Structure

8 vertices

Input sites of one MERA layer. Binary branching \(b=2\) and spatial dimension \(n=3\) give \(N=b^n=8\) — the vertices of \(Q_3\).

12 edges = \(C_1(Q_3)\)

The edge space. The curl-Laplacian \(L_1^{\uparrow} = \partial_2\partial_2^{\top}\) has spectrum {0⁽⁷⁾, 4⁽³⁾, 6⁽²⁾}, whose eigenspaces carry \(U(1)_Y\), \(SU(2)_L\), and \(SU(3)_c\).

6 faces — graviton

The proposed gravitational force carrier. The hyperoctahedral \(B_3\) symmetry of \(Q_3\) selects the two-dimensional irrep \(W_2\) on the hinge space of \(Q_3\times I\), with multiplicity 3, giving \(\dim W_2 \times m_{W_2} = 2\times 3 = 6\) as the spin-2 graviton mode.

Body diagonal — \(U(1)_Y\)

The transient open chain from \(v_{000}\) to \(v_{111}\). The hypercharge correction \(\delta_Y\) is the body-diagonal Wilson exponent.

Face cycles — \(SU(2)_L\)

The closed 4-cycles around the two \(xy\)-faces (\(z=0\) and \(z=1\)), co-located with the disentangler face-diagonals on those same two faces. \(SU(2)_L\) sits in the \(\lambda=4\) eigenspace of \(L_1^{\uparrow}\).

Disentanglers

The MERA unitaries that remove short-range entanglement at each layer. Proposed as the origin of the gauge force carriers, which sit in the curl-Laplacian eigenspaces of \(C_1(Q_3)\); they generate the \(B_3\)-equivariant correction on the twelve edges that splits the unified coupling into the distinct sector values. Placement: the four face-diagonal antipodal vertex pairs on the two \(xy\)-faces (\(z=0\) and \(z=1\)).

Output site

The single output of one MERA layer. The isometry \(w:\mathcal{H}^{\otimes 8}\to\mathcal{H}\) with \(w^{\dagger}w=\mathbb{I}\) coarse-grains the eight input vertices to one site, dropping the effective count from \(N+1=9\) to \(N=8\); the partition function becomes \(Z=\chi^{N}\) and the per-site vacuum entanglement is \(S^* = 2/9\).

MERA convergence

The eight dashed lines from the vertices to the centre indicate the binary MERA coarse-graining direction (\(b=2\), \(n=3\)).