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We explicitly solved the anomaly-free quantum constraints proposed by Tomlinand Varadarajan for the weak Euclidean model of canonical loop quantum gravity,in a large subspace of the model's kinematic Hilbert space which is the spaceof the charge network states. In doing so, we first identified the subspace onwhich each of the constraints acts convergingly, and then by explicitlyevaluating such actions we found the complete set of the solutions in theidentified subspace. We showed that the space of solutions consists of twoclasses of states, with the first class having a property that involves thecondition known from the Minkowski theorem on polyhedra, and the second classsatisfying a weaker form of the spatial diffeomorphism invariance.

Exploring the Tomlin-Varadarajan quantum constraints in $U(1)^3$ loop quantum gravity: solutions and the Minkowski theorem