The Surface Signature and Rough Surfaces

Parallel transport, or path development, provides a rich characterization of paths which preserves the underlying algebraic structure of concatenation. The path signature is universal among such maps: any (translation-invariant) parallel transport factors uniquely through the path signature. Furthermore, the path signature is a central object in the theory of rough paths, which provides an integration theory for highly irregular paths. A fundamental result is Lyons' extension theorem, which allows us to compute the signature of rough paths, and in turn provides a way to compute parallel transport of arbitrarily irregular paths. In this article, we consider the notion of surface holonomy, a generalization of parallel transport to the higher dimensional setting of surfaces parametrized by rectangular domains, which preserves the higher algebraic structures of horizontal and vertical concatenation. Building on work of Kapranov, we introduce the surface signature, which is universal among surface holonomy maps with respect to continuous 2-connections. Furthermore, we introduce the notion of a rough surface and prove a surface extension theorem, which allows us to compute the signature of rough surfaces. By exploiting the universal property of the surface signature, this provides a method to compute the surface holonomy of arbitrarily irregular surfaces.