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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.00047 |
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| _version_ | 1866909550371667968 |
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| author | Klinger-Logan, Kim Miller, Stephen D. Radchenko, Danylo |
| author_facet | Klinger-Logan, Kim Miller, Stephen D. Radchenko, Danylo |
| contents | We complete the program, initiated in a 2015 paper of Green, Miller, and Vanhove, of directly constructing the automorphic solution to the string theory $D^6 R^4$ differential equation $(Δ-12)f=-E_{3/2}^2$ for $SL(2,\Z)$. The construction is via a type of Poincaré series, and requires explicitly evaluating a particular double integral. We also show how to use double Dirichlet series to formally derive the predicted vanishing of one type of term appearing in $f$'s Fourier expansion, confirming a conjecture made by Chester, Green, Pufu, Wang, and Wen motivated by Yang-Mills theory (and later proved rigorously by Fedosova, Klinger-Logan, and Radchenko using the Gross-Zagier Holomorphic Projection Lemma.). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_00047 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The $D^6 R^4$ interaction as a Poincaré series, and a related shifted convolution sum Klinger-Logan, Kim Miller, Stephen D. Radchenko, Danylo Number Theory Mathematical Physics We complete the program, initiated in a 2015 paper of Green, Miller, and Vanhove, of directly constructing the automorphic solution to the string theory $D^6 R^4$ differential equation $(Δ-12)f=-E_{3/2}^2$ for $SL(2,\Z)$. The construction is via a type of Poincaré series, and requires explicitly evaluating a particular double integral. We also show how to use double Dirichlet series to formally derive the predicted vanishing of one type of term appearing in $f$'s Fourier expansion, confirming a conjecture made by Chester, Green, Pufu, Wang, and Wen motivated by Yang-Mills theory (and later proved rigorously by Fedosova, Klinger-Logan, and Radchenko using the Gross-Zagier Holomorphic Projection Lemma.). |
| title | The $D^6 R^4$ interaction as a Poincaré series, and a related shifted convolution sum |
| topic | Number Theory Mathematical Physics |
| url | https://arxiv.org/abs/2210.00047 |