I'm studying $\mathcal{N}=1$ 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 $\mathcal{N}=1^{*}$ theory where in each of the classical, massive vacuums you get an effective superpotential $W_{\text{eff}}(\tau)$ which depends holomorphically on the modular parameter $\tau$ 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 $\tau$-derivative of $W_{\text{eff}}$ vanishes) encode intersting data of the gauge theory. **But what about having a vanishing second derivative of $W_{\text{eff}}(\tau)$? 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?