Spinfoams in the holomorphic representation

TL;DR

This paper introduces a holomorphic representation for spinfoams using coherent state transforms, providing a clear geometric interpretation and analyzing the semiclassical limit related to Regge calculus.

Contribution

It derives the 4d spinfoam vertex in the holomorphic representation for both Euclidean and Lorentzian gravity, linking variables to classical geometry.

Findings

Variables have a clear interpretation in classical geometry.

The semiclassical limit reproduces the Regge action.

The representation simplifies the analysis of spinfoam asymptotics.

Abstract

We study a holomorphic representation for spinfoams. The representation is obtained via the Ashtekar-Lewandowski-Marolf-Mour\~ao-Thiemann coherent state transform. We derive the expression of the 4d spinfoam vertex for Euclidean and for Lorentzian gravity in the holomorphic representation. The advantage of this representation rests on the fact that the variables used have a clear interpretation in terms of a classical intrinsic and extrinsic geometry of space. We show how the peakedness on the extrinsic geometry selects a single exponential of the Regge action in the semiclassical large-scale asymptotics of the spinfoam vertex.