2025/01/30 by Colton Griffin, Griffin, Colton · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2501.18720
openalex publication_date 2025/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Vertex algebras (and their modules) can be described as vector spaces together with a linear operator-valued series in one parameter z. With the interpretation of z as a coordinate at a point on a curve, one can construct algebraic structures on the moduli space of curves from V-modules. Here we propose a generalization of vertex algebras involving linear operators in parameters z1,…,zn. One may interpret these as being the components of a set of coordinates on an n-dimensional algebraic variety. These are referred to as cohomological vertex algebras (CVAs): the formal punctured 1-disk underlying a vertex algebra is replaced by a ring modeling the cohomology of certain modifications of the formal n-disk. We prove several structural theorems for CVAs and give a definition of cohomological vertex operator algebras (CVOAs). Using a reconstruction theorem for CVAs, we provide basic examples such as the βγ-system, the Heisenberg CVA, and the affine Kac-Moody CVAs. We use these constructions to describe BRST reduction, leading to an analog of W-algebras.