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.

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory.

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We develop a categorical index calculus for elliptic symbol families. The categorified index problems we consider are a secondary version of the traditional problem of expressing the index class in K-theory in terms of differential-topological data. They include orientation problems for moduli spaces as well as similar problems for skew-adjoint and self-adjoint operators. The main result of this paper is an excision principle which allows the comparison of categorified index problems on different manifolds. Excision is a powerful technique for actually solving the orientation problem; applications appear in the companion papers Joyce-Tanaka-Upmeier [17], Joyce-Upmeier [18], and Cao-Joyce [8].

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