Rankin-Selberg methods for closed string amplitudes

Abstract

After integrating over supermoduli and vertex operator positions, scattering amplitudes in superstring theory at genus h ≤ 3 are reduced to an integral of a Siegel modular function of degree h on a fundamental domain of the Siegel upper half plane. A direct computation is in general unwieldy, but becomes feasible if the integrand can be expressed as a sum over images under a suitable subgroup of the Siegel modular group: if so, the integration domain can be extended to a simpler domain at the expense of keeping a single term in each orbit -a technique known as the Rankin-Selberg method. Motivated by applications to BPS-saturated amplitudes, Angelantonj, Florakis and I have applied this technique to one-loop modular integrals where the integrand is the product of a Siegel-Narain theta function times a weakly, almost holomorphic modular form. I survey our main results, and take some steps in extending this method to genus greater than one.

Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds

“…, p. In this representation, modular invariance is manifest, since a transformation Ω → (AΩ + B)(CΩ + D) −1 (A.2) can be compensated by a linear transformation (n, m) → (n, m) D ⊺ −B ⊺ −C ⊺ D ⊺ , y 1 → y 1 · (CΩ + D), under which the third line of (4.7) transforms as a weight p−q 2 modular form. We can therefore compute the integral using the orbit method [60,61,62,63,64], namely decompose the sum over (n, m) into various orbits under Sp(4, Z), and for each orbit O, retain the contribution of a particular element ς ∈ O at the expense of extending the integration domain F 2 = Sp(4, Z)\H 2 to Γ ς \H 2 , where Γ ς is the stabilizer of ς in Sp(4, Z). The integration domain is unfolded according to the formula γ∈Γς \Sp(4,Z)…”

D6ℛ4 amplitudes in various dimensions

“…As for the genus 2 modular integral, viewing SO(5, 5, Z) as the T-duality group in D = 5, we have to study the limit when the volume V 5 /l 5 s is scaled to infinity, and extract the term of order (r 4 /l 7 ) 3 = (V 5 /l 5 s ) 6/5 . The torus decompactification limit can be analyzed by applying the orbit method on the genus 2 Narain partition function Γ 5,5,2 , following [44]. The zero and rank one orbits reproduce the O(r 5 4 ) and O(r 4 4 ) terms in (B.46), while the rank 2 orbits contributes to the O(r 3 4 ) term.…”

Exact ∇4ℛ4 couplings and helicity supertraces

“…As in (A.1), the integral diverges if d ≥ h + 1, and must be regularized. A regularization scheme was described in detail for 1 ≤ h ≤ 3 in [46], and is easily generalized to arbitrary h. The regularized integral can then be expressed as a residue at s = d−3 2 of the maximal parabolic Eisenstein series E D d sΛ h attached to the rank h antisymmetric representation [46,53], or as a residue at s = h of the maximal parabolic Eisenstein series attached to the spinor representations [7].…”