Mathematical Problems Inspired by String Theory

研究IIA和IIB超共形场论的Calabi-Yau品种，试图给出一个严格的数学定义的顶点代数的理论。这个代数被认为是由Malikov，Schechtman和Vaintrob构造的手征de Rham复形的上同调的变形。该项目的第二个目标是将IIA和IIB模型的定义扩展到一些单一的品种。最迫切的问题是连接两个已有的奇异簇的椭圆亏格的定义，它是上同调McKay对应的推广。该项目的第三部分集中在从数论中出现的问题，与椭圆亏格的概念有关，并与爱森斯坦级数的产品有关。各种口味的弦理论是数学物理中“万物理论”的主要候选者。它的发展导致了快速增长的几个领域的数学，特别是代数几何，其中关注本身的空间的解决方案的多项式方程。不幸的是，仍然缺乏对一些潜在的丰富数学结构的严格理解，该项目旨在纠正这一点。这样的数学理解是重要的，因为它可能是弦理论未来发展的必要条件，而弦理论反过来又可能产生一个关于我们宇宙的基本过程和力量的更连贯的图像。该项目的一个单独部分涉及某些受弦理论启发的数论问题。这些都与椭圆曲线理论中长期存在的几何问题有关。椭圆曲线理论是费马大定理证明的关键部分，并在密码学中有实际应用。 该项目由代数，数论和组合程序以及拓扑和几何分析程序共同资助。

The investigator studies IIA and IIB superconformal field theories on a Calabi-Yau variety in an attempt to give a rigorous mathematical definition of the vertex algebra of the theories. This algebra is expected to be a deformation of the cohomology of the chiral de Rham complex constructed by Malikov, Schechtman and Vaintrob. The second aim of the project is to extend the definition of IIA and IIB models to some singular varieties. The most pressing question is to connect two existing definitions of elliptic genus of singular varieties, which is a generalization of cohomological McKay correspondence. Thethird part of the project focuses on problems from number theory thatarise in connection with the concept of elliptic genus and are relatedto products of Eisenstein series.String theory in its various flavors is a leading candidate for the``theory of everything'' in mathematical physics. Its development has led to rapid growth in several areas of mathematics, in particularalgebraic geometry, which concerns itself with spaces of solutions of polynomial equations. Unfortunately, there is still a lack of rigorous understanding of some of the underlying rich mathematical structures,which the project aims to rectify. Such mathematical understanding isimportant, because it may be a necessity for future development of string theory which in turn may yield a more coherent picture of thebasic processes and forces of our universe. A separate part of the project deals with certain number-theoretical questions inspired by string theory. These are related to long-standing mathematicalproblems in the theory of elliptic curves. The theory of ellipticcurves is a crucial part of the proof of Fermat's Last Theorem and has practical applications to cryptography. This project is jointly funded by the Algebra, Number Theory, and Combinatoric Program and the Topology and Geometric Analysis Program.