Bridgeland stability condition
In mathematics, and especially algebraic geometry, a Bridgeland stability condition is an algebro-geometric stability condition defined on elements of a triangulated category. Stability conditions serve a two-fold purpose in the theory - first, the space of all stability conditions on the triangulated category carries the structure of a complex manifold, thus furnishing an invariant of the category that is topological in nature. Second, each stability condition allows the construction of well-behaved moduli spaces parametrising objects in the category that are semistable with respect to it.
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. This concept was made precise by Tom Bridgeland, who defined stability conditions and initiated their study mathematically, and after whom the concept is named.
(Bridgeland) stability conditions remain an active area of research. The moduli spaces they furnish have yielded new constructions of hyperKähler varieties, and the stability manifold (i.e. the complex manifold formed by all stability conditions) has been used to study the autoequivalence groups of many triangulated categories. Wall-crossing, i.e. the analysis of how moduli spaces of semistable objects change as the stability condition varies in the stability manifold, has been employed to solve problems in enumerative geometry and Brill-Noether theory.