Hessian in the spinfoam models with cosmological constant

In this paper, we introduce a general method to prove the non-degeneracy of the Hessian in the spinfoam vertex amplitude for quantum gravity and apply it to the spinfoam models with a cosmological constant ($\mathrm{\Lambda }$-SF models). By reformulating the problem in terms of the transverse intersection of some submanifolds in the phase space of flat $\mathrm{S}\mathrm{L}(2,\mathbb{C})$ connections, we demonstrate that the Hessian is non-degenerate for critical points corresponding to non-degenerate, geometric 4-simplices in de Sitter or anti-de Sitter space. Non-degeneracy of the Hessian is an important necessary condition for the stationary phase method to be applicable. With a non-degenerate Hessian, this method not only confirms the connection of the $\mathrm{\Lambda }$-SF model to semiclassical gravity, but also shows that there are no dominant contributions from exceptional configurations as in the Barrett-Crane model. Given its general nature, we expect our criterion to be applicable to other spinfoam models under mild adjustments.