We construct integrality gap instances for SDP relaxation of the Maximum Cut and the Sparsest Cut problems. It is well-known that if the triangle inequality constraints were added to the SDP, then the SDP vectors naturally define an n-point negative type (or ell_2^2) metric where n is the number of vertices in the problem instance. Our gap-instances have the property that every sub-metric on t = O((log log log n)^{1/6}) points is isometrically embeddable into ell_1. The local ell_1-embeddability constraints are implied when the basic SDP relaxation is augmented with t rounds of the Sherali-Adams LP-relaxation. For the Maximum Cut problem, we obtain an optimal gap of alpha_{GW}^{-1} - eps, where alpha_{GW} is the Goemans-Williamson constant and eps > 0 is an arbitrarily small constant. For the Sparsest Cut problem, we obtain a gap of Omega((log log log n)^{1/13}). The latter result can be rephrased as a construction of an n-point negative type metric such that every t-point sub-metric is isometrically ell_1-embeddable, but embedding the whole metric into ell_1 incurs distortion Omega((log log log n)^{1/13}).