On spin structures and orientations for gauge-theoretic moduli spaces

Abstract

Let X be a compact manifold, G a Lie group, P→X a principal G-bundle, and BP the infinite-dimensional moduli space of connections on P modulo gauge, as a topological stack. For a real elliptic operator E• we previously studied orientations on the real determinant line bundle over BP, twisting E• by connections ∇Ad(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 [15–17].Here we consider complex elliptic operators F• and introduce the idea of spin structures, square roots of the complex determinant line bundle of F• twisted over BP. 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×S1. Thus, if P→X and Q→X×S1 are principal G-bundles with Q|X×{1}≅P, we relate spin structures on (BP,F•) to orientations on (BQ,E•) for a certain class of operators F• on X and E• on X×S1.Combined with [25], we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups G=U(m),SU(m). In a sequel [26] we will apply this to define canonical orientation data for all Calabi–Yau 3-folds X over C, as in Kontsevich and Soibelman [28, §5.2], solving a long-standing problem in Donaldson–Thomas theory.