Algebraic Geometry and Physics

This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics: algebraic geometry and quantum physics. I shall take a practical approach, concentrating more on explicit examples rather than formal developments. Topics covered are divided in three sections: (I) Algebraic geometry on two-dimensional exactly solvable statistical lattice models and its related Hamiltonians: I will report results on the algebraic geometry of rapidity curves appeared in the chiral Potts model, and the algebraic Bethe Ansatz equation in connection with quantum inverse scattering method for the related one-dimensional Hamiltonion chain, e.g., XXZ, Hofstadter type Hamiltonian. (II) Infinite symmetry algebras arising from quantum spin chain and conformal field theory: I will explain certain progress made on Onsager algebra, the relation with the superintegrable chiral Potts quantum chain and problems on its spectrum. In conformal field theory, mathematical aspects of characters of N=2 superconformal algebra are discussed, especially on the modular invariant property connected to the theory. (III). Algebraic geometry problems on orbifolds stemming from string theory: I will report recent progress on crepant resolutions of quotient singularity of dimension greater than or equal to three. The direction of present-day research of engaging finite group representations in the geometry of orbifolds is briefly reviewed, and the mathematical aspect of various formulas on the topology of string vacuum will be discussed.