Interactions of Derived Moduli Spaces and Gerbes with Elliptic Genera in Complex Geometry
Final Report Abstract
A connection A on a principal G-bundle P → X over a G2 -manifold (X, φ, ∗φ), where φ is a positive 3-form defining the G2-structure, is called a G2 -instanton if ∗FA ∧φ = −FA . Up to gauge, this is an elliptic non-linear differential equation whose solutions form the moduli space MP of G2 -instantons. They are analogous to the moduli space of flat connections on a 3-manifold, leading to a conjectural Casson invariant in the Donaldson–Segal programme that counts the number of G2-instantons. In order to get a well-behaved invariant, it is necessary to count each solution with an appropriate sign. This is because, under deformation of the structure, G2 -instantons may ‘bubble’ and degenerate into an associative submanifold. To balance this behaviour it is essential to have a precise understanding of the way signs are associated to G2 -instantons, in particular when the principal bundle P → X varies. The present project addressed this question by constructing orientations on all MP for G = U(m), SU(m) canonically from differential-topological data on X, a so-called flag structure. To do this, a categorified version of a powerful index theory technique, the excision principle, was established. Using it, some new results on orientations for more classical moduli space were also obtained and rediscovered. These include moduli spaces of flat connections on two- and three-manifolds, ASD-instantons, the Kapustin-Witten equations, the Vafa-Witten equations on four-manifolds, and the Haydys-Witten equations on five-manifolds.
Publications
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Canonical orientations for moduli spaces of G2-instantons with gauge group SU(m) or U(m)
D. Joyce and M. Upmeier
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A categorified excision principle for elliptic symbol families
M. Upmeier
