I'm studying N=1 supersymmetric gauge theories. To my understanding, if we can compute a superpotential (or effective superpotential in a given vacuum) then there is a holomorphic sector of observables which will arise as derivatives of the superpotential with respect to the parameters of the theory.
The main example I'm interested in is the N=1∗ theory where in each of the classical, massive vacuums you get an effective superpotential Weff(τ) which depends holomorphically on the modular parameter τ of an elliptic curve.
My question might have a general answer independent of any example, or perhaps not. Basically, I know that the critical points of the superpotential, (i.e. the points where the τ-derivative of Weff vanishes) encode intersting data of the gauge theory. But what about having a vanishing second derivative of Weff(τ)? Does this have important meaning in the physics? I know this should correspond to some expectation value of some observable vanishing, but I'm wondering if there's a more concrete interpretation/description available?