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Register a new userSpatially periodic instantons: Nahm transform and moduli
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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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The main result is a computation of the Nahm transform of a SU(2)-instanton over RxT^3, called spatially-periodic instanton. It is a singular monopole over T^3, a solution to the Bogomolny equation, whose rank is computed and behavior at the singular points is described.
In this paper, we complete the proof of an equivalence given by Nye and Singer of the equivalence between calorons (instantons on $S^1times R^3$) and solutions to Nahms equations over the circle, both satisfying appropriate boundary conditions. Many of the key ingredients are provided by a third way of encoding the same data which involves twistors and complex geometry.
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].
There exists a recursive algorithm for constructing BPST-type multi-instantons on commutative R^4. When deformed noncommutatively, however, it becomes difficult to write down non-singular instanton configurations with topological charge greater than one in explicit form. We circumvent this difficulty by allowing for the translational instanton moduli to become noncommutative as well. This makes possible the ADHM construction of t Hooft multi-instanton solutions with everywhere self-dual field strengths on noncommutative R^4.
Suppose $(X, g)$ is a compact, spin Riemannian 7-manifold, with Dirac operator $D$. Let $G$ be SU$(m)$ or U$(m)$, and $Eto X$ be a rank $m$ complex bundle with $G$-structure. Write ${mathcal B}_E$ for the infinite-dimensional moduli space of connections on $E$, modulo gauge. There is a natural principal ${mathbb Z}_2$-bundle $O^D_Eto{mathcal B}_E$ parametrizing orientations of det$,D_{{rm Ad }A}$ for twisted elliptic operators $D_{{rm Ad }A}$ at each $[A]$ in ${mathcal B}_E$. A theorem of Walpuski shows $O^D_E$ is trivializable. We prove that if we choose an orientation for det$,D$, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of $O^D_E$ for all such bundles $Eto X$, satisfying natural compatibilities. Now let $(X,varphi,g)$ be a compact $G_2$-manifold, with d$(*varphi)=0$. Then we can consider moduli spaces ${mathcal M}_E^{G_2}$ of $G_2$-instantons on $Eto X$, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with ${mathcal M}_E^{G_2}subset{mathcal B}_E$. The restriction of $O^D_E$ to ${mathcal M}_E^{G_2}$ is the ${mathbb Z}_2$-bundle of orientations on ${mathcal M}_E^{G_2}$. Thus, our theorem induces canonical orientations on all such $G_2$-instanton moduli spaces ${mathcal M}_E^{G_2}$. This contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of $G_2$-manifolds $(X,varphi,g)$ by counting moduli spaces ${mathcal M}_E^{G_2}$, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6. A third paper Cao-Gross-Joyce arXiv:1811.09658 studies orientations on moduli spaces in dimension 8.
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