Stability conditions for Slodowy slices and real variations of stability

TL;DR

This paper introduces the concept of real variations of stability conditions in triangulated categories, explores their relation to Bridgeland stability, and connects these ideas to quantum cohomology of symplectic resolutions, supported by explicit examples.

Contribution

It proposes a new variant of stability conditions called real variations, relates them to Bridgeland's framework, and verifies a conjecture linking these structures to quantum cohomology in specific cases.

Findings

Explicit submanifold in Bridgeland stability space for a local Calabi-Yau

Relation established between real variations and Bridgeland stability conditions

Conjecture connecting stability structures to quantum cohomology verified in some cases

Abstract

We provide examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau, and propose a new variant of definition of stabilities on a triangulated category, which we call a "real variation of stability conditions". We discuss its relation to Bridgeland's definition; the main theorem provides an illustration of such a relation. We also state a conjecture by the second author and Okounkov relating this structure to quantum cohomology of symplectic resolutions and establish its validity in some special cases. 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 of coherent sheaves on X and a collection of t-structures on this category permuted by the action have been constructed in arXiv:1101.3702 and arXiv:1001.2562 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. In the special case when dim (X)=2 a similar (in fact, stronger) result was obtained in arXiv:math/0508257.