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We study instanton solutions and superpotentials for the large number ofvacua of the plane-wave matrix model and a 2+1 dimensional Super Yang-Millstheory on $R\times S^2$ with sixteen supercharges. We get the superpotential inthe weak coupling limit from the gauge theory description. We study the gravitydescription of these instantons. Perturbatively with respect to a background,they are Euclidean branes wrapping cycles in the dual gravity background.Moreover, the superpotential can be given by the energy of the electric chargesystem characterizing each vacuum. These charges are interpreted as theeigenvalues of matrices from a reduction for the 1/8 BPS sector of the gaugetheories. We also discuss qualitatively the emergence of the extra spatialdimensions appeared on the gravity side.Comment: 29 pages, 3 figures, latex. v2: references added, comments added. v3:  accepted version in PR

Instantons, supersymmetric vacua, and emergent geometries