Identities from Vertex Operator Algebras on the Moduli of Curves

在代数几何中，人们的目标是理解变化，建立在多项式方程的解的基础上。通过改变多项式的系数，可以得到相关的物体族。通常，族本身可以被描述为一个种类，通过研究它，人们可以了解它所参数化的对象。曲线的模空间是这种方法的成功例子，它提供了对曲线族及其退化的洞察，并作为其他模空间的原型。经过类比和递归论证，它们达到了特殊变种的地位，在此基础上对代数几何理论进行了测试和探索。最重要的是，曲线的模空间得益于它们与数学和数学物理的许多不同领域的联系。这笔拨款还将支持费城高中生的数学丰富活动。更具体地说，顶点算子代数(VoA)及其表示定义了曲线的模空间上的协不变量层(和共形块的对偶层)。自20世纪80年代S以来，对这些特殊情况的研究，最近由PI和她的合著者推广到稳定的点曲线的模空间，他们证明了当由Voas定义的满足有限和半单性的假设时，协不变量的层具有因式分解和缝纫性质，并产生向量丛。该项目有三个主要目标：(1)将这些结果推广到VOA是有限但不是半简单的背景下，正如对数共形场理论中的结果所期望的那样；(2)研究这类轮子的重要性质和特征，包括它们的阶和更高的Chern类；以及(3)寻找VOA共形块向量空间的几何解释。该奖项反映了NSF的法定使命，并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，被认为值得支持。

In algebraic geometry, one aims to understand varieties, modeled on solutions of polynomial equations. By varying the coefficients of the polynomials, one obtains families of related objects. Often the family itself can be described as a variety, and by studying it, one learns about the objects it parameterizes. Moduli spaces of curves are successful examples of this approach, giving insight into families of curves and their degenerations, and serving as a prototype for other moduli spaces. Amenable to analogies and recursive arguments, they have achieved the status of special varieties on which the theory of algebraic geometry has been tested and explored. Most importantly, moduli spaces of curves have benefited from their connections to many different areas of mathematics and mathematical physics. The grant will also provide support for mathematics enrichment activities for high-school students in Philadelphia.More specifically, vertex operator algebras (VOAs) and their representations define sheaves of coinvariants (and dual sheaves of conformal blocks) on moduli spaces of curves. Studied in special cases since the 1980's, these were recently extended to the moduli space of stable pointed curves by the PI and her coauthors, who proved that when defined by VOAs satisfying finite and semi-simplicity assumptions, sheaves of coinvariants have factorization and sewing properties, and give rise to vector bundles. This project has three major objectives: (1) to generalize these results to the context where VOAs are finite but not semi-simple, as is expected from results in log-conformal field theory; (2) to study important properties and features of such sheaves in general, including their ranks and higher Chern classes; and (3) to find geometric interpretations of vector spaces of VOA conformal blocks.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.