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.
