Holonomy-flux spinfoam amplitude

Understand

We introduce a holomorphic representation for the Lorentzian EPRL spinfoam on arbitrary 2-complexes.

• The representation is obtained via the Ashtekar-Lewandowski-Marolf-Mour\~ao-Thiemann heat kernel coherent state transform.
• The new variables are classical holonomy-flux phase space variables $(h,X)\simeq \mathcal T^*SU(2)$ of Hamiltonian loop quantum gravity prescribing the holonomies of the Ashtekar connection $A=\Gamma + \gamma K$, and their conjugate gravitational fluxes.
• For small heat kernel `time' the spinfoam amplitude is peaked on classical space-time geometries, where at most countably many curvatures are allowed for non-zero Barbero-Immirzi parameter.