Bridgeland stability condition
In mathematics, and especially algebraic geometry, a Bridgeland stability condition, defined by Tom Bridgeland, is an algebro-geometric stability condition defined on elements of a triangulated category.The case of original interest and particular importance is when this triangulated category is the derived category of coherent sheaves on a Calabi–Yau manifold, and this situation has fundamental links to string theory and the study of D-branes.Such stability conditions were introduced in a rudimentary form by Michael Douglas called-stability and used to study BPS B-branes in string theory.[1] This concept was made precise by Bridgeland, who phrased these stability conditions categorically, and initiated their study mathematically.[2] The definitions in this section are presented as in the original paper of Bridgeland, for arbitrary triangulated categories.is a collection of full additive subcategoriessuch that The last property should be viewed as axiomatically imposing the existence of Harder–Narasimhan filtrations on elements of the categoryA Bridgeland stability condition on a triangulated category, called a central charge, satisfying It is convention to assume the categoryis essentially small, so that the collection of all stability conditions onforms a setIn good circumstances, for example whenis the derived category of coherent sheaves on a complex manifold, this set actually has the structure of a complex manifold itself.It is shown by Bridgeland that the data of a Bridgeland stability condition is equivalent to specifying a bounded t-structureof this t-structure which satisfies the Harder–Narasimhan property above.stable) with respect to the stability condition) exp ( i π φ (Recall the Harder–Narasimhan filtration for a smooth projective curveimplies for any coherent sheafWe can extend this filtration to a bounded complex of sheavesby considering the filtration on the cohomology sheaves, giving a functionThere is an analysis by Bridgeland for the case of Elliptic curves.{\displaystyle {\text{Stab}}(X)/{\text{Aut}}(X)\cong {\text{GL}}^{+}(2,\mathbb {R} )/{\text{SL}}(2,\mathbb {Z} )}is the set of stability conditions andis the set of autoequivalences of the derived category