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Parity and the Immirzi Parameter in Lorentzian Spinfoams

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The parity invariance of spinfoam gravity is an open question. Naively, parity
breaking should reside in the sign of the Immirzi parameter. I show that the
new Lorentzian vertex formula is in fact independent of this sign, suggesting
that the dynamics is parity-invariant. The situation with boundary states and
operators is more complicated. I discuss parity-related pieces of the
transition amplitude and graviton propagator in the large-spin 4-simplex
limit. Numerical results indicate patterns similar to those in the Euclidean
case. In particular, parity-related components of the graviton propagator
differ by a phase. I discuss possible resolutions of this issue.