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Algebraic Approaches to Cosmological Integrals

Claudia Fevola, Guilherme L. Pimentel, Anna-Laura Sattelberger, Tom Westerdijk

TL;DR

The paper addresses the problem of understanding cosmological integrals, modeled as Mellin-type integrals of the flat-space wavefunction, through algebraic structures such as the Weyl algebra, GKZ systems, and Mellin transforms. It develops a cohesive framework combining differential and shift techniques to derive equations satisfied by cosmological correlators, and demonstrates a graph-based multivariate partial fraction decomposition of the flat-space wavefunction. In the two-site chain example, the authors show that their differential equations arise from a restricted GKZ system, construct shift relations and canonical forms, and conjecture a general, graph-driven decomposition that encodes singularity structure via Euler discriminants. Overall, the work provides a bridge between D-module theory and cosmological calculations, offering a toolkit for systematic analysis of cosmological correlators and their singularities.

Abstract

Cosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools, we shed light on the differential and difference equations satisfied by these integrals. Moreover, we study a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition.

Algebraic Approaches to Cosmological Integrals

TL;DR

The paper addresses the problem of understanding cosmological integrals, modeled as Mellin-type integrals of the flat-space wavefunction, through algebraic structures such as the Weyl algebra, GKZ systems, and Mellin transforms. It develops a cohesive framework combining differential and shift techniques to derive equations satisfied by cosmological correlators, and demonstrates a graph-based multivariate partial fraction decomposition of the flat-space wavefunction. In the two-site chain example, the authors show that their differential equations arise from a restricted GKZ system, construct shift relations and canonical forms, and conjecture a general, graph-driven decomposition that encodes singularity structure via Euler discriminants. Overall, the work provides a bridge between D-module theory and cosmological calculations, offering a toolkit for systematic analysis of cosmological correlators and their singularities.

Abstract

Cosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools, we shed light on the differential and difference equations satisfied by these integrals. Moreover, we study a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition.

Paper Structure

This paper contains 14 sections, 3 theorems, 48 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical tools
  3. The Weyl algebra
  4. Connection matrices and gauge transformation
  5. Restrictions and GKZ systems
  6. Flat space wavefunctions from graphs
  7. Shift-relations
  8. Use in cosmology
  9. Two-site chain
  10. Shift relations
  11. GKZ system
  12. Canonical forms
  13. Partial fractioning of the flat space wavefunction
  14. Conclusion

Key Result

lemma 1

Let $p\in \mathbb{C}[\alpha_1,\ldots,\alpha_n]$ be a polynomial. Let $I$ be the $D_n$-ideal generated by $p\partial_{\alpha_i}+\frac{\partial p}{\partial \alpha_i}$, $i=1,\ldots,n$. Then the full annihilating $D$-ideal of $1/p$ equals the Weyl closure of $I$. As a formula, $\mathop{\mathrm{Ann}}\nol

Figures (5)

  • Figure 1: A single and a double exchange process, and the $4$-site star graph.
  • Figure 2: A real picture of the hyperplane arrangement $(L_1,L_2,L_3)$ for the two-site graph, here depicted for generic $X_1,X_2,Y$ with $X_1,X_2>|Y|$. Together with the coordinate axes, it encloses four bounded regions. To the triangle with label $i$, we have an associated differential operator $Q_i$ as resulting from \ref{['eq:gauge']}.
  • Figure 3: Some examples of graphs with $R_{H_i}=0$.
  • Figure 4: All oriented spanning trees for the $3$-site chain. The numbers represent the degree of each vertex. The one in gray is the only one which is not totally time-ordered in the sense of \ref{['def:tto']}.
  • Figure 5: The oriented spanning subgraphs for the one-loop bubble graph.

Theorems & Definitions (16)

  • Remark 2.1
  • Definition 2.2
  • lemma 1
  • lemma 2
  • proof
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • Definition 3.4
  • Conjecture 3.5
  • ...and 6 more