Deficit angles in 4D spinfoam with a cosmological constant: de Sitter-ness, anti–de Sitter-ness and more

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 semiclassical regime, the spinfoam amplitude is peaked at the real critical points that correspond to zero deficit angles (modulo $4\pi \mathbb{Z}/\gamma$) 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.

Figure 1

The decomposition of the ideal triangulation $\mathbf{T}(S^3\backslash {\mathrm{\Gamma }}_5)$ of $S^3\backslash {\mathrm{\Gamma }}_5$ 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 $\overline{1},\overline{2},\overline{3},\overline{4},\overline{5}$ with bars denote the 4-holed spheres on $\partial (S^3\backslash {\mathrm{\Gamma }}_3)$. In each ideal octahedron, $x$, $y$, $z$, $w$ (labeled in red) are chosen to form the equator of the octahedron. The same figure appears in [2, 14].

Figure 2

(a) An ideal tetrahedron whose edges are dressed with FGcoordinates $z$, $z^{\prime }$, or $z^{\prime \prime }$. Each pair of opposite edges are dressed with the same coordinate. The cusp boundaries are shown in gray. (b) An ideal octahedron. Choose the equator to be edges dressed with $x$, $y$, $z$, $w$. Adding an internal edge (in red) orthogonal to the equator separates the ideal octahedron into four ideal tetrahedra, each of which is dressed with different copies of coordinates $(x,x^{\prime },x^{\prime \prime })$, $(y,y^{\prime },y^{\prime \prime })$, $(z,z^{\prime },z^{\prime \prime })$, $(w,w^{\prime },w^{\prime \prime })$. For edges shared by different ideal tetrahedra, coordinates are multiplied together.

Figure 3

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 4

A complex critical point ${\overrightarrow{z}}_0(\overrightarrow{r})$ in the neighborhood of a real critical point ${\overrightarrow{x}}_0(\overrightarrow{r})$.

Figure 5

(a) Diagram of the 3-manifold corresponding to the ${\mathrm{\Delta }}_3$ 4-complex. The ambient 3-manifold (in black) has one noncontractable cycle. The (nonintersecting) blue lines denote the annuli and the red loop denotes the torus corresponding to the internal triangle shared by three 4-simplices. The 3-manifold on which the CS amplitude is defined is the graph (composed of the blue lines and the red loop) complement of the ambient 3-manifold. (b) The upper panel illustrates the gluing of ${\mathcal{S}}_2$ and ${\mathcal{S}}_1^{\prime }$. Numbers 1, 2, 3, 4 label the holes and the dotted lines denote the annuli dressed with FN coordinates that identify the holes pairwise. Edges in thick are dressed with the FG coordinates on the 4-holed spheres. The lower panel illustrates the quadrilateral to define $e^{-{\mathcal{Y}}_2}$ and $e^{{\mathcal{X}}_1^{\prime }}$ through (77).

Figure 6

A quadrilateral in a 2D ideal triangulation to define FG coordinate $x_E$ in terms of the framing flags $\{s_i{\}}_{i=1,\dots ,4}$ on four holes (represented by circles) parallel transported to a common point by (b2).

Figure 7

Two spacelike tetrahedra $T_a$ and $T_b$ forming a wedge (2 spacial dimensions are reduced). $N_a$ and $N_b$ are the outward-pointing normal to $T_a$ and $T_b$ respectively. (a) A thin wedge with dihedral angle ${\mathrm{\Theta }}_{ab}>0$. (b) A thick wedge with dihedral angle ${\mathrm{\Theta }}_{ab}<0$.