Phases of Lagrangian-invariant objects in the derived category of an abelian variety

Abstract

In this paper we study Lagrangian-invariant objects (LI objects for short) in the derived category $D^b\left(A\right)$ of coherent sheaves on an abelian variety. For every element of the complexified ample cone $D_A$ we construct a natural phase function on the set of LI objects, which in the case $dimA=2$ gives the phases with respect to the corresponding Bridgeland stability. The construction is based on the relation between endofunctors of $D^b\left(A\right)$ and a certain natural central extension of groups, associated with $D_A$ viewed as a Hermitian symmetric space. In the case when $A$ is a power of an elliptic curve, we show that our phase function has a natural interpretation in terms of the Fukaya category of the mirror dual abelian variety. As a by-product of our study of LI objects we show that the Bridgeland component of the stability space of an abelian surface contains all full stabilities.