  We prove exact identities for convolution sums of divisor functions of the
form $\sum_{n_1 \in \mathbb{Z} \smallsetminus
\{0,n\}}\varphi(n_1,n-n_1)\sigma_{2m_1}(n_1)\sigma_{2m_2}(n-n_1)$ where
$\varphi(n_1,n_2)$ is a Laurent polynomial with logarithms for which the sum is
absolutely convergent. Such identities are motivated by computations in string
theory and prove and generalize a conjecture of Chester, Green, Pufu, Wang, and
Wen from \cite{CGPWW}. Originally, it was suspected that such sums, suitably
extended to $n_1\in\{0,n\}$ should vanish, but in this paper we find that in
general they give Fourier coefficients of holomorphic cusp forms.
