# Curved Polyhedra

This week I have been studying a great paper:  SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry by  Haggard, Han, Kaminski, and Riello. This post looks at the section of the paper on curved tetrahedra. This carries on the literature review started in the following posts:

Curved Tetrahedra

Using critical point equations, together with the interpretation
of the holonomies Hab we can reconstruct a curved-tetrahedral geometry at every vertex of the graph. This post shows  how to recover the tetrahedral geometry from the holonomies in a constructive way.

The key equation is the closure condition

we will focus on the derived equation

where Ob ∈ SO(3) is the vectorial (spin 1) representation of Hb ∈ SU(2). The Ob are interpreted as parallel transports along specific, simple paths on the tetrahedron 1-skeleton.

The ordered composition of all the paths associated to a tetrahedron is equivalent to the trivial path, hence the identity on the right-hand side of the closure equation.

Flatly Embedded Surfaces

Consider a 4-dimensional spacetime (M4,  gαβ), with no torsion, constant curvature λ , and tetrad eαI . In this spacetime, consider a bounded 2-surface s that is flatly embedded in M4, i.e. such that the wedge product of its space- and time-like normal fields, nα and uβ respectively, is preserved by parallel transport on the surface s.45 Then, the holonomy around s of the torsionless spin connection   is given in the spinor representation by:

where the subscript + indicates the self-dual part of an object, as is the area of s, and we have defined uI and nI to be the internal spacelike and timelike normals to the surface. In the future pointing time gauge  and therefore,

where the last two equalities hold in time-gauge.

Finally we obtain in the future pointing time gauge,

and  in the vectorial representation,

Beyond the properties of area, curvature, and orientation, the shape of s is not defined for the moment. We can  further constrain its geometric degrees of freedom  by requiring each vertex of the graph to be identified with the simplest curved geometrical object with four faces, a homogeneously curved tetrahedron.
The cosmological constant or equivalently the curvature is totally free at each face. So this model cannot be considered a quantization of gravity with a fixed-sign cosmological constant: it is
rather a quantization of gravity with a cosmological constant, the sign of which is determined dynamically, and only semiclassically, by the imposed boundary conditions  the external jab and ξab.

Constant Curvature Tetrahedra

The faces of the curved tetrahedron are spherical or hyperbolic triangles, with a radius of curvature equal to . This means that their areas must lie in the interval orrespectively.

The spherical case is no problem, since SU(2) group elements have the right periodicity in their argument. By looking at the deformed SU(2)q representations with  and  , one only finds spins up to |k|/2, which translates into γj ≤ 6π/|Λ|.

The hyperbolic case, on the other hand, is more subtle.  These subtleties can give rise—in certain cases determined by the choice of the spins—to non-standard geometries that extend across the two sheets of the two-sheeted hyperboloid.

Consider the reconstruction at the vertex 5 of 􀀀Γ5. The closure
equation in the vector representation is

and the special side is 24. We will take the base point to be vertex 4. Because all the holonomies are based at vertex 4, all the nb are defined there, which we notate nb(4). However the property of being flatly embedded means having vanishing extrinsic curvature, and so this makes the normal to a face well-defined at any of its points. The faces 1, 2 and 3  contain vertex 4, and this means that n1(4), n2(4), n3(4) can be directly interpreted as normals to their respective faces, while n4(4) is the vector obtained after parallel-transporting n4 from its face to vertex 4, via the edge 24. That is,

where ocb is the vector representation of the holonomy from vertex b to vertex c, along the side cb.

It is possible to give an expression of the cosines of the dihedral angles directly in terms of the holonomies,

which are a sort of normalized, connected two-point functions of the holonomies.

Once we have unambiguously fixed the cosines of the dihedral angles, these can be used to construct the Gram matrix of the tetrahedron

The determinant of ³Gram determines whether the tetrahedron is hyperbolic or spherical. detG > 0 gives spherical geometry whilst
detG < 0 gives hyperbolic geometry, this therefore provides the crucial information

This fixes the sign of the cosmological constant at a given vertex. Consequently, there is no freedom, within a vertex, to change this sign, and a unique correspondence between the spinfoam and geometric data can finally be established. Note that flipping the sign of the cosmological constant does not change the Gram matrix, since it corresponds to flipping all the ±b. This fact is crucial, since it means that sgn () can actually be calculated.

Finally, from the Gram matrix one can fully reconstruct the curved tetrahedron. In practice this amounts to repeatedly applying the spherical (and/or hyperbolic) law of cosines to first calculate the face angles of the tetrahedron and then its side lengths.

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# A volume formula for hyperbolic tetrahedra in terms of edge lengths by Murakami and Ushijima

In this paper the authors  give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths.

Volume formulae for hyperbolic tetrahedra

For complex numbers z, a1, a2, . . . , a6, define a complex valued function U = U(a1, a2, a3, a4, a5, a6, z) as follows:

where Li2 is the dilogarithm function

Then define z+ and z- by,

where

Define a complex-valued function V(a1, a2, a3, a4, a5, a6,) as follows:

For the  tetrahedron

in this case:

where

and

is the Gram matrix defined in terms of dihedral angles.

Then

and

where,

and

Gl is the Gram matrix in terms of edge lengths defined as

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