Characterization of weight matrices that induce torus actions of different properties

Abstract

Symplectic manifolds arise as geometrical representations of classical mechanical systems. We focus on the study of symplectic quotients, which are quotients of symplectic manifolds by symmetries of the system. The simplest quotients are by finite groups of symmetries and are known as orbifolds. However, it has been observed that more complicated symplectic quotients can sometimes but not always be identified with orbifolds. In fact, previous work by HerbigSchwarz-Seaton has shown that if the group action of a torus on a complex space has certain properties, called 2-principal and stable, then there does not exist a symplectomorphism between the symplectic quotient and a linear symplectic orbifold. In another word, properties of the torus actions have implications on the connection between the symplectic quotients and orbifolds. My research focuses on characterizing the weight matrices that induce k-principal and stable torus actions. I will present progress towards determining such a characterization of the weight matrix.