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arXiv · 2510.12755

Hessian in the spinfoam models with cosmological constant

Wojciech Kamiński·Qiaoyin Pan
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Abstract

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 ($\Lambda$-SF models). By reformulating the problem in terms of the transverse intersection of some submanifolds in the phase space of flat ${\rm SL}(2,\mathbb{C})$ 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 $\Lambda$-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.

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Wojciech Kamiński, Qiaoyin Pan. 2025-10-14. Hessian in the spinfoam models with cosmological constant. https://doi.org/10.1007/s00023-026-01680-5

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