Derived categories of coherent sheaves on hyperkahler manifolds

代数几何，特别是HyperKahler流形的几何，研究一种高度约束的几何类型，它是非常严格的，几乎没有全局对称性。然而，来自物理学中弦理论对偶的神秘预测将这些空间与辛流形联系起来：这些几何要灵活得多，具有许多对称性。(例如，低维拓扑的对称群--辫子群、映射类群--作为许多辛流形的对称群出现。)这是因为，虽然对称不作用于HyperKahler流形本身，但它们作用于它们的D-膜范畴。代数几何与低维拓扑之间的这种联系令人惊讶，并允许人们利用代数几何的复杂而深刻的结构来构造低维拓扑中的不变量--即低维拓扑的代数几何数据编码性质，可用于区分不同的拓扑或形状。最简单的例子是从代数几何中产生纽结不变量，给出可以区分以不同方式嵌入到正常三维空间中的绳结环的数据。

Algebraic geometry, and particularly the geometry of HyperKahler manifolds, studies a highly constrained type of geometry which is very rigid with very few global symmetries. Mysterious predictions coming from dualities in string theory in physics, however, relate these spaces to symplectic manifolds: these are much more flexible geometries with many symmetries. (For instance the symmetry groups of low dimensional topology -- braid groups, mapping class groups -- occur as the symmetry groups of many symplectic manifolds.) The reason is that while the symmetries do not act on the HyperKahler manifolds themselves, they act on their categories of D-branes .This connection between algebraic geometry and low dimensional topology is a surprise and allows one to use the sophisticated and deep structures of algebraic geometry to construct invariants in low dimensional topology -- i.e. algebro-geometric data encoding properties of the low dimensional topology that can be used to distinguish different topologies or shapes . The simplest example is to produce knot invariants from algebraic geometry, giving data which can distinguish between knotted loops of string embedded into normal 3-dimensional space in different ways.