Symplectic geometry of unbiasedness and critical points of a potential. (English) Zbl 1453.53074

Bondal, Alexey; Zhdanovskiy, Ilya

Hori, Kentaro (ed.) et al., Primitive forms and related subjects – Kavli IPMU 2014. Proceedings of the international conference, University of Tokyo, Tokyo, Japan, February 10–14, 2014. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 83, 1-18 (2019).

Summary: The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential function, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of doubly stochastic matrices. Mirror symmetry interprets the polynomial as a Landau-Ginzburg potential for corresponding Fano variety and relates the symplectic geometry of the variety with systems of unbiased projectors.
For the entire collection see [Zbl 1446.53004].

MSC:

53D12	Lagrangian submanifolds; Maslov index
53D20	Momentum maps; symplectic reduction
53D37	Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14J33	Mirror symmetry (algebro-geometric aspects)
14J45	Fano varieties
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
81P45	Quantum information, communication, networks (quantum-theoretic aspects)
35Q56	Ginzburg-Landau equations

Keywords:

mutually unbiased bases; Lagrangian submanifold; momentum map; symplectic reduction; mirror symmetry; Fano variety; toric varieties; Newton polyhedra; quantum information; Landau-Ginzburg potential

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