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What are the stability conditions in the sense of Kontsevich-Soibelman.

I am reading Bridgeland's stability conditions and I've heard people talking about the Kontsevich-Soibelman Stability. I would appreciate a brief introduction on this, in particular my questions are :

  1. What are the Kontsevich-Soibelman Stability conditions ?

  2. How is it related to Bridgeland's Stability (or Douglas' $\pi$ - stability on D-branes) ?

  3. Why do we need to consider Kontsevich-Soibelman stability.

I've to admit my ignorance of the field. Please suggest some references. Thanks.

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Kontsevich-Soibelman's version is a version of Bridgeland's stability given for triangulated $A_\infty$ categories (with a few additional properties; Kontsevich and Soibelman call "non-commutative proper algebraic variety" such an $A_\infty$-category) rather than for triangulated categories as in the original Bridgeland's version.

The main point in Kontsevich-Soibelman definition is that once one thinks of the relevant $A_\infty$ categories as non-commutative analogues of algebraic varieties, one sees that a Bridgeland stability condition can be seen as the datum of a polarization on these non-commutative varieties. The reason for considering $A_\infty$ rather than ordinary categories is that categorical structures arising from branes are naturally $A_\infty$ categories (the most classical example to be done here is probably Fukaya's $A_\infty$ category of a symplectic manifold $X$).

The basic reference for Kontsevich-Soibelman's stability is obviously their paper "Stability structures, motivic Donaldson-Thomas invariants and cluster transformations", arXiv:0811.2435

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I haven't looked closely at the KS paper, but is the issue really $A_\infty$ vs triangulated? i.e. I would imagine a stability condition (like a t-structure) is really a structure on the underlying (triangulated) homotopy category in any case. KS do talk about stability structures for Lie algebras (abstracting Hall algebras attached to categories) and add a support condition.. –  David Ben-Zvi Dec 27 '10 at 22:01
    
@ Domenico - thanks for your response. What I understood from this is that : KS stability is the Bridgeland type stability defined on $A_{\infty}$ categories rather than triangulated categories. Is KS version a generalization of Bridgeland's in a sense that for triangulated categories KS = Bridgeland. –  J Verma Dec 27 '10 at 22:16
    
also KS categories are triangulated: they are $A_\infty$ triangulated categories. for the particular case of triangulated categories KS and Bridgeland definition basically coincide (there could be a few minor differences; e.g., as David points out, KS introduce a support condition) –  domenico fiorenza Dec 27 '10 at 22:45
    
moreover (again a David's comment) KS consider a decategorified version of the stability conditions: stability conditions on graded Lie algebras. This is not only a toy model: since wall corossing formulae basically "count" the "number" of nonisomorphic objects of some kind, thay have to hold at teh decategorified level. And indee a major point in KS is showing that stability condiions on graded Lie algebras are the natural setting for wall-crossing formulae. –  domenico fiorenza Dec 27 '10 at 22:48
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