On the Rankin–Selberg method for higher genus string amplitudes

Closed string amplitudes at genus $h \leq 3$ are given by integrals of Siegel modular functions on a fundamental domain of the Siegel upper half-plane. When the integrand is of rapid decay near the cusps, the integral can be computed by the Rankin–Selberg method, which consists of inserting an Eisenstein series $\mathcal{E}_h (s)$ in the integrand, computing the integral by the orbit method, and finally extracting the residue at a suitable value of $s$. String amplitudes, however, typically involve integrands with polynomial or even exponential growth at the cusps, and a renormalization scheme is required to treat infrared divergences. Generalizing Zagier’s extension of the Rankin–Selberg method at genus one, we develop the Rankin–Selberg method for Siegel modular functions of degree $2$ and $3$ with polynomial growth near the cusps. In particular, we show that the renormalized modular integral of the Siegel–Narain partition function of an even self-dual lattice of signature $(d, d)$ is proportional to a residue of the Langlands–Eisenstein series attached to the $h$-th antisymmetric tensor representation of the $\mathrm{T}$-duality group $O(d, d, \mathbb{Z})$.