Multi-orientable group field theory
type: Item
instance of: scholarly article
title: Multi-orientable group field theory
zbMATH Open document ID: 1246.81172
DOI: 10.1088/1751-8113/45/16/165401
author: Adrian Tanasa
published in: Journal of Physics A: Mathematical and Theoretical
publication date: 2012-06-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.0694
Mathematics Subject Classification ID: 81T18
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 6040418
zbMATH Keywords: group field theory
zbMATH Keywords: colored group field theory
zbMATH Keywords: multi-orientable group field theory
zbMATH Keywords: Boulatov model
zbMATH Keywords: colorable graph
zbMATH Keywords: multi-orientable graph
MaRDI profile type: Publication
OpenAlex ID: W3104393399
arXiv ID: 1109.0694
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review text: Group field theories (GFTs) are quantum field theories over group manifolds. They are introduced to generalize the role of the ribbon graphs (combinatorial maps) to two-dimensional quantum gravity to three- or four-dimensional space [\textit{L. Freidel}, Int. J. Theor. Phys. 44, No. 10, 1769--1783 (2005; Zbl 1100.83010)]. But, in general, combinatorial maps appear in GFT are not duals of manifolds. To eliminate a whole class of ``wrapping singularities'', the colored GFT has been introduced [\textit{R. Gurau}, Commun. Math. Phys. 304, No. 1, 69--93 (2011; Zbl 1214.81170)]. In this paper, an alternative class of GFT, multi-oriented GFT, is introduced. It is shown the class of multi-orientable GFT Feynman graphs (multi-orientable graphs) is wider than the class of colored GFT Feynman graphs (colorable graphs, prop. 3.1).NEWLINENEWLINE The definition of three-dimensional multi-oriented GFT uses the Boulatov model, whose field is a map \(\phi: SU(2)\to\mathbb{R}\) with the action NEWLINE\[NEWLINE\begin{multlined} S[\phi]={1\over 2} \int dg_1 dg_2 dg_3\phi(g_1, g_2, g_3) \phi(g_1, g_2, g_3)+\\ {\lambda\over 4!} \int dg_1\cdots dg_6\phi(g_1, g_2, g_3) \phi(g_3, g_5, g_5) \phi(g_5, g_2, g_6) \phi(g_6, g_4, g_1)\end{multlined}NEWLINE\]NEWLINE [\textit{D. V. Boulatov}, Mod. Phys. Lett. A 7, No. 18, 1629--1646 (1992; Zbl 1020.83539)]. In colored GFT, the real Boulatov field is replaced by the four complex-valued field \(\phi_p\); \(p= 0,\dots, 3\), being referred to as color. Then impose a clockwise cycle ordering at one of the types of vertices of the four colors at the vertex.NEWLINENEWLINE Multi-oriented GFT does not need copies of a complex field \(\phi\). The Boulatov interaction is restricted to vertices where each corner has \(a+\) or \(a-\) label. Each vertex has two corners labeled with \(+\) and two corners labeled with \(-\), which are cyclically ordered (explanation using a graph can be seen in Fig. 3). The action of this model is NEWLINE\[NEWLINES[\phi]= {1\over …
reviewed by: Akira Asada
description: scientific article; zbMATH DE number 6040418