Anomaly freedom of the vector modes with holonomy corrections in perturbative Euclidean loop quantum gravity

We study the perturbation of the effective Hamiltonian constraint with holonomy correction from Euclidean loop quantum gravity. The Poisson bracket between the corrected Hamiltonian constraint and the diffeomorphism constraint is derived for vector modes. Some specific form of the holonomy correction function fcdⁱ is found, which satisfies that the constraint algebra is anomaly free. This result confirms the possibility of nontrivial holonomy corrections from full theory while preserving anomaly-free constraint algebra in the perturbation framework. It also gives valuable hints on the possible form of holonomy corrections in the effective loop quantum gravity.