Skip to content
DraconiticResearch
Search Sources API
201,818 studies31 sourcesUpdated 24 min ago
PreprintCC BY 4.0arXiv

From MHV Gravity Amplitudes and Hodges map to Pentad Varieties

  • Joris Koefler
  • Emanuele Ventura
  • Hongmiao Yu
Open at sourcePDFDOICite

Abstract

We study some geometric aspects behind the Hodges matrix appearing in the context of scattering amplitudes for the 𝒩=8 supergravity. Through a vector bundle construction, we show that the Hodges matrix gives rise to a canonical rational map from the two-step flag variety to symmetric matrices, called the Hodges map. This construction allows to define the amplitude geometrically: its vanishing locus is a divisor - the MHV gravity divisor. Inspired by the Hodges map, we introduce the notion of Hadamard ratio of two varieties. In the case relevant to gravity, this is the ratio of two Grassmannians, which we call the pentad variety, connecting it to the pentad constraint from algebraic statistics. Moreover, we show that the pentad variety is closely related to the combinatorial algebraic geometry of Martin's affine slope variety, and the phirotopes of Below, Krummeck, and Richter-Gebert.

Related

  1. Segre determinants and MHV gravity amplitudes
    Joris Koefler, Emanuele Ventura, Hongmiao Yu
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryHigh-energy & particle physicsarXiv
  2. Subleading Collinear Limits of Yang-Mills Amplitudes from Gravity
    Jin Dong, Stephan Stieberger
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  3. Scattering Equations as the lowest order K-identities in the calculation of Stringy Scaling of Hard String Scattering Amplitudes
    Sheng-Hong Lai, Jen-Chi Lee, Yi Yang
    arXivPreprintCC0Full textMathematical physicsHigh-energy & particle physicsarXiv
  4. The Disordered Universe 2025: Analytic Properties of Amplitudes and Correlators
    Mattia Capuano, Chandramouli Chowdhury, Chantelle Esper +6
    arXivPreprintarXiv licenceFull textMathematical physicsGravitation & general relativityarXiv
  5. Six Particles, Infinite Optimism: Towards Positivity Bounds for Six-Point Amplitudes
    Stefan Kremminger, Andrew J. Tolley
    arXivPreprintarXiv licenceFull textMathematical physicsGravitation & general relativityarXiv

Cite

BibTeXRIS
BibTeX
@misc{koefler2026from,
  title = {{From MHV Gravity Amplitudes and Hodges map to Pentad Varieties}},
  author = {Joris Koefler and Emanuele Ventura and Hongmiao Yu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00864},
  eprint = {2611.00864},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00864},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - From MHV Gravity Amplitudes and Hodges map to Pentad Varieties
AU  - Joris Koefler
AU  - Emanuele Ventura
AU  - Hongmiao Yu
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00864
UR  - https://arxiv.org/abs/2611.00864
AB  - We study some geometric aspects behind the Hodges matrix appearing in the context of scattering amplitudes for the 𝒩=8 supergravity. Through a vector bundle construction, we show that the Hodges matrix gives rise to a canonical rational map from the two-step flag variety to symmetric matrices, called the Hodges map. This construction allows to define the amplitude geometrically: its vanishing locus is a divisor - the MHV gravity divisor. Inspired by the Hodges map, we introduce the notion of Hadamard ratio of two varieties. In the case relevant to gravity, this is the ratio of two Grassmannians, which we call the pentad variety, connecting it to the pentad constraint from algebraic statistics. Moreover, we show that the pentad variety is closely related to the combinatorial algebraic geometry of Martin's affine slope variety, and the phirotopes of Below, Krummeck, and Richter-Gebert.
ER  -