[Paper Review] A thin stringy moduli space for Slodowy slices

Rina Anno, Roman Bezrukavnikov+1 more|arXiv (Cornell University)|Aug 7, 2011

Homotopy and Cohomology in Algebraic Topology5 citations

TL;DR

This paper introduces a thin stringy moduli space within the Bridgeland stability conditions space for Slodowy slices—local Calabi-Yau threefolds arising from nilpotent orbit transversals in simple Lie algebras. It establishes that t-structures on the derived category D^b(Coh(X)) arise from points in a connected submanifold, which is a covering of a dual space to the Grothendieck group, acted upon by the affine braid group as deck transformations.

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 We discuss its relation to Bridgeland's definition; the main theorem provides an illustration of such a relation. 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^b(Coh(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. The dimension of our subset equals (in most cases) that of the second cohomology of X, so it may deserve the name of stringy moduli space; it is in a sense smaller than one may want, hence the attribute thin.

Motivation & Objective

• To define a new variant of stability conditions on triangulated categories, termed 'real variation of stability,' as a refinement of Bridgeland's framework.
• To identify a connected submanifold within the Bridgeland stability space that parametrizes t-structures on the derived category of a Slodowy slice.
• To demonstrate that this submanifold is a covering space of a dual to the Grothendieck group, with the affine braid group acting as deck transformations.
• To generalize earlier results in the dim(X)=2 case to higher-dimensional Slodowy slices, extending the geometric stability framework.

Proposed method

• Construct a submanifold inside the space of Bridgeland stability conditions using the geometry of Slodowy slices X, which are resolutions of transversal slices to nilpotent orbits.
• Leverage the known action of the affine braid group on D^b(Coh(X)) to analyze its action on t-structures and stability conditions.
• Show that the t-structures on D^b(Coh(X)) are parametrized by points in a connected submanifold of the stability space.
• Prove that this submanifold is a covering space of a submanifold in the dual of the Grothendieck group K0(X)∨.
• Use the affine braid group action to realize the monodromy of the covering, identifying it with deck transformations.
• Establish that the dimension of the submanifold matches the second Betti number b2(X), suggesting its role as a 'stringy moduli space'.

Experimental results

Research questions

• RQ1 How can a real variation of stability conditions be defined on a triangulated category, and how does it relate to Bridgeland stability?
• RQ2 What submanifold of the Bridgeland stability space parametrizes the t-structures on D^b(Coh(X)) for a Slodowy slice X?
• RQ3 How does the affine braid group act on the derived category and on the stability space, and what is its monodromy?
• RQ4 Why is the submanifold of stability conditions considered 'thin,' and what does this imply about its geometric and categorical significance?
• RQ5 In what way does the dimension of the submanifold relate to the cohomological invariants of X, particularly b2(X)?

Key findings

• The t-structures on D^b(Coh(X)) are realized as points in a connected submanifold of the Bridgeland stability space.
• This submanifold is a covering space of a submanifold in the dual of the Grothendieck group K0(X)∨.
• The affine braid group acts on the derived category and induces deck transformations on the covering, linking algebraic and geometric structures.
• The dimension of the submanifold equals the second Betti number b2(X), supporting its interpretation as a 'stringy moduli space'.
• The construction generalizes previous results in the dim(X)=2 case, extending them to higher-dimensional Slodowy slices.
• The submanifold is strictly smaller than expected, justifying the descriptor 'thin' in the title.