Gauge Theory, Geometry and the Large N Limit

We study the relationship between M theory with N units of momentum along a nearly lightlike circle and U(N) gauge theory in p+1 dimensions. We define large N limits of these theories in which low energy supergravity appears to be valid. The regularity of these limits implies an infinite series of nonrenormalization theorems for the gauge theory effective action. The apparent validity of supergravity also implies that the leading large N terms to all loops in the gauge theory sum up to a Born-Infeld form. We find certain nonplanar diagrams in the gauge theory that correspond to corrections to the 11 dimensional low-energy effective action that are suppressed by powers of the Planck length. Compatibility of two different large N limits that describe the same decompactified M theory leads to a conjecture for a relation between two limits of string theories. The relationship of our work to recent results in the finite N M(atrix) model is discussed