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The Barbero-Immirzi parameter $\gamma$ appears in the \emph{real} connectionformulation of gravity in terms of the Ashtekar variables, and gives rise to aone-parameter quantization ambiguity in Loop Quantum Gravity. In this paper weinvestigate the conditions under which $\gamma$ will have physical effects inEuclidean Quantum Gravity. This is done by constructing a well-definedEuclidean path integral for the Holst action with non-zero cosmologicalconstant on a manifold with boundary. We find that two general conditions mustbe satisfied by the spacetime manifold in order for the Holst action and itssurface integral to be non-zero: (i) the metric has to be non-diagonalizable;(ii) the Pontryagin number of the manifold has to be non-zero. The latter is astrong topological condition, and rules out many of the known solutions to theEinstein field equations. This result leads us to evaluate the on-shellfirst-order Holst action and corresponding Euclidean partition function on theTaub-NUT-ADS solution. We find that $\gamma$ shows up as a finite rotation ofthe on-shell partition function which corresponds to shifts in the energy andentropy of the NUT charge. In an appendix we also evaluate the Holst action onthe Taub-NUT and Taub-bolt solutions in flat spacetime and find that in thatcase as well $\gamma$ shows up in the energy and entropy of the NUT and boltcharges. We also present an example whereby the Euler characteristic of themanifold has a non-trivial effect on black-hole mergers.

Barbero-Immirzi parameter, manifold invariants and Euclidean path integrals