as is usual with a square root, one needs to determine its sign. the overall phase of the partition function is not an observable in quantum mechanics, and so for a given configuration this sign choice can be made arbitrarily. but one needs to check that the sign choice is consistent. to do this, let us deform the configuration through the configuration space, on a path which eventually returns to the original configuration. if the sign choice was consistent then, having returned to the original configuration, one will have the original sign. this is what needs to be checked.