SU(2) graph invariants, Regge actions and polytopes

Abstract

We revisit the the large spin asymptotics of 15j symbols in terms of cosines of the 4d Euclidean Regge action, as derived by Barrett and collaborators using a saddle point approximation. We bring it closer to the perspective of area-angle Regge calculus and twisted geometries, and compute explicitly the Hessian and phase offsets. We then extend it to more general SU(2) graph invariants, showing that saddle points still exist and have a similar structure. For graphs dual to 4d polytopes we find again two distinct saddle points leading to a cosine asymptotic formula, however a conformal shape-mismatch is allowed by these configurations, and the asymptotic action is thus a generalisation of the Regge action. The allowed mismatch correspond to angle-matched twisted geometries, 3d polyhedral tessellations with adjacent faces matching areas and 2d angles, but not their diagonals. We study these geometries, identify the relevant subsets corresponding to 3d Regge data and flat polytope data, and discuss the corresponding Regge actions emerging in the asymptotics. Finally, we also provide the first numerical confirmation of the large spin asymptotics of the 15j symbol. We show that the agreement is accurate to the per cent level already at spins of order 10, and the next-to-leading order oscillates with the same frequency and same global phase.

Figures (12)

• Figure 1: Numerical data versus the analytic leading order, equilateral configuration. See Section \ref{['SecLO']} for details.
• Figure 2: Evaluation times for the equilateral configuration of Fig. \ref{['FigLO']}. The log-log plot shows a power law increase in the needed time, with power law that can be fitted by $0.01 \lambda ^{4.36}$. The extrapolated time for $\lambda = 25$ which is the last point of Figure \ref{['FigLO']} is of about three hours and a half. Note that this estimation is done with $3jm$'s, $6j$'s and $9j$'s already calculated and pre-loaded into RAM.
• Figure 3: Left panel: An example of saddle-less configuration showing exponential decay of the amplitude. Log-log plot in the main picture, with data points in red and a $\lambda^{-6}$ plotted line for comparison. A linear plot is also shown in the upper picture. The boundary data are those of a Lorentzian 4-simplex, which satisfy closure but not the orientation condition (\ref{['CP1']}). Right panel: A vector geometry configuration with a single critical point, showing a power law decay $\lambda^{-6}$. Log-log plot in the main picture, with data points in red and the analytic result (\ref{['LO']}) and $\propto \lambda^{-6}$ plotted for comparison. A linear plot is also shown in the upper picture. The boundary data are $j_{ab}=2$ except $j_{23}=j_{45}=3$, and angles $\varphi_{53,42}=\frac{\pi}{4}$, $\varphi_{34,5}=\varphi_{25,4}=\varphi_{25,3}=\varphi_{34,2}=\frac{\pi}{8}$ in the notation developed in Section \ref{['SecVecGeo']}.
• Figure 4: Numerical data points versus the analytic leading order (\ref{['LO2']}) for equilateral Regge data, log-log (left panel) and linear (right panel) plots.
• Figure 5: Numerical data points versus the analytic leading order (\ref{['LO2']}) for isosceles Regge data, log-log (left panel) and linear (right panel) plots. We provide less data points than for the equilateral configuration of Fig. \ref{['FigLO']} because of the slower evaluation times (caused by the ratio 2-to-1 between the spins) and of the absence of half-integer spins, not allowed for this configuration.