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Derivatives of Superpotential in $\mathcal{N}=1$ Gauge Theories

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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?  

asked Jun 29 in Theoretical Physics by scpietromonaco (150 points) [ no revision ]

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