Spinfoams in the holomorphic representation

We study a holomorphic representation for spinfoams. The representation is obtained via the Ashtekar-Lewandowski-Marolf-Mourão-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.

Figure 1

The bulk holonomies $h_{vl}$ are associated to the internal face $f$ as well as to the external face $\tilde{f}$. The notation is the following: $v$ are spinfoam vertices and $l$ are links intersection of the faces with a small 3-sphere centered on the corresponding vertex. The boundary holonomy $h_{\tilde{l}}$ is associated to the external side (dashed line) of the boundary face $\tilde{f}$.