Irrational Vertex Algebras, Quantum Modular Forms, and Unrolled Quantum Groups

量子场论为现代粒子物理学提供了一个概念框架。共形场论是量子场论，它具有保角对称性。二维共形理论在弦理论中有着重要的应用，可以用所谓的顶点代数在数学上严格地构建。在过去的三十年里，在顶点代数和其他数学领域之间已经发现了深刻而令人惊讶的联系。在本专题中，将探讨顶点代数与代数和数论主题之间的相互作用。拨款还将支持研究生的培训。更详细地说，这个项目将加深我们对非理性顶点代数的理解，通过研究它们的表示，模块之间的缠结算子，以及它们的模块的特征的模变换性质。这将通过使用来自量子群理论和模形式理论的思想来完成。在一个方向上，研究了在单位根处某些无理数w代数与某些量子群之间的kazhdan - lusztig型对应。最近出现的非标准量子群被用来构造结点和3流形的新的强大不变量。在另一个方向上，将研究模形式与无理性顶点代数的模的性质有关的变换性质。在第三个方向上，将研究不合理理论中Verlinde公式的连续版本，以及来自筛选算子核的顶点代数的投影覆盖的构造。本项目对物理学中对数共形场论的研究也有启示意义。

Quantum field theories provide a conceptual framework for modern particle physics. Conformal field theories are quantum field theories that that have angle-preserving, or conformal, symmetries. Two dimensional conformal theories, which have important applications to string theory, can be rigorously constructed mathematically using what are known as vertex algebras. Over the the last thirty years, deep and surprising connections have been discovered between vertex algebras and other areas of mathematics. In this project, interactions between vertex algebras and topics in algebra and number theory will be investigated. The grant will also support the training of graduate students.In more detail, this project will deepen our understanding of irrational vertex algebras by investigating their representations, intertwining operators among modules, and modular transformation properties of characters of their modules. This will be done by using ideas coming from the theory of quantum groups and the theory of modular forms. In one direction, a Kazhdan-Lusztig-type correspondence between certain irrational W-algebras and certain quantum groups at roots of unity will be studied. The non-standard quantum groups that appear have recently been used to construct new powerful invariants of knots and 3-manifolds. In another direction, transformation properties of modular forms in connection with characters of modules of irrational vertex algebras will be investigated. In a third direction, a continuous version of the Verlinde formula in irrational theories and construction of projective covers for vertex algebras coming from kernels of screening operators will be studied. This project also has implications for the study of logarithmic conformal field theory in physics.