Title: Non-Hermitian Topology with Generalized Chiral Symmetry

Abstract: We study a generalization of chiral symmetry applicable to non-Hermitian systems and its topological consequences on one-dimensional chains. We uncover a rich family of topological phases hosting several chiral flavors characterized not by a single winding number, but a vector of of them. This, in turn, leads to a novel type of bulk-boundary correspondence, where -- in contrast with conventional chiral chains -- some flavors can have topologically stable non-zero charges on both ends. Moreover, we find that the total charge of each flavor can in some cases exceed the magnitude of the highest winding number in the vector invariant. Our work extends the topological classification of the non-Hermitian AIII class along a new axis.