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From the planar limit to M-theory

The large-N limit of gauge theories has been playing a crucial role in theoretical physics over the decades. Despite its importance, little is known outside the planar limit where the 't Hooft coupling $λ=g_{YM}^2N$ is fixed. In this Letter we consider more general large-N limit --- $λ$ grows with N, e.g., $g_{YM}^2$ is fixed. Such a limit is important particularly in recent attempts to find the nonpertubative formulation of M-theory. Based on various supporting evidence, we propose this limit is essentially identical to the planar limit, in the sense the order of the large-N limit and the strong coupling limit commute. For a wide class of large-N gauge theories, these two limits are smoothly connected, and the analytic continuation from the planar limit is justified. As simple examples, we reproduce a few properties of the six-dimensional N=(2, 0) theory on S^1 from the five-dimensional maximal super Yang-Mills theory, supporting the recent conjecture by Douglas and Lambert et al. that these two theories are identical.