The $D^6 R^4$ interaction as a Poincar\'e series, and a related shifted convolution sum - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... (Preprint) Year : 2022

## The $D^6 R^4$ interaction as a Poincar\'e series, and a related shifted convolution sum

Kim Klinger-Logan
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Stephen D. Miller
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#### Abstract

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 $(\Delta-12)f=-E_{3/2}^2$ for $SL(2,\mathbb{Z})$. The construction is via a type of Poincar\'e series, and requires explicitly evaluating a particular double integral. We also show how to 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.

### Dates and versions

hal-03954324 , version 1 (24-01-2023)

### Identifiers

• HAL Id : hal-03954324 , version 1
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### Cite

Kim Klinger-Logan, Stephen D. Miller, Danylo Radchenko. The $D^6 R^4$ interaction as a Poincar\'e series, and a related shifted convolution sum. 2022. ⟨hal-03954324⟩

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