Hessian in the spinfoam models with cosmological constant

TL;DR

This work proves the non-degeneracy of the Hessian in the stationary-phase analysis of the Λ-SF spinfoam vertex by recasting the condition as a transversal intersection of real Lagrangian submanifolds in the phase space of flat $SL(2,\mathbb{C})$ connections. It develops a general framework for relating Hessian non-degeneracy to intersections of real Lagrangian parts, transfers the problem to the Chern-Simons phase space, and then demonstrates non-degeneracy for critical points corresponding to non-degenerate 4-simplices embedded in de Sitter or anti-de Sitter spaces. The results ensure the validity of the semiclassical limit and exclude pathological exceptional configurations, while highlighting the method’s potential applicability to other spinfoam models. The work also outlines connections to FG-FN coordinates, holonomy descriptions, and the geometry of curved 4-simplices, offering a pathway to generalize non-degeneracy proofs beyond the Λ-SF model, with future considerations for EPRL-like models and stationary-phase conditions on non-compact domains.

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 ($Λ$-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 $Λ$-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.

Figures (3)

• Figure 1: $\Gamma_5$ graph projected on ${\mathbb R}^2$. It forms a triangulation of $S^3$, which is the boundary of a 4-simplex. Numbers $1,\cdots,5$ denote the vertices of the graph.
• Figure 2: Ideal octahedron $\mathrm{Oct}(a)$ ( in red), or equivalently 3-cell $\tilde{T}_a$. Here, $a\neq b\neq c \neq d \neq e$. Cusp boundaries of $\mathrm{Oct}(a)$ (on the tubular neighhorbood of edges of $T_a$) are shrunk to vertices of the octahedron. See Han:2021tzw for more details.
• Figure 3: 2-cells $F_b$ and $F_c$ on a fixed ${\mathcal{S}}_a$ ( shaded in blue). Each of them touches three out of four holes of ${\mathcal{S}}_a$. Their intersection gives 1-cell $E_{bc}$ ( in red).

Theorems & Definitions (33)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• Lemma 2.6
• proof
• proof : Proof of Proposition \ref{['prop:clean-0']}