Convolution identities for divisor sums and modular forms

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.