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What is 'large-N' ? From the planar limit to M-theory

The large-N limit of gauge theories has been playing a central role in theoretical physics over the decades. Despite its obvious importance, however, only very little had been known outside the planar limit, in which lambda=g_{YM}^2N is fixed. In this talk, we consider a more general large-N limit, in which lambda grows with N, e.g., with g_{YM} ^ 2 fixed. Such a limit plays a crucial role in particular in recent attempts to find the nonpertubative formulation of M-theory. We propose that such a limit is essentially identical to the planar limit, in the sense that the order of the large-N limit and the strong-coupling limit commute. Indeed, we show the validity of this conjecture in various concrete examples. 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. It enables us to calculate various quantities outside the planar limit. As simple examples, we reproduce a few properties of the six-dimensional (2, 0) theory from the five-dimensional maximal super Yang-Mills theory, which supports the conjecture by Douglas and Lambert et al. that these two theories are identical.