On the effective action in presence of local non-linear constraints

The conditions for the existence of the effective action in statistical field theory, the Legendre transform of the cumulant generating function, in presence of non-linear local constraints are discussed. This problem is of importance for non-perturbative approaches, such as the functional renormalization group. We show that the Legendre transform exists as long as the non-linear constraints do not imply linear constraints on the microscopic fields. We discuss how to handle the case of effectively linear constraints and we naturally obtain that the second derivative of the effective action is the Moore-Penrose pseudo-inverse of the correlation function. We illustrate our discussion with toy-models, and show that the correct counting of degrees of freedom in non-linearly constrained statistical field theories can be rather counter-intuitive.