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$\mathcal {W}$-constraints for the total descendant potential of a simple singularity

Part of: Groups and algebras in quantum theory Symplectic geometry, contact geometry Lie algebras and Lie superalgebras Singularities

Published online by Cambridge University Press:  07 February 2013

Bojko Bakalov  and
Todor Milanov
Bojko Bakalov
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA (email: bojko_bakalov@ncsu.edu)
Todor Milanov
Affiliation:
Kavli IPMU, University of Tokyo (WPI), Kashiwa 277-8583, Japan (email: todor.milanov@ipmu.jp)
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Abstract

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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 $\hat {\mathfrak {g}}$ 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.

Keywords

Frobenius manifoldsimple singularitytotal descendant potentialvertex algebra$\uppercase {\mathcal {w}}$-algebra

MSC classification

Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 17B69: Vertex operators; vertex operator algebras and related structures 32S30: Deformations of singularities; vanishing cycles 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 5 , May 2013 , pp. 840 - 888
Copyright
Copyright © 2013 The Author(s) 

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