$D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals

Abstract

Feynman integrals are solutions to linear partial differential equations with polynomial coefficients. Using a triangle integral with general exponents as a case in point, we compare D-module methods to dedicated methods developed for solving differential equations appearing in the context of Feynman integrals, and provide a dictionary of the relevant concepts. In particular, we implement an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series solutions of regular holonomic D-ideals, and compare them to asymptotic series derived by the respective Fuchsian systems.

Four lectures on Euler integrals

“…In fact, μ| X+ : X+ → int( P(s)) is a diffeomorphism. We now relate this to the moment map µ in (39). To identify the domains of µ and μ, we introduce the map κ…”

“…Sketch of proof of Theorem* 3.24. Consider the algebraic moment map in (39) with s i = 1. By Lemma 2.7, µ : n + → int(P) is a diffeomorphism, with P = ℓ i=1 ∆( f i ).…”

“…We revisit the Euler integral (7) of Example 0.1. The differential equations satisfied by this integral as a function of t 1 , t 2 , t 3 are the running example of [39]. In this case, they can be derived in an easy way from the GKZ differential equations for…”

Restrictions of Pfaffian systems for Feynman integrals

“…We use terminology on D-modules, such as regular holonomic, which is explained in the standard textbooks [22] and [57]. See [1, appendix B] for a short introduction to holonomic D-modules (for a pedagogical presentation which is tailored to Feynman integrals, see also the recent article [52]). Appendix A in the present manuscript gives an introduction to tensor products of modules, as they are used extensively throughout this text.…”

Holonomic representation of biadjoint scalar amplitudes

“…[10][11][12][13][14][15][16][17] (see also refs. [18][19][20][21][22][23][24]). The ideals associated with Feynman integrals are holonomic and thus Feynman integrals are holonomic functions [25].…”