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A double copy from twisted (co)homology at genus one

  • Regular Article - Theoretical Physics
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  • Published: 04 July 2024
  • Volume 2024, article number 40, (2024)
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A double copy from twisted (co)homology at genus one
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  • Rishabh Bhardwaj  ORCID: orcid.org/0000-0001-7624-44211,
  • Andrzej Pokraka  ORCID: orcid.org/0000-0003-1186-46241,
  • Lecheng Ren  ORCID: orcid.org/0000-0002-0846-80171 &
  • …
  • Carlos Rodriguez  ORCID: orcid.org/0000-0001-6523-08412 

  • 151 Accesses

  • 2 Citations

  • 1 Altmetric

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A preprint version of the article is available at arXiv.

Abstract

We study the twisted (co)homology of a family of genus-one integrals — the so called Riemann-Wirtinger integrals. These integrals are closely related to one-loop string amplitudes in chiral splitting where one leaves the loop-momentum, modulus and all but one puncture un-integrated. While not actual one-loop string integrals, they share many properties and are simple enough that the associated twisted (co)homologies have been completely characterized [1]. Using intersection numbers — an inner product on the vector space of allowed differential forms — we derive the Gauss-Manin connection for two bases of the twisted cohomology providing an independent check of [2]. We also use the intersection index — an inner product on the vector space of allowed contours — to derive a double-copy formula for the closed-string analogues of Riemann-Wirtinger integrals (one-dimensional integrals over the torus). Similar to the celebrated KLT formula between open- and closed-string tree-level amplitudes, these intersection indices form a genus-one KLT-like kernel defining bilinears in meromorphic Riemann-Wirtinger integrals that are equal to their complex counterparts.

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Acknowledgments

The authors would like to thank Oliver Schlotterer for many interesting discussions and participation in the early stages of this work. The authors would also like to thank Francis Brown, Eduardo Casali, Lorenz Eberhardt, Yoshiaki Goto, Saiei-Jaeyeong Matsubara-Heo, Sebastian Mizera, Franziska Porkert, Giulio Salvatori, Marcus Spradlin, Piotr Tourkine, Anastasia Volovich, and Federico Zerbini for useful discussions. AP would like to thank the institute for advanced study for its hospitality and the organizers of the String Amplitudes at Finite α′ workshop where this work was initiated. This work was supported in part by the US Department of Energy under contract DESC0010010 Task F (RB, AP, LR) and by Galkin Foundation Fellowship (LR). The research of CR is supported by the European Research Council under ERC-STG-804286 UNISCAMP.

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Authors and Affiliations

  1. Department of Physics, Brown University, Providence, RI, 02912, USA

    Rishabh Bhardwaj, Andrzej Pokraka & Lecheng Ren

  2. Department of Physics and Astronomy, Uppsala University, Box 516, 75120, Uppsala, Sweden

    Carlos Rodriguez

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  1. Rishabh Bhardwaj
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  2. Andrzej Pokraka
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  3. Lecheng Ren
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  4. Carlos Rodriguez
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Correspondence to Andrzej Pokraka.

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Bhardwaj, R., Pokraka, A., Ren, L. et al. A double copy from twisted (co)homology at genus one. J. High Energ. Phys. 2024, 40 (2024). https://doi.org/10.1007/JHEP07(2024)040

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  • Received: 09 January 2024

  • Revised: 26 May 2024

  • Accepted: 05 June 2024

  • Published: 04 July 2024

  • DOI: https://doi.org/10.1007/JHEP07(2024)040

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Keywords

  • Differential and Algebraic Geometry
  • Gauge-Gravity Correspondence
  • Scattering Amplitudes
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