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MPL—A program for computations with iterated integrals on moduli spaces of curves of genus zero

2016; Elsevier BV; Volume: 203; Linguagem: Inglês

10.1016/j.cpc.2016.02.033

1879-2944

Christian Bogner,

Mathematics, Computing, and Information Processing

We introduce the Maple program MPL for computations with multiple polylogarithms. The program is based on homotopy invariant iterated integrals on moduli spaces M0,n of curves of genus 0 with n ordered marked points. It includes the symbol map and procedures for the analytic computation of period integrals on M0,n. It supports the automated computation of a certain class of Feynman integrals. Program title: MPL Catalogue identifier: AFAE_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFAE_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU GPLv3 No. of lines in distributed program, including test data, etc.: 9477 No. of bytes in distributed program, including test data, etc.: 247861 Distribution format: tar.gz Programming language: Maple [1], version 16. Computer: Any computer supporting Maple. Operating system: Any system supporting Maple. Has the code been vectorized or parallelized?: Maple supports parallel computing for many but not for all of the commands used in the program. RAM: Highly problem dependent Classification: 4.4, 5. Nature of problem: The program serves for computations with a class of iterated integrals, defined on moduli spaces of curves of genus zero. It allows for the automated computation of period integrals of these spaces. Furthermore, based on these iterated integrals, it supports the automated, analytic computation of a certain class of Feynman integrals of perturbative quantum field theory. Solution method: The program computes with iterated integrals in terms of sequences of differential 1-forms. Feynman integrals are expressed in terms of coordinates of appropriate moduli spaces, such that the problem of their computation is reduced to the integration over members of the class of iterated integrals in these coordinates. Restrictions: The given Feynman integrals have to admit linear reducibility, unramifiedness and properly ordered polynomials at the tangential basepoint. These conditions are discussed in detail in Section 4. Unusual features: All iterated integrals are viewed as integrable words (or symbols). All computations are analytic and no numerical approximations are made. Additional comments: The program is obtained in one file but can be seen as divided into two parts. The first part is dedicated to computations with the mentioned class of iterated integrals on moduli spaces of curves of genus zero. This part alone already serves for the computation of a certain class of integrals, which may appear in various contexts. The second part is dedicated to the application to Feynman integrals and reduces the problem to computations which are accessible by the first part. An example Maple-worksheet UserManualExamples.mw and a supplementary user manual mmc1.pdf, commenting the examples of the worksheet are included (see Appendix C). Running time: The examples given in UserManualExamples.mw run in less than two minutes on a standard PC, but much longer computation times are expected for more complicated Feynman integrals. The running time strongly depends on the parametrization of the integrand. If the integrand is accessible by the part of the program discussed in Section 3, the integrations will be considerably faster than in cases, where the additional procedures of the part discussed in Section 4 are required. References: Maple 16. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.