Exact Thresholds and Instanton Effects in String Theory

Abstract

In this lecture we summarize some recent work on the understanding of instanton effects in string theories with 16 supersymmetries. In particular, we consider F4 couplings using the duality between the heterotic string on T4 and type IIA on K3 at an orbifold point, as well as higher and lower dimensional versions of this string‐string duality. At the perturbative level a non‐trivial test of the duality, requiring several miraculous identities, is presented by matching a purely one‐loop heterotic amplitude to a purely tree‐level type II result. A wide variety of non‐perturbative effects is shown to occur in this setting, including D‐brane instantons for type IIA on K3 × S1 and NS5‐brane instantons for type IIA on K3 × T2. Moreover, the analysis of the three‐dimensional case, which possesses a non‐perturbative SO(8, 24, Z) U‐duality, reveals the presence of Kaluza‐Klein 5‐brane instanton effects, both on the heterotic and the type II side.

Onℛ4Terms and MHV Amplitudes inN=5,6Supergravity Vacua of Type II Superstrings

“…Relying on previous results on vector boson scattering at one-loop in unoriented D-brane worlds [37], we have studied four graviton scattering amplitudes and derived explicit formulae for the one-loop threshold corrections in asymmetric orbifolds that realize the above vacua. In addition to a term 1/nf N =8 cR 4 , coming from the (0, 0) sector, contributions from non-trivial sectors of the orbifold to f N =5,6 cR 4 display a close similarity with Heterotic threshold corrections in the presence of Wilson lines [46,45,48]. For illustrative purposes, we have computed the relevant integrals for N = 6 in D = 5 exposing the expected SO(1, 5) T-duality symmetry.…”

“…Explicit computation is beyond the scope of the present investigation. It proceeds along the lines above and presents close analogy with threshold computations in N = 2 heterotic strings sectors in the present of Wilson lines [46,45,48] or, equivalently, N = 4 heterotic strings in D = 8 [44]. Rather than focussing on this interesting but rather technical aspect of the problem, let us turn our attention onto the non-perturbative dependence on the other R-R moduli as well as dilaton.…”

“…For N = 6, in the decompactification limit the relevant decomposition under SO * (12) → SU(5, 1) × R + is (47) so that the 15 particle charges in D = 6 satisfy 15 1/3 BPS 'purity' conditions in D = 5 ∂I 3 ∂Q [ij] = 0 (48) where…”

“…One-loop threshold integrals for toroidal compactifications have been briefly reviewed above and shown to represent the contribution of the (r, s) = (0, 0) un-twisted sector. For (r, s) = (0, 0) the threshold integrals involve shifted lattice sums as in heterotic strings with Wilson lines [46,45,47,48,49].…”

Perturbative higher-derivative terms in N=6 asymmetric orbifolds

“…1 This theory has been very useful in understanding dualities and non-perturbative effects in string theory [4,5], including D-brane and NS-five-brane instantons. Supersymmetry and U-duality give very stringent constraints on the low-energy effective action [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], whose exact couplings are known up to fourteen-derivative terms [26][27][28][29][30][31][32][33][34][35][36]. Such results offer an invaluable window into the non-perturbative regime of string theory, like supersymmetric D-branes and their bound states.…”