Representation Theory and Quantum Field Theory

这个提议是围绕着Malikov， Schechtman， Vaintrob和Gorbounov在之前的工作中发现和研究的几个顶点代数的例子。更具体地说，该建议的第一部分与计算这些轴在Calabi-Yau流形上的整体截面的代数有关。在环面和射影空间中出现的已知例子是有希望的。第二部分建立在最近的发现上，即在射影线的情况下产生的一类顶点代数与超代数Lie sl（2|1）和sl(2|1) - Toda场论密切相关。第三部分是试图将这些束的关系推广到椭圆属、量子上同调和在与Schechtman合著的著作中发现的模形式。这种关系是通过这样一个事实可见的：一个这样的束，一个手性德朗复，在一个偶维射影空间上的整体截面的空间特征等于相应环空间的等变特征。第四个是一个项目，旨在解释在一个简单李群的情况下产生的某种显著的对偶性，如果可能的话，用物理术语。整个建议可能与弦理论有关，例如，上述手性de Rham复合体允许再现光滑射影环变体的量子上同调。第五个是一个独立的项目，基于i.f arenkel和Malikov之前的工作；它与仿射平移函子在可容许表示的bgg型解析中的可能应用有关。这个项目同时在数学领域被称为“表征理论”，在数学物理领域被称为“量子场论”。表征理论通常被认为是数学中描述各种对称性最有用的领域，尤其是那些在自然界中出现的对称性。量子场论可以被认为是控制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，弦理论被广泛接受为解释所有自然基本定律的统一物理理论的唯一候选者。弦理论引人注目的思想是，物理学中的基本粒子可以用数学方法描述为一个环。这一思想的含义使以前被认为完全抽象的整个数学领域变成了理论物理研究的日常工具。这个项目的具体目标是更好地理解在这个观点中出现的所谓的“循环空间”。弦理论的数学思想已经在断层扫描中找到了重要的应用，当然还有更多的应用有待发现。

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.