$\mathcal{N}=1$ spectra, cubic couplings and the rigid fate of DGKT

Abstract

We show that in the DGKT scenario on a generic Calabi-Yau three-fold, a recently proposed holographic constraint on cubic couplings is satisfied if and only if the Calabi-Yau is rigid, i.e. when $h^{2,1}=0$. More generally, we illustrate how in 4d $\mathcal{N}=1$ supergravity, extremal cubic couplings are determined by the third derivatives of the real, Kähler-invariant superpotential, while the eigenvalues of its Hessian compute the conformal dimensions of the dual scalar operators. These results extend more broadly beyond 4d $\mathcal{N}=1$ supergravity. Applying them to supersymmetric DGKT vacua, we prove that extremal cubic couplings always vanish in the Kähler + universal CS/dilaton sector, whereas non-vanishing (super-)extremal couplings are always present in the complex structure sector. It follows that the holographic constraint is satisfied in DGKT if and only if the Calabi-Yau three-fold is rigid with $h^{2,1}=0$.