Wavefunction coefficients from amplitubes

Given a graph its set of connected subgraphs (tubes) can be defined in two ways: either by considering subsets of edges, or by considering subsets of vertices. We refer to these as binary tubes and unary tubes respectively. Both notions come with a natural compatibility condition between tubes which differ by a simple adjacency constraint. Compatible sets of tubes are referred to as tubings. By considering the set of binary tubes, and summing over all maximal binary-tubings, one is led to an expression for the flat-space wavefunction coefficients relevant for computing cosmological correlators. On the other hand, considering the set of unary tubes, and summing over all maximal unary-tubings, one is led to expressions recently referred to as amplitubes which resemble the scattering amplitudes of tr(ϕ3) theory. Due to the similarity between these constructions it is natural to expect a close connection between the wavefunction coefficients and amplitubes. In this paper we study the two definitions of tubing in order to provide a new formula for the flat-space wavefunction coefficient for a single graph as a sum over products of amplitubes. We also show how the expressions for the amplitubes can naturally be understood as a sum over orientations of the underlying graph. Combining these observations we are lead to an expression for the wavefunction coefficient given by a sum over terms we refer to as decorated amplitubes which matches a recently conjectured formula resulting from partial fractions. Motivated by our rewriting of the wavefunction coefficient we introduce a new definition of tubing which makes use of both the binary and unary tubes which we refer to as cut tubings. We explain how each cut tubing induces a decorated orientation of the underlying graph satisfying an acyclic condition and demonstrate how the set of all acyclic decorated orientations for a given graph count the number of basis functions appearing in the kinematic flow.