stability conditions in the sense of Kontsevich-Soibelman 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 :


*

*What are the Kontsevich-Soibelman Stability conditions ?

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

*Why do we need to consider Kontsevich-Soibelman stability.
I've to admit my ignorance of the field. Please suggest some references.
Thanks. 
 A: 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
