TY - JOUR

T1 - Canonical orientations for moduli spaces of G_2-instantons with gauge group SU(m) or U(m)

AU - Joyce, Dominic

AU - Upmeier, Markus

PY - 2021/9/9

Y1 - 2021/9/9

N2 - Suppose (X,g) is a compact, spin Riemannian 7-manifold, with Dirac operator D/g : C∞(X,S/) → C∞(X,S/). Let G be SU(m) or U(m), and E → X be a rank m complex bundle with G-structure. Write BE for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z2-bundle OD/g → BE parametrizing orientations of E detD/gAdA for twisted elliptic operators D/gAdA at each [A] in BE. A theorem of Walpuski [33] shows OD/g is trivializable. We prove that if we choose an orientation for detD/g, and a flag structure on X in the sense of [17], then we can define canonical trivializations of OD/g E for all such bundles E → X, satisfying natural compatibilities. Now let (X, φ, g) be a compact G2-manifold, with d(∗φ) = 0. Then we can consider moduli spaces MG2 of G2-instantons on E → X, which are smooth E manifolds under suitable transversality conditions, and derived manifolds in general, with MG2 ⊂ BE. The restriction of OD/g to MG2 is the Z2-bundle of orientations on MG2 . Thus, our theorem induces canonical orientations on E all such G2-instanton moduli spaces MG2 . E This contributes to the Donaldson–Segal programme [11], which proposes defining enumerative invariants of G2-manifolds (X, φ, g) by counting moduli spaces MG2, with signs depending on a choice of orientation.

AB - Suppose (X,g) is a compact, spin Riemannian 7-manifold, with Dirac operator D/g : C∞(X,S/) → C∞(X,S/). Let G be SU(m) or U(m), and E → X be a rank m complex bundle with G-structure. Write BE for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z2-bundle OD/g → BE parametrizing orientations of E detD/gAdA for twisted elliptic operators D/gAdA at each [A] in BE. A theorem of Walpuski [33] shows OD/g is trivializable. We prove that if we choose an orientation for detD/g, and a flag structure on X in the sense of [17], then we can define canonical trivializations of OD/g E for all such bundles E → X, satisfying natural compatibilities. Now let (X, φ, g) be a compact G2-manifold, with d(∗φ) = 0. Then we can consider moduli spaces MG2 of G2-instantons on E → X, which are smooth E manifolds under suitable transversality conditions, and derived manifolds in general, with MG2 ⊂ BE. The restriction of OD/g to MG2 is the Z2-bundle of orientations on MG2 . Thus, our theorem induces canonical orientations on E all such G2-instanton moduli spaces MG2 . E This contributes to the Donaldson–Segal programme [11], which proposes defining enumerative invariants of G2-manifolds (X, φ, g) by counting moduli spaces MG2, with signs depending on a choice of orientation.

M3 - Article

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

ER -