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。共形场理论是具有保角或共形对称性的量子场论。二维共形理论在弦理论中有重要的应用，它可以用所谓的顶点代数严格地数学构造。在过去的三十年里，在顶点代数和其他数学领域之间发现了深刻而令人惊讶的联系。在这个项目中，将研究顶点代数与代数和数论中的主题之间的相互作用。更详细地说，这个项目将通过研究无理顶点代数的表示、模之间的交织算子以及模的特征标的模变换性质来加深我们对无理顶点代数的理解。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，我们将研究某些无理W-代数与某些单位根处的量子群之间的Kazhdan-Lusztig型对应。最近出现的非标准量子群被用来构造纽结和3-流形的新的强大不变量。在另一个方向上，我们将研究无理顶点代数的模形式与模的特征有关的变换性质。在第三个方向，我们将研究无理理论中Verlinde公式的连续形式，以及来自屏蔽算子核的顶点代数的射影覆盖的构造。该项目对对数共形场理论在物理学中的研究也具有重要意义。

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.