Wavefunction coefficients from Amplitubes (2503.13596v2)

Abstract: 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 refered to as tubings. By considering the set of binary tubes, and summing over all maximal binary-tubings, one is lead 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 lead to expressions recently referred to as amplitubes which resemble the scattering amplitudes of $\text{tr}(\phi^3)$ theory. 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. 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.

Essay: Wavefunction Coefficients from Amplitubes

The paper "Wavefunction coefficients from Amplitubes" by Ross Glew explores the combinatorial structures underlying wavefunction coefficients and amplitubes, providing insights into their interconnectedness. It explores the implications of these constructs for cosmological correlators and scattering amplitudes, drawing connections between graph theory and particle physics.

Binary and Unary Tubes

At the core of the paper is the distinction between binary and unary tubes. Binary tubes are defined based on subsets of a graph's edges, leading to the computation of flat space wavefunction coefficients, denoted as $\Psi_G$. Unary tubes, conversely, focus on subsets of vertices and result in amplitubes, $A_G$, resembling scattering amplitudes in $\tr(\phi^3)$ theory. Despite differing definitions, the paper demonstrates a connection between these concepts through combinatorial graph structures.

Binary tubes form the basis of computing wavefunction coefficients by examining connected subgraphs' compatibility. A set of compatible binary tubes constitutes a tubing, and maximal binary tubings are pivotal in deriving expressions for $\Psi_G$. The paper elaborates on the mathematical framework, utilizing linear functions defined on graphs' vertices and edges to derive the wavefunction coefficients.

On the unary side, the paper explores how subgraphs formed by vertex sets can lead to amplitubes. By considering maximal unary tubings, expressions mimicking canonical forms associated with positive geometries, such as graph associahedra, are obtained.

Connection Between Wavefunctions and Amplitubes

The author proposes that the expressions for wavefunction coefficients can be decomposed into sums of amplitubes, revealing a shared underlying mathematical structure. This is achieved by considering graph cuts—subsets of edges—and summing over terms related to connected components in these modified graphs.

The connection is nuanced and involves understanding graph orientations induced by tubings. These orientations provide a decomposition of amplitubes into sums over valid graph orientations. Such decompositions are framed within the context of decorated amplitubes, where edges can be assigned distinct orientations or depicted as broken edges.

Implications and Future Directions

This paper provides a novel perspective on the relationship between graph theory constructs used in physics. The link between $\Psi_G$ and $A_G$ paves the way for new interpretations of particle interactions and the structures governing wavefunction coefficients.

Implications for cosmological correlators suggest potential advances in computational techniques used in particle physics, particularly in evaluating wavefunctions within specific theories. The paper's findings might influence future methodologies in handling complex graph-based systems in theoretical physics, encouraging exploration of higher-dimensional analogs and more intricate connectivity conditions.

Future research could focus on leverage the insights gathered here to extend these combinatorial frameworks to more complex graph structures, potentially improving understanding in quantum field theory or cosmological settings. Moreover, exploring the role of decorated orientations within broader graph theory domains could inspire new directions in both mathematics and physics.

In summary, Ross Glew's paper presents a significant advancement in comprehending the fundamental relations between graph constructs and their physical interpretations. This work suggests avenues for further research and computational strategies, contributing to the ongoing development of theoretical physics and related mathematical frameworks.