Ktheoretic DonaldsonThomas theory and the Hilbert scheme of points on a surface
Abstract
Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the Ktheoretic DonaldsonThomas theory of certain toric CalabiYau threefolds to study Ktheoretic variants of such expressions. We study limits of the Ktheoretic DonaldsonThomas partition function of a toric CalabiYau threefold under certain oneparameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under socalled preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kosçaz and Vafa. Applying these results to specific CalabiYau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on $\mathbb{C}^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.04567
 Bibcode:
 2019arXiv190504567A
 Keywords:

 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematical Physics;
 14C05;
 14N35;
 14C35
 EPrint:
 41 pages, 10 figures