Description
I will discuss vertex algebras defined over commutative rings, and as special cases the universal W_{\infty}-algebras of types W(2,3,4,...) and W(2,4,6,...), which are defined over the polynomial ring in two variables. The existence and uniqueness of these algebras was conjectured in the physics literature, and was recently established in my papers arXiv:1710.02275 and arXiv:1805.11031 (joint with S. Kanade). All one-parameter vertex algebras of type W(2,3,...,N) or W(2,4,...,2N) for some N satisfying some mild hypotheses, can be obtained as quotients of these algebras. This includes the principal W-algebras of types A, B, and C, as well as many others arising as cosets of affine vertex algebras inside larger structures. Each of these one-parameter vertex algebras corresponds to a certain curve in the plane, and the intersection points of these curves give rise to nontrivial isomorphisms between these vertex algebras. Finally, we will describe some remarkable infinite families of such curves whose singular points and pairwise intersection points are all rational.
