Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 43 min ago
PreprintCC BY 4.0arXiv

Magic Relations and Critical Varieties of Feynman Integrals

  • Giulio Crisanti
  • Hjalte Frellesvig
  • Andrzej Pokraka
  • Sid Smith
Open at sourcePDFDOICite

Abstract

Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break down, they are important to detect and understand. In this paper, we take a first step toward a systematic characterization of such identities. Specifically, we observe and argue that the occurrence of magic relations always coincides with the presence of higher-dimensional critical varieties in the generating sector. This provides a practical computational test to check if a family of Feynman integrals can contain magic relations and to find them, which we implement in the ancillary Mathematica file Magic-Test.m. Additionally, we discuss how to count the number of master integrals in the presence of higher-dimensional critical varieties, classify the behavior of magic relations under symmetries, and we discuss their interplay with cuts.

Related

  1. Canonical differential equations for Feynman integrals from 𝒜-hypergeometric systems in the Schwinger representation
    Mateo Jimenez-Santacruz, Cristhiam Lopez-Arcos, Alexander Quintero Velez
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryMathematical physicsarXiv
  2. All-multiplicity monodromy and KLT relations for AdS string integrals
    Maria Nocchi, Rodrigo Schmidt Pitombo, Aurélie Strömholm Sangaré +1
    arXivPreprintCC BY 4.0Full textMathematical physicsHigh-energy & particle physicsarXiv
  3. Hypercube Richardson varieties from magic words
    Balázs Elek
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  4. Minuscule Relations in Quantum K-Theory of Flag Varieties
    Koushik Brahma, Yicen Huang, Takeshi Ikeda +2
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Feynman Propagators on Exponentially Decaying Perturbations of Stationary Spacetimes with Compact Cauchy Surface
    Selim Amar
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{crisanti2026magic,
  title = {{Magic Relations and Critical Varieties of Feynman Integrals}},
  author = {Giulio Crisanti and Hjalte Frellesvig and Andrzej Pokraka and Sid Smith},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.29789},
  eprint = {2605.29789},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.29789},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Magic Relations and Critical Varieties of Feynman Integrals
AU  - Giulio Crisanti
AU  - Hjalte Frellesvig
AU  - Andrzej Pokraka
AU  - Sid Smith
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2605.29789
UR  - https://arxiv.org/abs/2605.29789
AB  - Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break down, they are important to detect and understand. In this paper, we take a first step toward a systematic characterization of such identities. Specifically, we observe and argue that the occurrence of magic relations always coincides with the presence of higher-dimensional critical varieties in the generating sector. This provides a practical computational test to check if a family of Feynman integrals can contain magic relations and to find them, which we implement in the ancillary Mathematica file Magic-Test.m. Additionally, we discuss how to count the number of master integrals in the presence of higher-dimensional critical varieties, classify the behavior of magic relations under symmetries, and we discuss their interplay with cuts.
ER  -