4-dimensional BF gravity on the lattice

Abstract

We propose the lattice version of BF gravity action whose partition function leads to the product of a particular form of 15-j symbol which corresponds to a 4-simplex. The action is explicitly constructed by lattice B field defined on triangles and link variables defined on dual links and is shown to be invariant under lattice local Lorentz transformation and Kalb-Ramond gauge transformation. We explicitly show that the partition function is Pachner move invariant and thus topological. The action includes the vanishing holonomy constraint which can be interpreted as a gauge fixing condition. This formulation of lattice BF theory can be generalized into arbitrary dimensions. = 2 2 J m 1 J m1 3 J m3 + J m 4 4 + J .In general any 3nj-symbols (n = 2, 3, 4, 5, ..) can be graphically represented by the closed trivalent graph. For example 6-j symbol can be decomposed into four 3-j symbols by the formula ,where the factor (−1) i (J i −m i ) can be understood as a sign factor coming from the invariant metric. The graphical presentation of the above 6-j symbol can be given 1 J

Graphical calculus of volume, inverse volume and Hamiltonian operators in loop quantum gravity

“…Moreover, the 3 j-symbol has the symmetric property 28) which reflects the fact that the 3 j-symbol takes real numbers (dues to the real CGCs), i.e.,…”

“…Graphic calculus has been introduced in LQG in a few papers (see e.g., [13,14,18,[21][22][23]28,29]). Here we focus on the original Brink's graphical method and its suitable extension to LQG.…”

“…As a powerful tool for practical calculation, graphic calculus has been introduced in LQG in a few papers (see e.g., [13,14,18,[21][22][23]28,29]). These graphical methods are based on the graphical methods developed by Yutsis in [30], Brink in [31], Varshalovich in [32], and Kauffman in [33].…”

“…Brink's graphical method was only recalled in the appendix of [13]. Then Varshalovich's method was adopted in [14,23,28]. Brink's graphical method was also taken to study the propagator of spinfoam models in [29], in which the graphical method was only used to calculate the action of the right-invariant vector field (the "grasping operator") on the intertwiners but not the action of holonomy operator.…”

Generalized Gauge Theories and the Weinberg–salam Model With Dirac–kähler Fermions

“…In fact three-dimensional lattice gravity and four-dimensional BF lattice gravity have been formulated by using the leading terms of the generalized Chern-Simons action which are the standard Chern-Simons action in three dimensions and BF action in four dimensions. 6,7 These experiences of the formulation of lattice gravity including 0-, 1-and 2-form field variables provide us a feeling that the form variables may play an important role in the formulation of lattice gravity.…”