对四维N=2超对称理论BPS谱以及Wall Crossing现象研究，是当前关于量子场论、弦理论以及相关数学理论的研究中一个非常重要而活跃的交叉领域。申请人将利用toric几何，规范线性sigma模型，超对称场论局域化技术等方法，着重研究一般BPS quiver的内禀Higgs态，包括内禀Higgs指标的间接和直接计算，内禀Higgs指标的Wall Crossing不变性的证明，内禀Higgs指标在Kontsevich-Soibelman代数形式中扮演的角色，以及在N=2超引力框架下内禀Higgs指标与BPS黑洞熵的关系等课题。由此，进一步加深我们对Wall Crossing现象以及N=2超对称理论中的物理性质和数学结构的理解。

BPS spectrum of 4 dimensional N=2 supersymmetric theory and wall crossing phenomena are very important and active interdisciplinary topics in recent studies of quantum field theory, string theory and the corresponding mathematics. By using toric geometry, gauged linear sigma model and localization technique of supersymmetric field theory, we shall study the intrinsic Higgs states in general BPS quiver gauge theories, including the indirect and direct computations of intrinsic Higgs index, the proof of its wall crossing invariance, its role in Kontsevich-Soibelman algebraic formula as well as the relation between intrinsic Higgs index and BPS black hole entropy in N=2 supergravity. The studies on these topics will enhance our understandings of wall crossing phenomena as well as the physical properties and mathematical structures in N=2 supersymmetric theories.