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# On the Rankin-Selberg method for higher genus string amplitudes

Abstract : 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 $E_h(s)$ 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 T-duality group $O(d,d,Z)$.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-01271308
Contributor : Boris Pioline Connect in order to contact the contributor
Submitted on : Tuesday, February 9, 2016 - 9:15:37 AM
Last modification on : Friday, December 3, 2021 - 12:07:46 PM

### Citation

Ioannis Florakis, Boris Pioline. On the Rankin-Selberg method for higher genus string amplitudes. Communications in Number Theory and Physics, International Press, 2017, 11, pp.337-404. ⟨10.4310/CNTP.2017.v11.n2.a4⟩. ⟨hal-01271308⟩

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