Building Blocks for W-algebras

University Of Denver, Denver CO

Investigators

Abstract

Vertex operator algebras (VOAs) arose in physics in the 1980s as the symmetry algebras of two-dimensional conformal field theories (CFTs) and were first defined mathematically by Borcherds. They have turned out to be fundamental objects with connections to many subjects including finite groups, Lie theory, combinatorics, integer partitions, modular forms, and algebraic geometry. W-algebras are an important class of VOAs that are associated to a Lie (super)algebra g and a nilpotent element f in the even part of g. They appear in various settings including integrable systems, CFT to higher spin gravity duality, the Allday-Gaiotto-Tachikawa correspondence, and the quantum geometric Langlands program. In this project, the PI will investigate the structure and representation theory of W-algebras. This will advance the subject and provide research training and collaborative opportunities for graduate students and postdocs. In more detail, principal W-algebras (the case where f is a principal nilpotent) are the best understood class of W-algebras. They satisfy Feigin-Frenkel duality, and in classical Lie types they also admit a coset realization which has numerous applications to representation theory. It turns out that both Feigin-Frenkel duality and the coset realization are special cases of a more general duality which was conjectured by Gaiotto and Rapcak and proven recently by the PI and Creutzig. It says that the affine cosets of certain triples of W-algebras are isomorphic as 1-parameter VOAs. These cosets are known as Y-algebras in type A, and orthosymplectic Y-algebras in types B, C, and D. The Y-algebras can all be obtained as 1-parameter quotients of a universal 2-parameter VOA, and they are conjectured to be the building blocks for all W-algebras in type A. The orthosymplectic Y-algebras are quotients of another universal 2-parameter VOA, but they are not all the necessary building blocks for W-algebras in types B, C, and D. The main goals of this project are (1) to identify the missing building blocks, which we expect to arise as quotients of a third universal 2-parameter VOA; (2) to prove that W-algebras of all classical types can be obtained as conformal extensions of tensor products of building blocks. Special cases will involve W-algebras with N=1 and N=2 supersymmetry, and the PI hopes to prove some old conjectures from physics on coset realizations of these structures. Finally, the Y-algebras and other building blocks admit many levels where their simple quotients are lisse and rational. Exhibiting W-algebras at special levels as extensions of building blocks will lead to many new rationality results. 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.