Localized Donaldson-Thomas theory of surfaces

Abstract

Let S be a projective simply connected complex surface and L be a line bundle on S. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of L. The moduli space admits a C *action induced by scaling the fibers of L. We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on S. We define the localized Donaldson-Thomas invariants of L by virtual localization in the case that L twisted by the anticanonical bundle of S admits a nonzero global section. When p g (S) > 0, in combination with Mochizuki's formulas, we are able to express the localized DT invariants in terms of the invariants of the nested Hilbert schemes defined by the authors in [GSY17a], the Seiberg-Witten invariants of S, and the integrals over the products of Hilbert schemes of points on S. When L is the canonical bundle of S, the Vafa-Witten invariants defined recently by Tanaka-Thomas, can be extracted from these localized DT invariants. VW invariants are expected to have modular properties as predicted by S-duality.

Hilbert schemes, Donaldson–Thomas theory, Vafa–Witten and Seiberg–Witten theory

“…Their rich geometric structures have proved to have many applications in mathematics and physics (see [N99] for a survey). The current article is a survey of the articles [GSY17a] and [GSY17b], where the authors studied the enumerative geometry of "nested Hilbert schemes" of points and curves on algebraic surfaces.…”

“…In [GSY17a] and [GSY17b], in order to define invariants for the nested Hilbert schemes (see [GSY17a, Definitions 2.13, 2.14]), the authors constructed a virtual fundamental class [S [n] β ] vir and then considered cases where one integrates appropriate cohomology classes against it. More precisely, they constructed a natural perfect obstruction theory over S [n] β .…”

“…In this section we give an overview of the results of [GSY17b]. Let (S, h) be a nonsingular simply connected projective surface with h = c 1 (O S (1)).…”

“…X is a noncompact Calabi-Yau threefold and one can define the DT invariants of X by using C * -localization, where C * -acts on X by scaling the fibers of ω S . More precisely, let v = (r, γ, m) ∈ ⊕ 2 i=0 H 2i (S, Q) be a Chern character vector, and M ω S h (v) be the moduli space of compactly supported 2-dimensional stable sheaves E on X such that ch(q * E) = v. Here stability is defined by means of the slope of q * E with respect to the polarization h. In [GSY17b] the authors provided M ω S h (v) with a perfect obstruction theory by reducing the natural perfect obstruction theory given by [T98]. The fixed locus M ω S h (v) C * of the moduli space is compact and the reduced obstruction theory gives a virtual fundamental class over it, denoted by [M ω S h (v) C * ] vir red .…”

“…The focus of SW theory in physics is the study of moduli space of vacua in N = 2, D = 4 super Yang-Mills theory, and in particular certain dualities of the theory such as electric-magnetic duality (Montonen-Olive duality). In [GLSY17] Gukov-Liu-Sheshmani-Yau conjectured a relation between SW invariants (associate to SU (2) gauge invariant theory) of a projective surface S and DT invariants of a noncompact threefold X obtained by total space of a line bundle L on S. In a later work [GSY17a,GSY17b] , which will be elaborated below, Gholampour-Sheshmani-Yau proved the conjecture in [GLSY17,Equation 56].…”