From non-unitary wheeled PROPs to smooth amplitudes and generalised convolutions

TL;DR

This paper introduces TRAPs, a new algebraic structure generalizing wheeled PROPs, and demonstrates their applications in constructing smooth amplitudes and generalized convolutions on manifolds.

Contribution

It defines TRAPs as wheeled PROP-like structures without units, constructs free TRAPs from graphs, and shows how they can be completed to wheeled PROPs with applications in smooth kernels.

Findings

TRAPs can be completed to wheeled PROPs.

Graph-based TRAPs facilitate the construction of smooth amplitudes.

Generalized convolutions arise from vertical concatenation in TRAPs.

Abstract

We introduce the concept of TRAP (Traces and Permutations), which can roughly be viewed as a wheeled PROP (Products and Permutations) without unit. TRAPs are equipped with a horizontal concatenation and partial trace maps. Continuous morphisms on an infinite dimensional topological space and smooth kernels (resp. smoothing operators) on a closed manifold form a TRAP but not a wheeled PROP. We build the free objects in the category of TRAPs as TRAPs of graphs and show that a TRAP can be completed to a unitary TRAP (or wheeled PROP). We further show that it can be equipped with a vertical concatenation, which on the TRAP of linear homomorphisms of a vector space, amounts to the usual composition. The vertical concatenation in the TRAP of smooth kernels gives rise to generalised convolutions. Graphs whose vertices are decorated by smooth kernels (resp. smoothing operators) on a closed manifold form a TRAP. From their universal properties we build smooth amplitudes associated with the graph.