[Paper Review] Canonical orientations for moduli spaces of $G_2$-instantons with gauge group SU(m) or U(m)

Dominic Joyce, Markus Upmeier|arXiv (Cornell University)|Nov 6, 2018

Geometric Analysis and Curvature Flows28 references9 citations

TL;DR

This paper establishes canonical orientations for moduli spaces of $G_2$-instantons with structure group $Σμ(m)$ or $Σμ(m)$ on compact $G_2$-manifolds by constructing canonical trivializations of the orientation double bundle associated to the determinant of the Dirac operator. Using a fixed orientation of the Dirac determinant and a flag structure on the manifold, the authors induce canonical orientations on the moduli spaces of $G_2$-instantons, enabling the definition of signed enumerative invariants in the Donaldson–Segal programme.

ABSTRACT

Suppose $(X, g)$ is a compact, spin Riemannian 7-manifold, with Dirac operator $D$. Let $G$ be SU$(m)$ or U$(m)$, and $E o X$ 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 $O^D_E o{\mathcal B}_E$ parametrizing orientations of det$\,D_{{ m Ad }A}$ for twisted elliptic operators $D_{{ m Ad }A}$ 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 $E o X$, satisfying natural compatibilities. Now let $(X,φ,g)$ be a compact $G_2$-manifold, with d$(*φ)=0$. Then we can consider moduli spaces ${\mathcal M}_E^{G_2}$ of $G_2$-instantons on $E o X$, 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 $(X,φ,g)$ 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.

Motivation & Objective

• To resolve the problem of choosing consistent orientations for moduli spaces of $G_2$-instantons with unitary or special unitary gauge groups.
• To provide a canonical, geometrically natural choice of orientation for these moduli spaces, independent of arbitrary choices.
• To support the Donaldson–Segal programme by enabling the construction of signed enumerative invariants of $G_2$-manifolds.
• To extend the theory of canonical orientations from lower-dimensional gauge theories to dimension seven, specifically for $G_2$-instantons.
• To establish a framework compatible with derived geometry and excision theorems in gauge-theoretic moduli problems.

Proposed method

• Constructs the orientation double bundle $O^{ ot{ ext{D}}^g}_E$ over the moduli space ${\mathcal{B}}_E$ of connections, using the determinant line bundle of the twisted Dirac operator $\not{\text{D}}^{g}_{\mathop{\rm Ad}A}$.
• Uses a fixed orientation of $\mathop{\rm det}\not{\text{D}}^g$ and a flag structure on the spin 7-manifold $X$ to define canonical trivializations of $O^{ ot{\text{D}}^g}_E$.
• Applies the excision theorem and functoriality of orientation isomorphisms to ensure compatibility across different bundles and manifolds.
• Transfers the canonical trivialization from the full moduli space ${\mathcal{B}}_E$ to the subspace ${\mathcal{M}}_E^{G_2}$ of $G_2$-instantons via restriction.
• Employs a gluing construction using tubular neighborhoods, spin diffeomorphisms, and bundle isomorphisms to compare orientations across different charts and manifolds.
• Extends results from $\mathop{\rm SU}(m)$ to $\mathop{\rm U}(m)$ bundles via the map $E \mapsto E \oplus \Lambda^m E^*$, preserving canonical orientation structures.

Experimental results

Research questions

• RQ1 How can one define a canonical, geometrically natural orientation on the moduli space of $G_2$-instantons with gauge group $\mathop{\rm SU}(m)$ or $\mathop{\rm U}(m)$?
• RQ2 What geometric data on a compact spin 7-manifold allows for the construction of a canonical trivialization of the orientation double bundle associated to the Dirac operator determinant?
• RQ3 How do canonical orientations behave under gluing or excision in the context of $G_2$-instanton moduli spaces?
• RQ4 Can the canonical orientation construction be extended from $\mathop{\rm SU}(m)$ to $\mathop{\rm U}(m)$ bundles in a way that preserves compatibility and functoriality?
• RQ5 What role does a flag structure play in resolving the ambiguity in orientation choices for $G_2$-instanton moduli spaces?

Key findings

• The orientation double bundle $O^{ ot{\text{D}}^g}_E$ over the moduli space ${\mathcal{B}}_E$ is canonically trivialized when a fixed orientation of $\mathop{\rm det}\not{\text{D}}^g$ and a flag structure on $X$ are chosen.
• The canonical trivialization is compatible with the excision isomorphism, ensuring consistency across different bundles and manifolds.
• The restriction of the canonical orientation to the subspace ${\mathcal{M}}_E^{G_2}$ of $G_2$-instantons yields a canonical orientation on each such moduli space.
• The canonical orientation on ${\mathcal{M}}_E^{G_2}$ is independent of arbitrary choices and is functorial under diffeomorphisms and bundle operations.
• The construction extends from $\mathop{\rm SU}(m)$ to $\mathop{\rm U}(m)$ bundles via the $E \mapsto E \oplus \Lambda^m E^*$ construction, preserving canonical orientation properties.
• The canonical orientation induces a well-defined signed count of $G_2$-instantons, supporting the Donaldson–Segal programme for enumerative invariants of $G_2$-manifolds.