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Register a new userOn the moduli space of Donaldson-Thomas instantons
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Authors
Yuuji Tanaka
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In alignment with a programme by Donaldson and Thomas [DT], Thomas [Th] constructed a deformation invariant for smooth projective Calabi-Yau threefolds, which is now called the Donaldson-Thomas invariant, from the moduli space of (semi-)stable sheaves by using algebraic geometry techniques. In the same paper [Th], Thomas noted that certain perturbed Hermitian-Einstein equations might possibly produce an analytic theory of the invariant. This article sets up the equations on symplectic 6-manifolds, and gives the local model and structures of the moduli space coming from the equations. We then describe a Hitchin-Kobayashi style correspondence for the equations on compact Kahler threefolds, which turns out to be a special case of results by Alvarez-Consul and Garcia-Prada [AG].
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The contents of this article are now presented in the appendix of arXiv:0805.2195v2.
In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact Kahler threefolds, and study some of analytic properties of solutions to them, using analytic methods in higher-dimensional Yang-Mills theory developed by Nakajima and Tian with some additional arguments concerning an extra non-linear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson-Thomas equation of a unitary vector bundle over a compact Kahler threefold has a converging subsequence outside a closed subset whose real 2-dimensional Hausdorff measure is finite, provided that the L^2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-compactness theorem of solutions to the equations on compact Kahler threefolds.
We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact Kahler threefolds. The equation could be thought of as a generalization of the Hitchin equation on Riemann surfaces to Kahler threefolds. In the appendix of arXiv:0805.2192, following an analogy with the Hitchin equation, we introduced a stability condition for a pair consisting of a locally-free sheaf over a compact Kahler threefold and a section of the associated sheaf of the endomorphisms tensored by the canonical bundle of the threefold. In this article, we prove a Hitchin--Kobayashi-type correspondence for this and the Donaldson-Thomas instanton on compact Kahler threefolds.
We study higher rank Donaldson-Thomas invariants of a Calabi-Yau 3-fold using Joyce-Songs wall-crossing formula. We construct quivers whose counting invariants coincide with the Donaldson-Thomas invariants. As a corollary, we prove the integrality and a certain symmetry for the higher rank invariants.
This paper establishes that the Nahm transform sending spatially periodic instantons (instantons on the product of the real line and a three-torus) to singular monopoles on the dual three-torus is indeed a bijection as suggested by the heuristic. In the process, we show how the Nahm transform intertwines to a Fourier-Mukai transform via Kobayashi-Hitchin correspondences. We also prove existence and non-existence results.
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