$\mathcal {W}$-constraints for the total descendant potential of a simple singularity. (May 2013)

• Record Type: Journal Article
• Authors: Bakalov, Bojko; Milanov, Todor
• Is Part Of: Compositio mathematica. Volume 149:Number 5(2013)
• Page Start: 840
• Page End: 888
• Publication Date: 2013-05
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X12000668
• Languages: English
• ISSNs: 0010-437X

Abstract

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