Holonomy-flux spinfoam amplitude

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 $(h,X)\simeq {\mathcal{T}}^{\ast }SU(2)$ of Hamiltonian loop quantum gravity prescribing the holonomies of the Ashtekar connection $A=\mathrm{\Gamma }+\gamma K$, 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. We briefly comment on the possibility to use the alternative flipped classical limit.