The Higher Riemann-Hilbert Correspondence and Multiholomorphic Mappings

This thesis consists in two chapters. In the first part we describe an $A_\infty$-quasi-equivalence of dg-categories between Block's $\Perf$, corresponding to the de Rham dga $\As$ of a compact manifold $M$ and the dg-category of infinity-local systems on $M$. We understand this as a generalization of the Riemann-Hilbert correspondence to $\Z$-graded connections (superconnections in some circles). In one formulation an infinity-local system is an $(\infty,1)$-functor between the $(\infty,1)$-categories ${\pi}_{\infty}M$ and a repackaging of the dg-category of cochain complexes by virtue of the simplicial nerve and Dold-Kan. This theory makes crucial use of Igusa's notion of higher holonomy transport for $\Z$-graded connections which is a derivative of Chen's main idea of generalized holonomy. In the appendix we describe some alternate perspectives on these ideas and some technical observations. The second chapter is concerned with the development of the theory of \emph{multiholomorphic maps}. This is a generalization in a particular direction of the theory of pseudoholomorphic curves. We first present the geometric framework of compatible $n$-triads, from which follows naturally the definition of a multiholomorphic mapping. We develop some of the essential analytic and differential-geometric facts about these maps in general, and then focus on a special case of the theory which pertains to the calibrated geometry of $G_2$-manifolds. This work builds toward the realization of invariants generated by the topology of the moduli spaces of multiholomorphic maps. Because this study is relatively fundamental, there will be many instances where questions/conjectures are put forward, or directions of further research are described.