Holomorphic Simplicity Constraints for 4d Spinfoam Models

TL;DR

This paper introduces a new set of holomorphic simplicity constraints for 4d Riemannian gravity within spinfoam models, utilizing spinor reformulations and coherent states to improve the quantum imposition of constraints.

Contribution

It develops holomorphic simplicity constraints using spinor and U(N) frameworks, enabling strong quantum imposition and leading to a new spinfoam model for 4d gravity.

Findings

Holomorphic constraints are equivalent to classical ones.

Coherent intertwiners solve the new constraints.

A new spinfoam model is constructed from these constraints.

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.