Andrew Linshaw: Universal vertex algebras beyond the W_∞-algebras

The principal W-algebras W^k(𝔰𝔩_n), as well as the cosets Com(V^k(𝔤𝔩_n), V^k(𝔰𝔩_n+1)) for n ≥ 2, are all quotients of a universal 2-parameter vertex operator algebra (VOA) which is freely generated of type W(2,3,4, …). This VOA admits many other interesting 1-parameter quotients which can be identified (up to an extra Heisenberg field) with the Gaiotto-Rapcak Y-algebras. We consider a similar construction in type C, namely, the cosets Com(V^k(𝔰𝔭_2n), V^k(𝔰𝔭_2n+2)), for n ≥ 2. This gives rise to a 2-parameter VOA which is freely generated of type W(1³, 2, 3³, 4,…), which we expect to be the universal VOA of this type. The universal algebra admits 8 infinite families of 1-parameter quotients, which are analogues of the Gaiotto-Rapcak Y-algebras. Assuming that the universal algebra has exactly two parameters, which is ongoing work to prove, we present some applications including new rationality results for W-(super)algebras.

This is a joint work with Thomas Creutzig and Vlad Kovalchuk.

This video was produced by the International Centre for Mathematical Sciences, as part of the workshop Geometric Representation Theory and W-algebras.