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Deficit Angles in 4D Spinfoam with Cosmological Constant: (Anti) de Sitter-ness and More

Muxin Han, Qiaoyin Pan

Abstract

This paper investigates the critical behaviors of the 4-dimensional spinfoam model with cosmological constant for a general 4-dimensional simplicial complex as the discretization of spacetime. We find that, at the semi-classical regime, the spinfoam amplitude is peaked at the real critical points that correspond to zero deficit angles (modulo $4π\mathbb{Z}/γ$) hinged by internal triangles of the 4-complex. Since the 4-simplices from the model are of constant curvature, the discrete geometry with zero deficit angle manifests a de Sitter (dS) spacetime or an anti de Sitter (AdS) spacetime depending on the sign of the cosmological constant fixed by the boundary condition. The non-(A)dS spacetimes emerge from the complex critical points by an analytic continuation to complex configurations.

Deficit Angles in 4D Spinfoam with Cosmological Constant: (Anti) de Sitter-ness and More

Abstract

This paper investigates the critical behaviors of the 4-dimensional spinfoam model with cosmological constant for a general 4-dimensional simplicial complex as the discretization of spacetime. We find that, at the semi-classical regime, the spinfoam amplitude is peaked at the real critical points that correspond to zero deficit angles (modulo ) hinged by internal triangles of the 4-complex. Since the 4-simplices from the model are of constant curvature, the discrete geometry with zero deficit angle manifests a de Sitter (dS) spacetime or an anti de Sitter (AdS) spacetime depending on the sign of the cosmological constant fixed by the boundary condition. The non-(A)dS spacetimes emerge from the complex critical points by an analytic continuation to complex configurations.
Paper Structure (15 sections, 1 theorem, 131 equations, 10 figures)

This paper contains 15 sections, 1 theorem, 131 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminary: 4D spinfoam with $\Lambda\neq 0$ from boundary Chern-Simons theory
  3. Classical theory
  4. Quantum theory -- the vertex amplitude
  5. Gluing $S^3\backslash\Gamma_5$'s -- the edge amplitude
  6. The full finite amplitude
  7. Stationary analysis for vertex amplitude and curved Regge action
  8. Stationary analysis for spins and the critical deficit angle
  9. Away from the (A)dS-ness
  10. An example: spinfoam amplitude of the $\Delta_3$ 4-complex
  11. Conclusion and outlook
  12. Fock-Goncharov coordinates and the Fenchel-Nielsen coordinates
  13. Geometrical interpretations of the Fenchel-Nielson coordinates
  14. Fock-Goncharov coordinates on ${\mathcal{S}}_2$ and ${\mathcal{S}}'_1$ in $\Delta_3$ 4-complex
  15. Proof of Theorem \ref{['theorem:hormander']}

Key Result

Theorem 7.1

Let $\vec{x}_0\in{\mathbb R}^n$ be a real critical point of the action $S(\vec{x},\vec{r})$ defined in eq:effective_action with $\vec{x}$ and $\vec{r}$ defined above where the Hessian is non-degenerate at the critical points, i.e. $\left.\det\left( \partial^2_{\vec{x}\vec{x}} S \right)\right|_{\vec{ Analytic continue $\vec{x}$ to $\vec{z}=\vec{x}+i\vec{y}\in{\mathbb C}^n$ near the critical point $

Figures (10)

  • Figure 1: The decomposition of the ideal triangulation ${\bf T}(S^3\backslash\Gamma_5)$ of $S^3\backslash\Gamma_5$ into 5 ideal octahedra ( in red), each of which can be decomposed into 4 ideal tetrahedra. The cusp boundaries of the ideal octahedra are shrunk to vertices in this figure. Numbers $\bar{1},\bar{2},\bar{3},\bar{4},\bar{5}$ with bars denote the 4-holed spheres on $\partial (S^3\backslash\Gamma_3)$. In each ideal octahedron, $x, y, z, w$ ( labelled in red) are chosen to form the equator of the octahedron. The same figure appears in Han:2015gmaHan:2021tzw.
  • Figure 2:
  • Figure 3:
  • Figure 5: FG coordinate dressing the edge on $T_a$ connecting hole $i$ and $j$ defined from framing flags $\{s_i,s_j, s_k, s_l\}$ parallel transported from holes of ${\mathcal{S}}_a$ before and after flipping the orientation of ${\mathcal{S}}_a$.
  • Figure 6: A complex critical point $\vec{z}_0(\vec{r})$ in the neighbourhood of a real critical point $\vec{x}_0(\vec{r})$.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Theorem 7.1