}(\mathbb{C})$ can be explicitly determined as the solution of the recurrence relations for $G_{2,4}(\mathbb{C})$ given by Hara and one of the authors (A. Sako). To provide the solution to the recurrence relations, it is necessary to solve a system of linear equations in each order. However, to give a concrete expression of the general term is not simple because the variables increase with the order of the differentiation of the star product. For this reason, there has been no formula to express the general term of the recurrence relations. In this paper, we overcome this problem by transforming the recurrence relations into simpler ones. We solve the recurrence relations using creation and annihilation operators on a Fock space. From this solution, we obtain an explicit formula of a star product with separation of variables on $G_{2,4}(\mathbb{C})$.

Key words: noncommutative differential geometry; deformation quantization; complex Grassmannians; Kähler manifolds; locally symmetric spaces.

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