Abstract

Anything about mirror symmetry refers to some mysterious relationships between complex and symplectic geometry. One form of it says that if you have some complex geometry X, there is a mirror symplectic geometry Y such that (in some sense) the derived category of coherent sheaves on X, denoted D^b(Coh X), is equivalent to the Fukaya category of Y, denoted Fuk(Y).

In fact, starting from a complex geometry (think an algebraic variety) there isn’t just one mirror. There’s a family of mirrors living over a parameter space, which is sometimes referred to as the Stringy Kähler Moduli Space (SKMS). The fundamental group of the SKMS acts naturally on Fuk(Y) via monodromy, and by mirror symmetry, we expect to see this action carry over. My goal is to explain some details of this story in the context of Calabi Yau toric geometric invariant theory, where it’s conjectured that the fundamental group acts on the derived category via spherical twists. We’ll start by introducing the derived category, geometric invariant theory, and see where we get to!