stability conditions in the sense of Kontsevich-Soibelman - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:39:29Z http://mathoverflow.net/feeds/question/50513 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50513/stability-conditions-in-the-sense-of-kontsevich-soibelman stability conditions in the sense of Kontsevich-Soibelman J Verma 2010-12-27T21:16:05Z 2010-12-27T21:49:41Z <p>What are the stability conditions in the sense of Kontsevich-Soibelman. </p> <p>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 :</p> <ol> <li><p>What are the Kontsevich-Soibelman Stability conditions ?</p></li> <li><p>How is it related to Bridgeland's Stability (or Douglas' $\pi$ - stability on D-branes) ?</p></li> <li><p>Why do we need to consider Kontsevich-Soibelman stability.</p></li> </ol> <p>I've to admit my ignorance of the field. Please suggest some references. Thanks. </p> http://mathoverflow.net/questions/50513/stability-conditions-in-the-sense-of-kontsevich-soibelman/50515#50515 Answer by domenico fiorenza for stability conditions in the sense of Kontsevich-Soibelman domenico fiorenza 2010-12-27T21:49:41Z 2010-12-27T21:49:41Z <p>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.</p> <p>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 $X$).</p> <p>The basic reference for Kontsevich-Soibelman's stability is obviously their paper "Stability structures, motivic Donaldson-Thomas invariants and cluster transformations", arXiv:0811.2435</p>