We show that the monodromy theorem holds on arbitrary connected free sets for noncommutative free analytic functions. Applications are numerous-- pluriharmonic free functions have globally defined pluriharmonic conjugates, locally invertible functions are globally invertible, and there is no nontrivial cohomology theory arising from analytic continuation on connected free sets. We describe why the Baker-Campbell-Hausdorff formula has finite radius of convergence in terms of monodromy, and solve a related problem of Martin-Shamovich. We generalize the Dym-Helton-Klep-McCullough-Volcic theorem-- a uniformly real analytic free noncommutative function is plurisubharmonic if and only if it can be written as a composition of a convex function with an analytic function. The decomposition is essentially unique. The result is first established locally, and then Free Universal Monodromy implies the global result. Moreover, we see that plurisubharmonicity is a geometric property-- a real analytic free function plurisubharmonic on a neighborhood is plurisubharmonic on the whole domain. We give an analytic Greene-Liouville theorem, an entire free plurisubharmonic function is a sum of hereditary and antihereditary squares.
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