Logarithmic exotic conformal Galilean algebras

Logarithmic representations of the conformal Galilean algebra (CGA) and the Exotic Conformal Galilean algebra (ecga) are constructed. This can be achieved by non-decomposable representations of the scaling dimensions or the rapidity indices, specific to conformal Galilean algebras. Logarithmic representations of the non-exotic CGA lead to the expected constraints on scaling dimensions and rapidities and also on the logarithmic contributions in the co-variant two-point functions. On the other hand, the ecga admits several distinct situations which are distinguished by different sets of constraints and distinct scaling forms of the two-point functions. Two distinct realisations for the spatial rotations are identified as well. This is the first concrete example of a reducible, but non-decomposable representation, without logarithmic terms. Such cases had been anticipated before.