Wick rotation for spin foam quantum gravity

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Models of spin foam quantum gravity come in different spacetime signatures, either Euclidean or Lorentzian. The choice is reflected in the use of $Spin(4)$ or $SL(2,\mathbb{C})$ as gauge groups for the holonomies of the connection. In this work we show that a rotation of the Immirzi parameter to purely imaginary values maps the Euclidean Engle-Pereira-Rovelli-Livine spin foam model to its Lorentzian version and vice versa. Our methods provide a general recipe for relating spin foam models of different signature through analytic continuation of the gauge groups and their unitary irreducible representations.