Skip to main content
SpringerLink
Log in

On Feynman graphs, matroids, and GKZ-systems

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We show in several important cases that the A-hypergeometric system attached to a Feynman diagram in Lee–Pomeransky form, obtained by viewing the coefficients of the integrand as indeterminates, has a normal underlying semigroup. This continues a quest initiated by Klausen and studied by Helmer and Tellander. In the process, we identify several relevant matroids related to the situation and explore their relationships.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

This manuscript has no associated data.

Notes

  1. We will typically use E and reserve \(E_G\) for cases where extra clarity is needed, for example when several graphs are around.

  2. Since the total momentum sum is zero, both 2-forest components give the same coefficient.

  3. A more general class of polynomials arises from realizations of matroids, see for example [1, 4, 5, 16].

References

  1. Bloch, S., Esnault, H., Kreimer, D.: On motives associated to graph polynomials. Commun. Math. Phys. 267(1), 181–225 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. de Alfaro, V., Jakšić, B., Regge, T.: Differential Properties of Feynman Amplitudes, Lecture Semester on High-Energy Physics and Elementary Particles (Trieste, 1965), pp. 263–272. International Atomic Energy Agency, Vienna (1965)

    Google Scholar 

  3. de la Cruz, L.: Feynman integrals as A-hypergeometric functions. J. High Energy Phys. 2019(12), 1–45 (2019)

    MathSciNet  MATH  Google Scholar 

  4. Denham, G., Pol, D., Schulze, M., Walther, U.: Graph hypersurfaces with torus action and a conjecture of Aluffi. Commun. Number Theory Phys. 15(3), 455–488 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Denham, G., Schulze, M., Walther, U.: Matroid connectivity and singularities of configuration hypersurfaces. Lett. Math. Phys. 111(1), Paper No. 11, p. 67 (2021)

  6. Gel’fand, I.M., Goresky, R.M., MacPherson, R.D., Serganova, V.V.: Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63(3), 301–316 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Golubeva, V.A.: The family of partial differential equations for an analytic function of Feynman type. Dokl. Akad. Nauk SSSR 212, 71–74 (1973)

    MathSciNet  Google Scholar 

  8. Hochster, M.: Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes. Ann. Math. (2) 96, 318–337 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Howard, B.J.: Edge unimodular polytopes. In: Maclagan, D., Haase, C., Hibi, T. (eds.) Mini-Workshop:Projective Normality of Smooth Toric Varieties. Oberwolfach Reports, no. Report No. 39/2007, pp. 2291–2293 (2007)

  10. Howard, B.J.: Matroids and geometric invariant theory of torus actions on flag spaces. J. Algebra 312(1), 527–541 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Helmer, M., Tellander, F.: Cohen–Macaulay property of Feynman integrals. Commun. Math. Phys. (2022) (to appear)

  12. Klausen, R.P.: Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems. J. High Energy Phys. 121(4), 41 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Klausen, R.P.: Kinematic singularities of Feynman integrals and principal A-determinants. J. High Energy Phys. (2), Paper No. 004, p. 44 (2022)

  14. Matusevich, L., Miller, E., Walther, U.: Homological methods for hypergeometric families. J. Am. Math. Soc. 18(4), 919–941 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Oxley, J.: Matroid Theory. Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. Oxford University Press, Oxford (2011)

    Google Scholar 

  16. Patterson, E.: On the singular structure of graph hypersurfaces. Commun. Number Theory Phys. 4(4), 659–708 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reichelt, T., Schulze, M., Sevenheck, C., Walther, U.: Algebraic aspects of hypergeometric differential equations. Beitr. Algebra Geom. 62(1), 137–203 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schultka, K.: Toric geometry and regularization of Feynman integrals (2018). arXiv:1806.01086

  19. Saito, M., Sturmfels, B., Takayama, N.: Gröbner Deformations of Hypergeometric Differential Equations. Algorithms and Computation in Mathematics, vol. 6. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  20. Tsuchiya, A.: Cayley sums and Minkowski sums of 2-convex normal lattice polytopes (2019). arXiv:1804.10538v3

  21. White, N.L.: The basis monomial ring of a matroid. Adv. Math. 24(3), 292–297 (1977)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I am much indebted to René-Pascal Klausen for enlightening discussions on Feynman amplitudes, Hypothesis 1.2 and QFTs. Both he and Mathias Schulze provided valuable criticism on earlier versions of this article. My sincere thanks go to Martin Helmer and Felix Tellander for writing their article and sharing their insights. I am also grateful to Diane MacLagan, Christian Haase and Karen Yeats for helpful explanations on polytopal yoga. Praise goes to the referees for careful reading and suggestions for improvement.

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN, 47907, USA

    Uli Walther

Authors
  1. Uli Walther
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Uli Walther.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

UW was supported by NSF Grant DMS-2100288 and by Simons Foundation Collaboration Grant for Mathematicians #580839.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Walther, U. On Feynman graphs, matroids, and GKZ-systems. Lett Math Phys 112, 120 (2022). https://doi.org/10.1007/s11005-022-01614-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-022-01614-2

Keywords

Mathematics Subject Classification

Search

Navigation