A quantum field comonad

We encapsulate the basic notions of the theory of vertex algebras into the construction of a comonad on an appropriate category of formal distributions. Vertex algebras are recovered as coalgebras over this comonad.Comment: 13 page

Notes on affine W-algebras

These are expanded and revised notes for a minicourse entitled "Affine W-algebras", which took place as part of the thematic month "Quantum Symmetries" at the Centre de Recherches Mathematiques in Montreal, Canada in October 2022. The first few sections consist of rapid introductions to vertex algebras, affine Kac-Moody algebras and their integrable and admissible modules, and the homological BRST procedure for quantum Hamiltonian reduction. The affine W-algebras are defined using these ingredients. The remainder of the notes treats the structure and representation theory of the exceptional affine W-algebras, with emphasis on modular tensor categories of representations.Comment: 52 page

Higher level twisted Zhu algebras

The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper we consider the general set-up of a vertex algebra $V$, graded by \G/\Z for some subgroup \G of $\R$ containing $\Z$, and with a Hamiltonian operator $H$ having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level $p$ Zhu algebras \zhu_{p, \G}(V), and we prove the following theorems: For each $p$ there is a bijection between the irreducible \zhu_{p, \G}(V)-modules and the irreducible \G-twisted positive energy $V$-modules, and $V$ is (\G, H)-rational if and only if all its Zhu algebras \zhu_{p, \G}(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for $H$. We provide an explicit description of the level $p$ Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra \vir^c and the universal affine Kac-Moody vertex algebra V^k(\g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.Comment: 47 pages, no figure

A Hodge filtration on chiral homology and Poisson homology of associated schemes

We introduce filtrations in chiral homology complexes of smooth elliptic curves, exploiting the mixed Hodge structure on cohomology groups of configuration spaces. We use these to relate the chiral homology of a smooth elliptic curve with coefficients in a vertex algebra with the Poisson homology of the associated Poisson scheme. As an application we deduce finite dimensionality results for chiral homology in low degrees.Comment: 28 page