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تسجيل مستخدم جديدOn spin structures and orientations for gauge-theoretic moduli spaces
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Let $X$ be a compact manifold, $G$ a Lie group, $P to X$ a principal $G$-bundle, and $mathcal{B}_P$ the infinite-dimensional moduli space of connections on $P$ modulo gauge. For a real elliptic operator $E_bullet$ we previously studied orientations on the real determinant line bundle over $mathcal{B}_P$. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson. Here we consider complex elliptic operators $F_bullet$ and introduce the idea of spin structures, square roots of the complex determinant line bundle of $F_bullet$. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures. Our main result identifies spin structures on $X$ with orientations on $X times S^1$. Thus, if $P to X$ and $Q to X times S^1$ are principal $G$-bundles with $Q|_{Xtimes{1}} cong P$, we relate spin structures on $(mathcal{B}_P,F_bullet)$ to orientations on $(mathcal{B}_Q,E_bullet)$ for a certain class of operators $F_bullet$ on $X$ and $E_bullet$ on $Xtimes S^1$. Combined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups $G=U(m), SU(m)$. In a sequel arXiv:2001.00113 we apply this to define canonical orientation data for all Calabi-Yau 3-folds $X$ over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory.
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Let $X$ be a compact manifold, $D$ a real elliptic operator on $X$, $G$ a Lie group, $Pto X$ a principal $G$-bundle, and ${mathcal B}_P$ the infinite-dimensional moduli space of all connections $ abla_P$ on $P$ modulo gauge, as a topological stack. F
or each $[ abla_P]in{mathcal B}_P$, we can consider the twisted elliptic operator $D^{ abla_{Ad(P)}}$ on X. This is a continuous family of elliptic operators over the base ${mathcal B}_P$, and so has an orientation bundle $O^D_Pto{mathcal B}_P$, a principal ${mathbb Z}_2$-bundle parametrizing orientations of Ker$D^{ abla_{Ad(P)}}oplus$Coker$D^{ abla_{Ad(P)}}$ at each $[ abla_P]$. An orientation on $({mathcal B}_P,D)$ is a trivialization $O^D_Pcong{mathcal B}_Ptimes{mathbb Z}_2$. In gauge theory one studies moduli spaces $mathcal M$ of connections $ abla_P$ on $P$ satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds $(X, g)$. Under good conditions $mathcal M$ is a smooth manifold, and orientations on $({mathcal B}_P,D)$ pull back to orientations on $mathcal M$ in the usual sense under the inclusion ${mathcal M}hookrightarrow{mathcal B}_P$. This is important in areas such as Donaldson theory, where one needs an orientation on $mathcal M$ to define enumerative invariants. We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on $({mathcal B}_P,D)$, after fixing some algebro-topological information on $X$. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, Kapustin-Witten and Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds. Two sequels arXiv:1811.02405, arXiv:1811.09658 discuss orientations in 7 and 8 dimensions.
We calculate the homomorphism of the cohomology induced by the Krichever map of moduli spaces of curves into infinite-dimensional Grassmannian. This calculation can be used to compute the homology classes of cycles on moduli spaces of curves that are defined in terms of Weierstrass points.
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 connecti
ons 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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