Graviton scattering in matrix theory and supergravity

I briefly review recent work on the comparison between two and three graviton scattering in supergravity and matrix theory 1 Motivations In the low-energy regime, M-theory is D = 11, N = 1 supergravity. In the matrix model the fundamental degrees of freedom of M-theory are 0-branes (that is, Derichlet particles). For this model to be a correct description of M-theory, it must then reproduce supergravity in the long-distance regime. In particular, 0-brane scattering amplitudes in D = 10 must reproduce those of compactified (fromD = 11 down to 10) supergravity, for which the gravitons carry momentum in the compactified direction. Such a correspondence between amplitudes in these two different-looking theories plays an important role because it can be computed explicitely. It has now been succesfully checked in the twoand three-graviton scattering amplitudes. 2 Two-graviton scattering The scattering of two graviton carrying momentum in a compactified direction has been studied several times in the literature [1]. The simplest way to compute it is by means of the effective lagrangian [2] L = −p−ẋ − = −p− √ 1− h−−v − 1 h−− , (1) where h−− = f(r)/2πR11 and f(r) = 2κ M/7Ω r for the space-time of the shock wave generated by the graviton moving with momentum p− = N2/R11. Actually, this is a special case of shock wave in which the 11-th dimension has been smeared. By expanding in the relative velocity v, we find L = −p− { v 2 + a1 v r + a2 v r · · · } , (2) where the exact values of the coefficients a1 and a2 are known. 2 Marco Fabbrichesi The corresponding amplitude in matrix theory can derived from the gauge fixed action, the bosonic part of which reads S = ∫