Multi-orientable group field theory.

*(English)*Zbl 1246.81172Group 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 [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 [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).

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 \[ \begin{split} S[\phi]={1\over 2} \int dg_1 dg_2 dg_3\phi(g_1, g_2, g_3) \phi(g_1, g_2, g_3)+\\ {\lambda\over 4!} \int dg_1\cdots dg_6\phi(g_1, g_2, g_3) \phi(g_3, g_5, g_5) \phi(g_5, g_2, g_6) \phi(g_6, g_4, g_1)\end{split} \] [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.

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 \[ S[\phi]= {1\over 2} \int\overline\phi \phi+{\lambda\over 4!} \int\overline\phi \phi\overline\phi \phi. \] The author says the notion of corners of a vertex is natural within the framework of non-local QFT as in the Moyal noncommutative QFT.

In §3, colorable GFT graphs are shown to be multi-orientable (Prop. 3.1). Then examples of multi-orientable graphs, which are not colorable are presented (Fig. 4 and 6). More precisely, generalized “non-planar” tadpoles and tadfaces are shown to be not multi-orientable, but there exists multi-orientable generalized planar tadpoles, while colorable generalized planar tadpoles do not exist. Since the models on the Moyal space, orientability on the vertex discards the “non-planar” like tadpoles, comparison with the noncommutative QFT is remarked at the end of this section.

In §4, some Feynman amplitudes of non-colorable, but multi-orientable, colorable and non-multi-orientable graphs are computed (Examples of such graphs are given in Fig. 9, 10 and 11). The author remarks the presence of the “middle” strand of these QFT edges is necessary for defining the bubbles [A. Tanasa, “Generalization of the BollobĂˇs-Riordan polynomial for tensor graphs”, J. Math. Phys. 52, Article ID 073514 (2011), arXiv:1012.1798].

Related to renormalizability, it is shown two-point functions of type \(\phi\phi\) or \(\overline\phi\overline\phi\) and four point functions of a type distinct of \(\phi\overline\phi \phi\overline\phi\) are not permitted by multi-orientabillity (Th. 5.1). The author remarks these phenomena appear as some kind of consequence of the fact that the restrictivity of the colorability condition is more significant than the restrictivity of the multi-orientability condition.

§6 discusses generalization of multi-orientability to four-dimensional GFT models [H. Ooguri, Mod. Phys. Lett. A 7, No. 30, 2799–2810 (1992; Zbl 0968.57501)]. Since interaction is odd in this case (cf. Figs. 13, 14 and 15), the following action is proposed \[ S[\phi]={1\over 2} \int\overline\phi \phi+{\lambda_1\over 5!} \int \overline\phi \phi\overline\phi \phi\overline\phi+ {\lambda_2\over 5!} \int\phi\overline\phi \phi\overline\phi\phi. \] In conclusion (§7), the author says that multi-orientable (or colorable) GFT can be seen as a laboratory for testing various QFT.

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 \[ \begin{split} S[\phi]={1\over 2} \int dg_1 dg_2 dg_3\phi(g_1, g_2, g_3) \phi(g_1, g_2, g_3)+\\ {\lambda\over 4!} \int dg_1\cdots dg_6\phi(g_1, g_2, g_3) \phi(g_3, g_5, g_5) \phi(g_5, g_2, g_6) \phi(g_6, g_4, g_1)\end{split} \] [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.

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 \[ S[\phi]= {1\over 2} \int\overline\phi \phi+{\lambda\over 4!} \int\overline\phi \phi\overline\phi \phi. \] The author says the notion of corners of a vertex is natural within the framework of non-local QFT as in the Moyal noncommutative QFT.

In §3, colorable GFT graphs are shown to be multi-orientable (Prop. 3.1). Then examples of multi-orientable graphs, which are not colorable are presented (Fig. 4 and 6). More precisely, generalized “non-planar” tadpoles and tadfaces are shown to be not multi-orientable, but there exists multi-orientable generalized planar tadpoles, while colorable generalized planar tadpoles do not exist. Since the models on the Moyal space, orientability on the vertex discards the “non-planar” like tadpoles, comparison with the noncommutative QFT is remarked at the end of this section.

In §4, some Feynman amplitudes of non-colorable, but multi-orientable, colorable and non-multi-orientable graphs are computed (Examples of such graphs are given in Fig. 9, 10 and 11). The author remarks the presence of the “middle” strand of these QFT edges is necessary for defining the bubbles [A. Tanasa, “Generalization of the BollobĂˇs-Riordan polynomial for tensor graphs”, J. Math. Phys. 52, Article ID 073514 (2011), arXiv:1012.1798].

Related to renormalizability, it is shown two-point functions of type \(\phi\phi\) or \(\overline\phi\overline\phi\) and four point functions of a type distinct of \(\phi\overline\phi \phi\overline\phi\) are not permitted by multi-orientabillity (Th. 5.1). The author remarks these phenomena appear as some kind of consequence of the fact that the restrictivity of the colorability condition is more significant than the restrictivity of the multi-orientability condition.

§6 discusses generalization of multi-orientability to four-dimensional GFT models [H. Ooguri, Mod. Phys. Lett. A 7, No. 30, 2799–2810 (1992; Zbl 0968.57501)]. Since interaction is odd in this case (cf. Figs. 13, 14 and 15), the following action is proposed \[ S[\phi]={1\over 2} \int\overline\phi \phi+{\lambda_1\over 5!} \int \overline\phi \phi\overline\phi \phi\overline\phi+ {\lambda_2\over 5!} \int\phi\overline\phi \phi\overline\phi\phi. \] In conclusion (§7), the author says that multi-orientable (or colorable) GFT can be seen as a laboratory for testing various QFT.

Reviewer: Akira Asada (Takarazuka)