Studying Special Holonomy Manifolds Using Conformal Field Theory Tools

该奖项资助了奥尔巴尼大学丹尼尔·罗宾斯教授的研究活动。共形场论（CFT’s）在物理学中被用来描述相变（比如冰融化成水），或者用来理解量子理论在非常低或非常高的能量下会发生什么。CFT在弦理论中也发挥着核心作用，弦理论是统一所有自然基本力的量子引力理论的主要候选理论。在不同的几何空间中运动的弦用不同的CFT来描述，从而导致不同的物理现象和性质。这个研究项目将探索CFT、几何和由此产生的物理定律之间的联系。这项工作通过促进物理学和数学之间跨学科边界的基础科学的进步来促进国家利益，从而加强美国的基础科学。研究生和本科生也将参与该项目，接受罗宾斯教授的研究培训，并有机会向其他研究人员展示他们的研究成果。在更技术性的层面上，本项目将把最近开发的CFT技术应用于一类由特殊完整流形构造的超共形场论（SCFT）。共形自举程序和模自举程序的哲学和工具都利用CFT一致性条件来推导CFT定义数据上的约束，将用于寻找在弦理论中起重要作用的一类流形几何数据上的约束，并探索它们的模空间和相应弦理论结构的物理性质。该项目还将把CFT缺陷研究的最新思想和工具应用于同一类SCFT，研究由畴壁连接的理论之间的联系，边界SCFT，相关弦理论中的d膜，以及特殊完整流形的数学不变量的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award funds the research activities of Professor Daniel Robbins at the University at Albany. Conformal field theories (CFT's) are used in physics to describe phase transitions (such as ice melting into water), or to understand what happens in a quantum theory at very low or very high energies. CFT's also play a central role in string theory, which is the leading candidate for a quantum theory of gravity that unifies all of the fundamental forces of nature. Strings moving in different geometrical spaces are described by different CFT's that in turn lead to different physical phenomena and properties. This research program will explore the connections between CFT's, geometry, and the resulting physical laws. This work advances the national interest by fostering the progress of fundamental science at the interdisciplinary boundary between physics and mathematics, and thereby strengthening fundamental science within the United States. Students, both graduate and undergraduate, will also be involved in this project, receiving research training from Professor Robbins and receiving opportunities to present their work to other researchers.On a more technical level, this project will apply recently developed CFT techniques to a class of superconformal field theories (SCFT's) constructed from special holonomy manifolds. The philosophy and tools of the conformal bootstrap program and the modular bootstrap program, both of which exploit CFT consistency conditions in order to derive constraints on the defining data of a CFT, will be used to find constraints on the geometrical data of a class of manifolds that play an important role in string theory, and to explore aspects of their moduli spaces and the physical properties of the corresponding string theory constructions. The project will also apply recent ideas and tools from the study of CFT defects to the same class of SCFT's, investigating connections between theories connected by domain walls, boundary SCFT's, D-branes in the associated string theories, and the study of mathematical invariants of special holonomy manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.