Representation Theory of Vertex Superalgebras and W-algebras

• Applicant: Professor Dr. Thomas Creutzig
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 551865932

Representation theories of vertex algebras are a principal source of non semi-simple ribbon categories. Such categories are needed for the construction of non semi-simple topological field theories and they appear as categories of line operators in higher dimensional supersymmetric quantum field theories. Currently only few such representation categories are completely understood. My research goal is to improve this situation substantially. In many respects the representation theory of quasi Hopf algebras is much clearer and a major goal is to prove equivalences between categories of modules of vertex algebras and quasi Hopf algebras (like quantum supergroups). The first objective is to establish structural results on rigidity and the Verlinde formula for vertex tensor categories. These findings will be important for the following objectives. Then the theory of Kazhdan-Lusztig correspondences will be developed further. The aim is an effective theory for showing that certain vertex tensor categories are braided tensor equivalent to categories of corresponding quasi Hopf algebras. The last three projects will apply the findings of the first two. Firstly, the category of weight modules of the universal affine vertex algebra of sl(2) will be studied. In particular an equivalence to a modification of a quantum supergroup of sl(2|1) will be shown. Then similar statements will be shown for subregular W-algebras, i.e. weight modules will be classified, existence and rigidity of vertex tensor categories and equivalences to modifications of certain quantum supergroups will be shown. Finally, Kazhdan-Lusztig correspondences for affine vertex superalgebras of type I will be established.

• DFG Programme: Research Grants