On 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 [14-16].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 × S 1 . Thus, if P → X and Q → X × S 1 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 × S 1 .Combined with [24], we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups G = U(m), SU(m). In a sequel [25] we will apply this to define canonical orientation data for all Calabi-Yau 3-folds X over C, as in Kontsevich and Soibelman [27, §5.2], solving a long-standing problem in Donaldson-Thomas theory.

Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds

“…Here is our first main result. It will be proved in §2 using a wide range of ideas and techniques, including much of the general theory of orientations in [32,33,57], some surgery theory, some special geometry of SU(4), and the classification of compact simply-connected 5-manifolds in Crowley [11]. Theorem 1.11.…”

“…The main reason we do not do this in [32] is that to relate orientations on different moduli spaces we consider direct sums of connections, which give a morphism Φ : B P × B Q → B P ⊕Q , but this and similar morphisms do not make sense for the spaces B irr P , B P , B irr P , so we prefer to work with the B P . We define orientation bundles O E• P ,Ō E• P on the moduli spaces B P , B P :…”

“…In gauge theory one studies moduli spaces M ga P of (irreducible) connections ∇ P on a principal bundle P → X (perhaps plus some extra data, such as a Higgs field) satisfying a curvature condition. Under suitable genericity conditions, these moduli spaces M ga P will be smooth manifolds, and the ideas of [32] can often be used to prove M ga P is orientable, and construct a canonical orientation on M ga P . These orientations are important in defining enumerative invariants such as Casson invariants, Donaldson invariants, and Seiberg-Witten invariants.…”

“…We will do this in the following steps: Step 1. Use results of Joyce, Tanaka and Upmeier [32] to show that B Q is orientable for any principal U(m)-or SU(m)-bundle Q → X, and also C α is orientable for all α ∈ K 0 (X), if and only if B P is orientable when P = X ×SU(4) is the trivial SU(4)-bundle over X.…”

“…Applying this to semistable derived moduli schemes (M ss α , ω) (at least in the case when semistable=stable), using the orientations this paper provides, and pairing the virtual cycle [M ss α ] virt with certain coho-mology classes on M ss α , should yield the proposed DT4 invariants. Sections 1.1-1.4 summarize background material on the general theory of orientations in gauge theory from Joyce, Tanaka and Upmeier [32], and on Spin(7)-manifolds and Spin(7)-instantons, and on Calabi-Yau m-folds and moduli spaces of (complexes of) coherent sheaves. The main results are stated in §1.5, and the proofs of Theorems 1.11 and 1.15 are given in §2 and §3.…”