Hidden gauge symmetry in holomorphic models

•We have found a new gauge symmetry in holomorphic models.•This complex gauge symmetry connects different real systems.•The gauge condition determines the type of hermiticity of the variables.•The procedure is generalizable to any dimension.

We study the effect of a hidden gauge symmetry on complex holomorphic systems. For this purpose, we show that intrinsically any holomorphic system has this gauge symmetry. We establish that this symmetry is related to the Cauchy–Riemann equations, in the sense that the associated constraint is a first class constraint only in the case that the potential be holomorphic. As a consequence of this gauge symmetry on the complex space, we can fix the gauge condition in several ways and project from the complex phase-space to real phase space. Different projections are gauge related on the complex phase-space but are not directly related on the real physical phase-space.