W-Constraints for the Total Descendant Potential of a Simple Singularity

Abstract

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finite-dimensional Lie algebra, and W its Weyl group. The set of g-invariants in the basic representation of the affine Kac-Moody algebra g^ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F. Using period integrals, we construct an analytic continuation of the twisted representation of F. Our construction yields a global object, which may be called a W-twisted representation of F. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the W-algebra.

61 pages, 2 figures

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Cited by in corpus (17)

• W-algebras as coset vertex algebras
• The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy
• Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies
• Twisted logarithmic modules of vertex algebras
• The Eynard--Orantin recursion for the total ancestor potential
• Orbifolds of lattice vertex algebras under an isometry of order two
• Additional symmetries of the extended bigraded Toda hierarchy
• Higher Airy structures, W algebras and topological recursion
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy
• The Period map for quantum cohomology of ${\mathbb{P}}^2$
• Matrix model for the total descendant potential of a simple singularity of type $D$
• Virasoro constraints in Drinfeld-Sokolov hierarchies
• Twisted logarithmic modules of free field algebras
• Orbifolds of Lattice Vertex Algebras
• Darboux transformations and Fay identities for the extended bigraded Toda hierarchy
• A New Class of Higher Quantum Airy Structures as Modules of $\mathcal{W}({\mathfrak{gl}}_r)$-Algebras