On the Rankin–Selberg method for higher genus string amplitudes

Abstract

Closed string amplitudes at genus h ≤ 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).

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)…”

String theory integrands and supergravity divergences

“…The 'renormalized' integral is then defined by subtracting both divergent pieces before taking the limit L → ∞ [22,35],…”

“…Using the truncated fundamental domain F 3 (L) defined in [35], and specializing to the dimensions where logarithmic terms arise, we get (Eq. (4.72) in [35], noting that the measure dµ h used in that reference differs from the one used here by a factor 1/2 h(h+1)/2 ) Unfortunately, the genus-three contributions to higher derivative interactions E (d,3) (p,q) with 2p + 3q > 3 are not known at present, due to difficulties in regulating the integral over the pure spinor ghosts [38]. However, by requiring that coefficients of divergences recombine into U-duality invariant combinations, we shall see that the next-to-leading term must produce the following divergences in D = 6 and D = 4,…”

“…Indeed, the local modular Laurent polynomials A i,j (S) arising in the limit V → 0 are integrable around S 1 = 0, even though they are not smooth on this locus 6. The domain F 1 (L 0 ) and S 1 (L 0 , L) correspond to the boundaries at t = L of regions I and II, in the terminology of[22,35].…”

Supersymmetry breaking in heterotic strings and the decompactification problem

“…More recently, in the context of string phenomenology, there has been an increased interest in nonsupersymmetric string constructions [16,17,18,35,19,20,21,22,23,24,25,26,27,28,29,30] and, in particular, there has been considerable progress in understanding radiative corrections to gauge and gravitational couplings in such setups. Part of this progress was possible thanks to the development of new mathematical techniques for studying string loop amplitudes [31,32,33,34,35,36].…”