Gauge gravity duality and new geometrical structures

该项目将侧重于测量重力对应对几何学的影响。特别是，它暗示了代数结构（如Calabi-Yau代数）与新的微分几何结构（希钦广义几何概念的扩展）之间的关系。一个具体的猜想是代数的循环上同调等于流形上Dolbeault上同调的新推广。这将为从希尔伯特系列颤振理论，到拓扑弦，到黑洞状态计数，到新的超卡勒商结构等主题提供新的见解。学生将从学习物理对应双方的一些基础知识和基础数学开始，然后专注于一些新的例子：这些包括Ypq几何的一般类的边缘变形，Plich-Warner几何和Pilch-Warner和T11之间的流动，特别是几何对对缠绕的m5膜理论。

The project will focus on the implications of the gauge gravity correspondence for geometry. In particular, it implies a relationship between algebraic structures (such as Calabi-Yau algebras) on the one hand and new differential geometrical structures (extensions of Hitchin's notion of generalised geometry) on the other. One concrete conjecture is that the cyclic cohomology of the algebra is equal to a new generalisation of Dolbeault cohomololgy on the manifold. This will give new insight into topics ranging from Hilbert series of quiver theories, to topological strings, to black hole state counting, to new hyper-Kahler quotient constructions. The student will start by learning some of the basics of both sides of the physical correspondence, and the underlying mathematics, then focus on a number of new examples: these include marginal deformations of the general class of Ypq geometries, the Plich-Warner geometries and flows between Pilch-Warner and T11, and in particular, geometries dual to wrapped M5-brane theories.