The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

Abstract

The closure constraint is a central piece of the mathematics of loop quantum gravity. It encodes the gauge invariance of the spin network states of quantum geometry and provides them with a geometrical interpretation: each decorated vertex of a spin network is dual to a quantized polyhedron in R 3 . For instance, a 4-valent vertex is interpreted as a tetrahedron determined by the four normal vectors of its faces. We develop a framework where the closure constraint is re-interpreted as a Bianchi identity, with the normals defined as holonomies around the polyhedron faces of a connection (constructed from the spinning geometry interpretation of twisted geometries). This allows us to define closure constraints for hyperbolic tetrahedra (living in the 3-hyperboloid of unit future-oriented spacelike vectors in R 3,1 ) in terms of normals living all in SU(2) or in SB(2, C). The latter fits perfectly with the classical phase space developed for q-deformed loop quantum gravity supposed to account for a non-vanishing cosmological constant Λ > 0. This is the first step towards interpreting q-deformed twisted geometries as actual discrete hyperbolic triangulations. IntroductionLoop quantum gravity (for monographs see [1][2][3]) is based on a reformulation of general relativity as an SU (2) gauge theory of the Ashtekar-Barbero connection [4][5][6][7] together with its canonically conjugated field, the densitized triad 1 . At the quantum level, the kinematics of the theory is well-understood: the space of quantum states of geometry are wave-functions of the Ashtekar-Barbero connection and admits a canonical basis labelled by spin networks [8]. Spin networks are (abstract) graphs (which might have knotting information in 3+1d) colored with spins on the edges (hence the name) and intertwiners, which are invariant representations of the gauge group, on the vertices. The spin network basis diagonalize the geometrical operators. The spins carried by the edges corresponding to quanta of surfaces [9]. The volume operator is a bit more involved but we still can find a basis of intertwiners that diagonalize the volume operator.The spin network states have a natural geometrical interpretation in the twisted geometry framework [12]. Each node of the graph is interpreted as a flat convex polyhedron. Each edge coming out of the vertex corresponds to a face of the polyhedron and the area of this face is given by the spin carried by the edge. The intertwiner should encode the remaining quantum degrees of freedom. Then, the polyhedra corresponding to the vertices are glued together on their faces. By construction, the area of two glued faces match. Still, the geometry is said to be twisted since the shapes of the faces do not have to match. In the continuum, this corresponds to a discontinuous metric [13]. There is a notable second geometrical parametrization that was proposed: spinning geometries [14]. The construction is roughly the same but the faces and edges of the polyhedra are not assumed to be (extrinsically) flat. As a consequence,...

Bubble networks: framed discrete geometry for quantum gravity

“…The 3d geometry is constructed as a cellular complex from 3d flat cells whose boundary 2-cells are minimal surfaces between 1-cells. The "normal vectors" J's of the surfaces are defined as an integrated angular momentum, computed as the holonomy of a specific connection around the surfaces (see also [25]). This holonomy is matched across the boundary when gluing two bubbles and is not generically the normal vector to a flat 2d face.…”

“…Nevertheless, when coarse-graining loop quantum gravity, curvature naturally builds up at the spin network nodes [21][22][23] and it becomes necessary to allow for "curved nodes", corresponding to quantum curved polyhedra (e.g. embedded in spherical or hyperbolic space) [24,25], and to allow for the 3d embedding of the boundary surface to fluctuate, thereby allowing for bubbles with arbitrary geometry. The line of research we pursue in the present work parallels the logic developed in [3,26], which couples surface geometry degrees of freedom to the Ashtekar triad-connection variables leading to coupled spin networks and conformal field theories at the quantum level.…”

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…A possible reason could be polar duality [76]. Another possible link is the interpretation of the Gauß constraints (or closure constraints) as a Bianchi identity and a possible re-construction of a new kind of connection proposed in [86,87].…”

Fusion basis for lattice gauge theory and loop quantum gravity

“…Hence, we effectively obtain torsion, defined as a violation of the Gauß constraint, due to the presence of curvature. Such an effect, which was named curvature-induced torsion in [14], is strictly related to the need of deforming the Gauß constraint in phase spaces describing piecewise homogeneouslycurved (instead of piecewise flat) geometries [55,[79][80][81][82][83] (see also [84][85][86][87], for an analysis in four dimensions). In terms of defect excitations discussed in this paper, torsion excitations interpreted as spinning particles can arise from the fusion of two spinless defects, since two particles can have orbital angular momentum.…”