2012/03/15 by Bojko Bakalov, Todor Milanov · 2 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebra over a field #Algebraic structures and combinatorial models #Descendant #Gravitational singularity #Lie algebra #Quantum Mechanics and Non-Hermitian Physics #Simple (philosophy) #Singularity #Subalgebra #Type (biology) #Vertex (graph theory) #math-ph #math.AG #math.MP #math.QA #msc:17B69 #msc:32S30 #msc:53D45 #msc:81R10
paper · pdf · doi:10.1112/s0010437x12000668
published as Compositio Math. 149 (2013) 840-888 · 61 pages, 2 figures
arxiv created 2012/03/15 · openalex publication_date 2013/02/07 · openalex created_date 2016/06/24 · arxiv updated 2019/02/20 · openalex updated_date 2026/08/05
Abstract Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE . Let \mathfrak g be the corresponding finite-dimensional Lie algebra, and W its Weyl group. The set of \mathfrak g -invariants in the basic representation of the affine Kac–Moody algebra \mathfrak g is known as a \mathcal W -algebra and is a subalgebra of the Heisenberg vertex algebra \mathcal F . Using period integrals, we construct an analytic continuation of the twisted representation of \mathcal F . Our construction yields a global object, which may be called a W -twisted representation of \mathcal F . Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the \mathcal W -algebra.