A double copy from twisted (co)homology at genus one
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910515457949696 |
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| author | Bhardwaj, Rishabh Pokraka, Andrzej Ren, Lecheng Rodriguez, Carlos |
| author_facet | Bhardwaj, Rishabh Pokraka, Andrzej Ren, Lecheng Rodriguez, Carlos |
| contents | 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 [Goto2022,arXiv:2206.03177]. 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 [Mano&Watanabe2012]. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_02148 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A double copy from twisted (co)homology at genus one Bhardwaj, Rishabh Pokraka, Andrzej Ren, Lecheng Rodriguez, Carlos High Energy Physics - Theory Mathematical Physics Algebraic Geometry Algebraic Topology 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 [Goto2022,arXiv:2206.03177]. 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 [Mano&Watanabe2012]. 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. |
| title | A double copy from twisted (co)homology at genus one |
| topic | High Energy Physics - Theory Mathematical Physics Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2312.02148 |