Holomorphic Simplicity Constraints for 4d Spinfoam Models

TL;DR

The paper develops a holomorphic formulation of the 4d Riemannian gravity simplicity constraints in the spinfoam framework by exploiting a spinor description of SU(2) intertwiners and the Spin(4) factorization. It introduces holomorphic/anti-holomorphic $F$-simplicity constraints, proves their classical equivalence to standard constraints, and solves them quantum mechanically with new coherent intertwiners that diagonalize the annihilation operators in the U(N) framework, enabling a Gupta–Bleuler-type treatment. A full spin-network formalism is constructed with these simple intertwiners, and a novel spinfoam model is proposed where vertex amplitudes are holomorphic in spinor data and amplitudes are obtained from evaluating coherent spin networks; the BF theory limit is discussed via a spinor-based discretized path integral. The work offers a route to strongly impose simplicity constraints, avoids strict diagonal simplicity in the EPRL–FK sense, and opens avenues for analyzing semiclassical behavior, renormalization, and extensions to Lorentzian signature, as well as possible connections to group-field theories and spinfoam cosmology.

Abstract

Within the framework of spinfoam models, we revisit the simplicity constraints reducing topological BF theory to 4d Riemannian gravity. We use the reformulation of SU(2) intertwiners and spin networks in term of spinors, which has come out from both the recently developed U(N) framework for SU(2) intertwiners and the twisted geometry approach to spin networks and spinfoam boundary states. Using these tools, we are able to perform a holomorphic/anti-holomorphic splitting of the simplicity constraints and define a new set of holomorphic simplicity constraints, which are equivalent to the standard ones at the classical level and which can be imposed strongly on intertwiners at the quantum level. We then show how to solve these new holomorphic simplicity constraints using coherent intertwiner states. We further define the corresponding coherent spin network functionals and introduce a new spinfoam model for 4d Riemannian gravity based on these holomorphic simplicity constraints and whose amplitudes are defined from the evaluation of the new coherent spin networks.

Figures (4)

• Figure 1: From left ro right, plots of the modified Bessel function $I_1(x)$, of its logarithm $\ln I_1(x)$ which illustrates its asymptotic behavior for large $x\gg 1$, and the ratio $I_2(x)/I_1(x)$ which quickly converges to $1$ as $x\rightarrow+\infty$.
• Figure 2: Plots of the (un-normalized) probability distribution $P_A[J]$ for the area $J$ for various values of $A(z)$: from left to right $A(z)$ is 5, 20 and 40. We see that these distributions are approximatively Gaussians centered around the classical value $A(z)$.
• Figure 3: On the left, plot of the Poisson distribution $P[j_f]$ describing the probability distribution of the spin label $j_f$ for a value $x=\langle z_f|z_f\rangle=50$. The x-axis gives the value of $2j_f$. On the right, its superposition with its Gaussian approximation around its maximum $2j_f\sim x=50$.
• Figure 4: From the left to the right: the boundary graph of a 4-simplex $\sigma$ (nodes are tetrahedra $\tau$ and links are triangles $\Delta$), a tetrahedron $\tau$ shared by the two 4-simplices $S(\tau)$ and $T(\tau)$, and a plaquette around a triangle $\Delta$ with all the 4-simplices and tetrahedra sharing the same triangle.

Theorems & Definitions (4)

• Proposition 1.1
• proof
• Proposition 1.2
• proof