D-BRANE MODULI SPACES IN MATHEMATICS AND PHYSICS

采用d膜模空间作为一个统一的主题，本研究计划包括三个项目，位于代数几何和弦理论之间的接口。第一个是围绕一个新的数学结构（ADHM束不变量），产生了关于局部BPS不变量、颤振变体的量子上同调和黑洞对应的新的数学猜想。第二个项目研究了局部表面上bridgeeland稳定物体模空间的相机结构和壁穿越行为，强调了Donaldson-Thomas型不变量与黑洞物理之间的联系。第三章提出了一种系统的方法，通过对抛物希格斯束的虚拟循环和局部化来研究拓扑规范理论中表面算子的相关子。更广泛的影响：数学和弦理论之间的相互作用一直是现代数学物理的主要推动力。这里的研究计划旨在揭示这种相互作用的新方面，为Donaldson-Thomas理论、颤振变体的量子上同调以及理论物理的基础领域（如黑洞物理和弦对偶性）开辟新的研究方向。其目的是通过揭示数学和物理之间的新联系来刺激几个领域的发展。在教育方面，本研究计划为各级学生和博士后提供多种培训机会，促进各种技术和沟通技能的发展，这些技能在学术环境之外具有广泛的适用性。

Employing D-brane moduli spaces as a unifying theme, this research program consists of three projects lying at the interface between algebraic geometry and string theory. The first is centered around a new mathematical construction (ADHM sheaf invariants) yielding new mathematical conjectures concerning local BPS invariants, quantum cohomologyof quiver varieties and black hole correspondence. The second project investigates the cameral structure and wallcrossing behavior of moduli spaces of Bridgeland stable objects on local surfaces emphasizing a connection between Donaldson-Thomas type invariants and black hole physics. The third proposes a systematic approach to correlators of surface operators in topological gauge theories via virtual cycles and localization for parabolic Higgs bundles. Broader Impact: The interaction between mathematics and string theory has been a major driving force in modern mathematical physics. The research planned here aims at revealing new aspects of this interaction opening new directions of research in Donaldson-Thomas theory, quantum cohomology of quiver varieties as well as fundamental areas of theoretical physics such as black hole physics and string duality. This aims to stimulate the development of several areas of mathematics and physics by unraveling new connections between them. With respect to education, this research program offers multiple training opportunities for students of all levels as well as postdoctoral fellows, stimulating the development of a variety of technical and communication skills with a wide range of applicability beyond the academic environment.