$\mathcal {W}$-constraints for the total descendant potential of a simple singularity. (May 2013) - Bakalov, Bojko; Milanov, Todor
Abstract Simple, or Kleinian, singularities are classified by Dynkin diagrams of type <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8whtw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$ADE$]]> </alternatives>. Let <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wjsv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathfrak {g}$]]> </alternatives> be the corresponding finite-dimensional Lie algebra, and <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wkpq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$W$]]> </alternatives> its Weyl group. The set of <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wm6h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathfrak {g}$]]> </alternatives>-invariants in the basic representation of the affine Kac–Moody algebra <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wkrt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\hat {\mathfrak {g}}$]]> </alternatives> is known as a <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8whjg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathcal {W}$]]> </alternatives>-algebra and is a subalgebra of the Heisenberg vertex algebra <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8whmk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathcal {F}$]]> </alternatives>. Using period integrals, we construct an analytic continuation of the twisted representation of <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wktx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathcal {F}$]]> </alternatives>. Our construction yields a global object, which may be called a <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8whcq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$W$]]> </alternatives>-twisted representation of <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wj71" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathcal {F}$]]> </alternatives>. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the <alternatives><inline-graphic xlink:href="ark:/27927/pgh1gg8wj5x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> <![CDATA[$\mathcal {W}$]]> </alternatives>-algebra.