Representation Theory and Quantum Field Theory

这个建议是围绕着Malikov，Scheck htman，Vaintrob和GorBounov在以前的工作中发现和研究的几个顶点代数层的例子来提出的。更具体地说，该提案的第一部分涉及到计算Calabi-Yau流形上这些轮子的全局截面的代数。在环面和射影空间的情况下出现的已知例子是有希望的。第二部分是建立在最近的发现基础上的，即在射影直线的情况下产生的一族这样的顶点代数与超代数Lie sl(2|1)和sl(2|1)-Toda场论密切相关。第三个是试图将这种层与椭圆格、量子上同调和模形式之间的关系推广到与谢克特曼共同工作中发现的模形式。这种关系可以通过这样一个事实可见一斑，即偶数维射影空间上的一个这样的层--手征de Rham复形--的整体截面空间的特征等于相应环空间的等变签名。第四个是一个项目，旨在解释在简单李群的情况下出现的某种显著的对偶性，如果可能的话，用物理学的术语。整个提议可能与弦理论有关，例如，上述手征德罗姆络合物允许重现光滑射影环簇的量子上同调。第五个项目是在I.Frenkel和Malikov之前工作的基础上进行的一个单独的项目；它涉及到仿射平移函子在BGG型可接受表示的分解中的可能应用。这是一个同时在数学领域被称为“表示理论”的项目，在数学物理领域被称为“量子场论”。表示论通常被认为是描述各种对称性，特别是自然界中出现的对称性的最有用的数学领域。量子场论可以被认为是支配亚原子粒子行为的量子力学的逻辑基础。这个项目的一个重要灵感来源是弦理论，它被广泛认为是解释所有自然基本定律的统一物理理论的唯一候选者。弦理论的惊人思想是，物理学中的基本粒子可以在数学上描述为一个回路。这一思想的含义使以前被认为完全抽象的整个数学领域成为理论物理中日常研究的工具。这个项目的具体目标是为了更好地了解在这一观点下出现的所谓“循环空间”。弦理论的数学思想已经在层析成像中得到了重要的应用，当然还有更多的应用等待着去发现。

This proposal is centered around several examples of sheaves of vertex algebras that Malikov, Schechtman, Vaintrob, and Gorbounov discovered and investigated in the previous work. More specifically, the first part of the proposal has to do with computation of the algebras of global sections of these sheaves on Calabi-Yau manifolds. The known examples arising in the case of tori and projective spaces are promising. The second part is built on the recent discovery that a family of such vertex algebras arising in the case of the projective line is closely related to the super-algebra Lie sl(2|1) and to sl(2|1) - Toda field theory. The third is an attempt to generalize the relation of such sheaves to ellptic genera, quantum cohomology, and modular forms discovered in a joint work with Schechtman. This relation is visible through the fact that the character of the space of global sections of one such sheaf, a chiral de Rham complex, over over an even-dimensional projective space equals the equivariant signature of the corresponding loop space. The fourth is a project aiming at an explanation of a certain remarkable duality arising in the case of a simple Lie group, in physics terms if possible. That the whole proposal may be related to string theory is revealed, for example, in the fact that the above-mentioned chiral de Rham complex allows to reproduce the quantum cohomology of smooth projective toric varieties. The fifth is a separate project based on the previous work of I.Frenkel and Malikov; it has to do with a possible application of affine translation functors to BGG-type resolutions of admissible representations.This is a project simultaneously in the area of mathematics known as "Representation Theory," and in the area of mathematical physics called "Quantum Field Theory." Representation theory is generally considered to be the most useful area of mathematics for describing symmetries of all kinds, especially those that arise in nature. Quantum field theory may be thought of as the logical foundation for the quantum mechanics that govern the behavior of subatomic particles. An important source of inspiration for this project is string theory, which is widely accepted as the only candidate for a unified physical theory explaining all fundamental laws of Nature. The striking idea of string theory is that an elementary particle in physics can be described mathematically as a loop. The implications of this idea have made entire areas of mathematics previously thought of as totally abstract into everyday tools of research in theoretical physics. The specific goal of this project is to obtain a better understanding of the so-called "loop spaces" that arise in this point of view. Mathematical ideas from string theory have already found important applications in tomography, and surely there are more applications waiting to be found.