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 $f^{i}_{cd}$ is found, which satisfies that the constraint algebra is
anomaly-free. This result confirms the possibility of non-trivial 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 effective loop quantum gravity.
