Noncommutative plurisubharmonicity, existence of pluriharmonic conjugates, and free universal monodromy

We fully characterize free noncommutative plurisubharmonic functions as compositions of a convex function with an analytic function, completing a long-standing program. The decomposition is essentially unique when the convex function is chosen to be of a natural form. 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 anti-hereditary squares. Our monodromic considerations imply that pluriharmonic free functions have globally defined pluriharmonic conjugates. We also describe why the Baker-Campbell-Hausdorff formula has finite radius of convergence in terms of monodromy, and, in doing so, solve a related problem of Martin-Shamovich.