Stability Conditions for Slodowy Slices and Real Variations of Stability

Abstract

The paper provides new examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau. More precisely, let X be the standard resolution of a transversal slice to an adjoint nilpotent orbit of a simple Lie algebra over C. An action of the affine braid group on the derived category D[superscript b] (Coh(X)) and a collection of t-structures on this category permuted by the action have been constructed in [BR] and [BM] respectively. In this note we show that the t-structures come from points in a certain connected submanifold in the space of Bridgeland stability conditions. The submanifold is a covering of a submanifold in the dual space to the Grothendieck group, and the affine braid group acts by deck transformations. We also propose a new variant of definition of stabilities on a triangulated category, which we call a ”real variation of stability conditions” and discuss its relation to Bridgeland’s definition. The main theorem provides an illustration of such a relation. We state a conjecture by the second author and A. Okounkov on examples of this structure arising from symplectic resolutions of singularities and its relation to equivariant quantum cohomology. We verify this conjecture in our examples.