Anomaly Matching Condition In Two-Dimensional Systems

Based on the Son-Yamamoto relation obtained for the transverse part of the triangle axial anomaly in QCD(4), we derive its analog in a two-dimensional system. It connects the transverse part of the mixed vector-axial current two-point function with the diagonal vector and axial current two-point functions. Being fully nonperturbative, this relation may be regarded as anomaly matching for conductivities or certain transport coefficients depending on the system. We consider the holographic renormalization group flows in holographic Yang-Mills-Chern-Simons theory via the Hamilton-Jacobi equation with respect to the radial coordinate. Within this holographic model, it is found that the renormalization group flows for the following relations are diagonal: the Son-Yamamoto relation and the left-right polarization operator. Thus, the Son-Yamamoto relation holds at a wide range of energy scales.