Differential equations for correlation functions in cosmology are studied in [ABHJLP]. Therein, a basis of master integrals is constructed from the underlying hyperplane arrangement via canonical forms in the setup of positive geometries, which results in a matrix differential equation for the cosmological correlator that is in $\varepsilon$-factorized form. For the $2$-site chain (mathematically, the path graph on $2$ vertices), the underlying $D$-ideal was investigated in [FPSW].
Here we revisit the $D$-ideal $I = \langle \nabla_1+\nabla_3,\nabla_2+\nabla_3,H\rangle$ (see Equation (11) in [FPSW]), and carry out the gauge transformation to write the connection matrices in $\varepsilon$-factorized form. This form is especially useful, as it allows for the construction of formal power series solutions in the variable $\varepsilon$ of such systems via the ``path-ordered exponential formalism.’’
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First, we check that the system has finite holonomic rank using holonomicRank.
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Then, we compute the system in connection form and verify that it meets the integrability conditions.
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Next, we use gauge matrix for changing base to a base given by suitable set of standard monomials, and compute the gauge transform with respect to this gauge matrix.
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Now we are ready to perform the gauge transform from this basis to the $\epsilon$-factorized form.
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Finally, we verify that only the last system is in the $\epsilon$-factorized form using isEpsilonFactorized.
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[FPSW] C. Fevola, G. L. Pimentel, A.-L. Sattelberger, and T. Westerdijk. Algebraic Approaches to Cosmological Integrals. Preprint arXiv:2410.14757. To appear in Le Matematiche.
[ABHJLP] N. Arkani-Hamed, D. Baumann, A. Hillman, A. Joyce, H. Lee, and G. L. Pimentel. Differential Equations for Cosmological Correlators. Preprint arXiv:2312.05303.
The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/ConnectionMatrices/examples.m2:71:0.