Gauge theory, geometry and the large-N limit

We study the relationship between M-theory on a nearly light-like circle and U(N) gauge theory in p + 1 dimensions. We define large-N limits of these theories in which low-energy supergravity is valid. The regularity of these limits implies an infinite series of non-renormalization theorems for the gauge theory effective action, and the leading large-N terms sum to a Born-Infeld form. 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.