U - duality and D - brane

We investigate D-brane instanton contributions to R4 couplings in any toroidal compactification of type II theories. Starting from the 11D supergravity one-loop fourgraviton amplitude computed by Green, Gutperle and Vanhove, we derive the nonperturbative O(e−1/λ) corrections to R4 couplings by a sequence of T-dualities, and interpret them as precise configurations of bound states of D-branes wrapping cycles of the compactification torus. Dp-branes explicitely appear as fluxes on D(p + 2)branes, and as gauge instantons on D(p+4)-branes. Specific rules for weighting these contributions are obtained, which should carry over to more general situations. Furthermore, it is shown that U-duality in D ≤ 6 relates these D-brane configurations to O(e−1/λ 2 ) instantons for which a geometric interpretation is still lacking. Laboratoire Propre du CNRS UPR A.0014. CERN–TH/97-272, CPTH–S563.1097 October 1997 –2– Although a non-perturbative definition of superstring theory is still elusive, the discovery of string dualities has made it clear that this theory should include various p-brane objects in its BPS spectrum. Under compactification, these objects can wrap on r-cycles of the compactification manifold to yield (p− r)-branes in lower dimensions, or instanton configurations if r = p+1 [1]. An exact calculation of physical couplings should take these instanton effects into account, but the rules for weighting them are still largely unknown. In some cases, the constraints of duality are strong enough to determine the exact nonperturbative completion of these couplings [2, 3, 4, 5, 6, 7], thereby opening a window on the string theory instanton rules. This way, D-instantons [3], D-particles [8, 9] and (p, q) strings [10] contributions to R couplings in toroidal compactifications of type II string have been brought under control, as well as general D-brane contributions to four derivative couplings in K3 compactifications of the same [5]. In this letter, we want to take a more systematic approach to this problem, and derive the contributions of general Dp-brane instantons to R couplings in toroidally compactified type II theory, by applying a sequence of perturbative T-dualities on the well understood D-particle contribution. Imposing S-duality will force us to include additional contributions, the origin of which remains to be elucidated. The general contribution of D-particles to R couplings in toroidally compactified type IIA theory has been obtained in Ref. [9] from the one-loop scattering amplitude of four supergravitons in 11-dimensional supergravity compactified on a (N + 1)-torus: A4 = 2πV11 ∫ ∞ 0 dt t5/2 ˆ ∑ nI e π t ngIJn J , (1) where gIJ is the volume V11 metric of the torus in eleven-dimensional Planck units. The sum runs over (N+1)-uplets of non-zero integers (as denoted by the hat over the sum) dual to the momentum of the supergraviton running in the loop. In terms of type IIA variables, the metric gIJ decomposes as ds11 = R 2 11(dx 11 +Aidx) + 1 R11 dxgijdx j , (2) where the eleven-dimensional radius R11 is related to the type IIA coupling [11] through –3– R11 = e 2φ/3 = λ, and gij denotes the metric in the string frame. The Kaluza–Klein connection A coincides with the Ramond–Ramond (RR) one-form gauge potential of type IIA superstring (RR potentials will be denoted by curl letters in the following). Going to the string frame and Poisson-resumming on n → m (the details of the procedure are explained at length in Ref. [10]), the amplitude (1) can be expanded at weak coupling as A4 = 2ζ(3)Ve+2V ˆ ∑ ni 1 ngijn +4πVe ˆ ∑