Foundations of F-Theory

弦理论为粒子物理的理论模型提供了一致的高能外推，并自然地将这些模型与引力统一起来，并为数学物理提供了许多应用于理论物理的工具。一种有前途的弦理论模型被称为f理论，近年来引起了人们的兴趣，它指出了一些基础研究的必要性。这些研究涉及数学的一个复杂分支——代数几何，它为构建物理模型提供了背景。PI将确定f理论模型的规范群和物质含量的限制，包括Calabi-Yau三倍和Calabi-Yau四倍；是否每一个椭圆纤维的Calabi-Yau四倍体都属于等维族，如果不是，大维纤维的物理解释是什么；椭圆光纤Calabi-Yau流形的基底结构；Calabi-Yau流形的f理论解释，该流形具有非椭圆的属1纤化（即没有截面）；椭圆纤维Calabi-Yau四重结构的总空间的双空间几何如何与四重结构上的通量选择相互作用，特别是f理论真空在多大程度上必然被通量提升；t膜理论的m理论对偶解释；f理论与非几何异质性弦的对偶性是否可以推广；以及其他可能与f理论相关的Calabi-Yau几何主题，包括对最近发现的进一步探索，即描述Calabi-Yau流形的测量线性sigma模型通过S2上的配分函数包含了关于Calabi-Yau流形的丰富信息。更广泛的影响该项目还将通过PI在本科生、研究生和博士后层面的教育活动，以及PI通过尽可能多地提供电子信息来简化科学交流的持续努力，产生更广泛的影响。

String theory provides consistent high-energy extrapolations of theoretical models of particle physics, as well as naturally unifying those models with gravity, and provides tools of mathematical physics with a number of applications in theoretical physics. One promising type of string theory model known as F-theory has seen a resurgence of interest in recent years which points out the need for some basic foundational studies. These studies involve a sophisticated branch of mathematics, algebraic geometry, which provides the background for constructing the physical models. The PI will determine what are the restrictions for gauge groups and matter content for F-theory models, both on Calabi-Yau threefolds and Calabi-Yau fourfolds; whether every elliptically-fibered Calabi-Yau fourfold is birational to an equidimensional family, and if not, what is the physical interpretation of the fibers of large dimension; the structure of the base of an elliptically fibered Calabi-Yau manifold; the F-theory interpretation of Calabi-Yau manifolds which have a genus one fibration that is not elliptic (i.e., there is no section); how the birational geometry of the total space of an elliptically fibered Calabi-Yau fourfold interacts with the choices of flux on that fourfold, and in particular, to what extent F-theory vacua are necessarily lifted by fluxes; the M-theory dual interpretation of the theory of T-branes; can the duality between F-theory and nongeometric heterotic strings be extended; and other topics in Calabi-Yau geometry which may be relevant to F-theory, including further exploration of the recent discovery that the gauged linear sigma models which describe Calabi-Yau manifolds contain via a partition function on S2 a wealth of information about the Calabi-Yau manifold. Broader Impact The project will also have a broader impact through educational activities of the PI, at the undergraduate, graduate and postdoctoral levels, as well as through the PI's ongoing efforts to streamline scientific communication by making as much of it available electronically as possible.