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The $D^6 R^4$ interaction as a Poincaré series, and a related shifted convolution sum

Kim Klinger-Logan, Stephen D. Miller, Danylo Radchenko

TL;DR

This work completes the direct construction of the automorphic solution to the inhomogeneous equation $(\Delta-12) f = -\left(2\zeta(3) E_{3/2}\right)^2$ for $SL(2,\mathbb{Z})$ via a Poincaré-series framework, building on the program initiated in GMV2015. It explicitly evaluates the central double integral $\mathcal{I}(n_1,n_2;y)$ in the Fourier expansion for the critical case $a=\tfrac{3}{2}$, $\lambda=12$, and shows the resulting Fourier modes align with the previously known structure, including the homogeneous part. A formal double-Dirichlet-series analysis yields the predicted vanishing of the total homogeneous coefficient sum over nonzero modes, matching physics-inspired conjectures (and later proven rigorously in follow-up work). The methods extend to both half-integer and integer $a$, offering a concrete route to solving similar automorphic differential equations arising in string theory and related automorphic-form contexts.

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 $(Δ-12)f=-E_{3/2}^2$ 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.).

The $D^6 R^4$ interaction as a Poincaré series, and a related shifted convolution sum

TL;DR

This work completes the direct construction of the automorphic solution to the inhomogeneous equation for via a Poincaré-series framework, building on the program initiated in GMV2015. It explicitly evaluates the central double integral in the Fourier expansion for the critical case , , and shows the resulting Fourier modes align with the previously known structure, including the homogeneous part. A formal double-Dirichlet-series analysis yields the predicted vanishing of the total homogeneous coefficient sum over nonzero modes, matching physics-inspired conjectures (and later proven rigorously in follow-up work). The methods extend to both half-integer and integer , offering a concrete route to solving similar automorphic differential equations arising in string theory and related automorphic-form contexts.

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 differential equation for . 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 '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.).
Paper Structure (8 sections, 1 theorem, 73 equations)

This paper contains 8 sections, 1 theorem, 73 equations.

Table of Contents

  1. Introduction
  2. Solution to $(\Delta-\lambda)f=-E_a^2$
  3. Half-integral $a$
  4. Integral $a$
  5. The integral $\mathcal{I}$
  6. The first term $T_1$
  7. The second term $T_2$
  8. A conjecture of Chester, Green, Pufu, Wang and Wen

Key Result

Lemma 4.1

For $d\ge 1$ and $\text{Re}(s)$ sufficiently large one has and where $\Lambda$ is the von Mangoldt function and $v_p(n)$ denotes the valuation of $n$ at prime $p$. In particular, the product is equal to $d^k$ when $s+k=0$.

Theorems & Definitions (2)

  • Lemma 4.1
  • proof