On orientations for gauge-theoretic moduli spaces

DOI10.1016/J.AIM.2019.106957arXiv1811.01096MaRDI QIDQ6309153

Dominic Joyce, Markus Upmeier, Yuuji Tanaka

Publication date: 2 November 2018

Abstract: Let $X$ be a compact manifold, $D$ a real elliptic operator on $X$, $G$ a Lie group, $PoX$ a principal $G$-bundle, and ${mathcalB}_P$ the infinite-dimensional moduli space of all connections $abl{a}_P$ on $P$ modulo gauge, as a topological stack. For each $[abl{a}_P]in{mathcalB}_P$, we can consider the twisted elliptic operator $D^{abl{a}_{Ad(P)}}$ on X. This is a continuous family of elliptic operators over the base ${mathcalB}_P$, and so has an orientation bundle $O_P^{D}o{mathcalB}_P$, a principal ${mathbbZ}_2$-bundle parametrizing orientations of Ker$D^{abl{a}_{Ad(P)}}oplus$Coker$D^{abl{a}_{Ad(P)}}$ at each $[abl{a}_P]$. An orientation on $({mathcalB}_P,D)$ is a trivialization $O_P^{D}cong{mathcalB}_{P}imes{mathbbZ}_2$. In gauge theory one studies moduli spaces $mathcalM$ of connections $abl{a}_P$ on $P$ satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds $(X,g)$. Under good conditions $mathcalM$ is a smooth manifold, and orientations on $({mathcalB}_P,D)$ pull back to orientations on $mathcalM$ in the usual sense under the inclusion $mathcalMhookrightarrow{mathcalB}_P$. This is important in areas such as Donaldson theory, where one needs an orientation on $mathcalM$ to define enumerative invariants. We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on $({mathcalB}_P,D)$, 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.

Mathematics Subject Classification ID

Moduli problems for differential geometric structures (58D27) Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) (53C07)

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