Higher Holonomy for Higher Lie Algebroid Connections on Graded Vector Bundles

Abstract

We prove the existence of a Riemann--Hilbert $A_\infty$-functor from the curved dg-category of $\mathscr{A}$-connections on graded vector bundles, where $\mathscr{A}$ is a higher Lie algebroid, to the curved dg-category of $Π_\infty(\mathscr{A})$-connections on graded vector bundles, where $Π_\infty(\mathscr{A})$ is the homotopy groupoid of $\mathscr{A}$. When restricted to the sub dg-categories of flat objects, it induces an $A_\infty$-functor from the dg-category of representations up to homotopy of $\mathscr{A}$ to the dg-category of representations up to homotopy of $Π_\infty(\mathscr{A})$, which recovers the one from arXiv:1011.4693 and arXiv:0908.2843 when $\mathscr{A}$ is a Lie algebroid. The key ingredient for defining such functors is a non-abelian Stokes's theorem for endomorphism-valued differential forms. Furthermore, for a graded vector bundle $E$, we introduce the notion of its higher general linear groupoid in such a way that $G$-connections on $E$, for a higher Lie groupoid $G$, become smooth functors from $G$ to the higher general linear groupoid of $E$. We combine this with the Riemann--Hilbert functor to define higher parallel transport functors for $\mathscr{A}$-connections. Finally, we use these functors to define the higher holonomy groups of $\mathscr{A}$-connections and endow them with the structure of augmented simplicial Lie groups.