On Feynman graphs, matroids, and GKZ-systems

@article{Walther2022OnFG,
  title={On Feynman graphs, matroids, and GKZ-systems},
  author={Uli Walther},
  journal={Letters in Mathematical Physics},
  year={2022},
  volume={112}
}
  • U. Walther
  • Published 11 June 2022
  • Mathematics
  • Letters in Mathematical Physics
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. 

Feynman Integral Relations from GKZ Hypergeometric Systems

  • Henrik J. Munch
  • Mathematics
    Proceedings of Loops and Legs in Quantum Field Theory — PoS(LL2022)
  • 2022
We study Feynman integrals in the framework of Gel’fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems. The latter defines a class of functions wherein Feynman integrals arise as special cases, for

FeynGKZ: a Mathematica package for solving Feynman integrals using GKZ hypergeometric systems

In the Lee-Pomeransky representation, Feynman integrals can be identified as a sub-set of Euler-Mellin integrals, which are known to satisfy Gel ' fand-Kapranov-Zelevinsky (GKZ) system of partial

References

SHOWING 1-10 OF 29 REFERENCES

Graph hypersurfaces with torus action and a conjecture of Aluffi

Generalizing the star graphs of Muller-Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that

Cohen-Macaulay Property of Feynman Integrals

: The connection between Feynman integrals and GKZ A -hypergeometric systems has been a topic of recent interest with advances in mathematical techniques and computational tools opening new

Algebraic aspects of hypergeometric differential equations

We review some classical and modern aspects of hypergeometric differential equations, including A -hypergeometric systems of Gel $$'$$ ′ fand, Graev, Kapranov and Zelevinsky. Some recent advances in

Matroid connectivity and singularities of configuration hypersurfaces

This work shows that configuration polynomials, forms and schemes are reduced and describes the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3.

Feynman integrals as A-hypergeometric functions

  • L. Cruz
  • Mathematics
    Journal of High Energy Physics
  • 2019
Abstract We show that the Lee-Pomeransky parametric representation of Feynman integrals can be understood as a solution of a certain Gel’fand-Kapranov-Zelevinsky (GKZ) system. In order to define

Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems

We show that almost all Feynman integrals as well as their coefficients in a Laurent series in dimensional regularization can be written in terms of Horn hypergeometric functions. By applying the

On Motives Associated to Graph Polynomials

The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization

On the Singular Structure of Graph Hypersurfaces

We show that the singular loci of graph hypersurfaces correspond set-theoretically to their rank loci. The proof holds for all configuration hypersurfaces and depends only on linear algebra. To make

Cayley sums and Minkowski sums of $2$-convex-normal lattice polytopes

In the present paper, we consider the integer decomposition property for Minkowski sums and Cayley sums. In particular, we focus on these constructions arising from $2$-convex-normal lattice