The Hodge theory of Knizhnik-Zamolodchikov equations and Rigid Local Systems

克尼日尼克-扎莫洛奇科夫微分方程联系了数学和物理的许多领域，包括表示论（研究对称性）、代数几何（研究多项式方程的解）和共形场论（来自数学物理）。这个项目的目标是发展霍奇理论技术，这是复杂的分析性质，在研究这些微分方程。这项工作将产生新的结构有关的表示理论，霍奇理论，理论的动机，和数学物理，这将进一步发展这些领域之间的关系和他们的计算方面。该项目将为研究生提供研究培训机会，具体而言，将开展三个相互关联的研究项目。第一，是开发一个motivic因式分解公式的不变量和共形块局部系统（这是局部系统的模光滑点曲线相关的一个简单的李代数和表示）亏格为零。第二是确定Hodge-Galois融合环和表示环的结构，它们是motivic分解的结果。最后，第三个是计算的例子，特别是对于情况下，当共形块本地系统是刚性的全球monodromy是有限的，使用霍奇theoretical criterions.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The Knizhnik-Zamolodchikov differential equations link many areas of mathematics and physics, including representation theory (the study of symmetries), algebraic geometry (study of solutions of polynomial equations), and conformal field theory (from mathematical physics). The goal of this project is to develop Hodge theoretic techniques, which are complex analytic in nature, in the study of these differential equations. This work will produce new structures relating representation theory, Hodge theory, the theory of motives, and mathematical physics, which will further develop the relations between these fields and their computational aspects. The project will provide research training opportunities for graduate students.In more detail, three interrelated research projects will be undertaken. The first, is to develop a motivic factorization formula for invariants and conformal block local systems (which are local systems on the moduli of smooth pointed curves associated to a simple Lie algebra and representations) in genus zero. The second is to determine the structure of the Hodge-Galois fusion rings and representation rings that arise as a consequence of motivic factorization. Finally, the third is to compute examples, particularly for cases when the conformal blocks local system is rigid where the global monodromy is finite, using Hodge theoretic criteria.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.