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K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface

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Title: K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface
Authors: Arbesfeld, N
Item Type: Working Paper
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 K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions. We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter 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 so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kos\c{c}az and Vafa. Applying these results to specific Calabi-Yau 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.
Issue Date: 8-Oct-2020
URI: http://hdl.handle.net/10044/1/87682
Publisher: arXiv
Copyright Statement: © 2020 The Author(s).
Keywords: math.AG
math.AG
hep-th
math-ph
math.MP
14C05, 14N35, 14C35
math.AG
math.AG
hep-th
math-ph
math.MP
14C05, 14N35, 14C35
Notes: 56 pages, 10 figures, exposition expanded and section on rank two vector bundles added
Publication Status: Published
Appears in Collections:Pure Mathematics
Mathematics