Infinite Linear Quivers and Continuous Rank Functions

We introduce continuous rank functions for field theories with linear quivers, and demonstrate they describe infinitely long quivers. In the context of AdS/CFT, these 'continuous quivers' have well-defined supergravity dual descriptions with an infinite number of smeared NS5 and D-branes. These continuous rank functions generalise the supergravity solutions one can consider to be the dual descriptions of quiver field theories. We demonstrate this construction explicitly in the context of the half-BPS classes of AdS$_5$ and AdS$_7$ geometries, where we construct examples of a continuous parabolic and sinusoidal rank function.