Anomalous Charge Conjugation and Parity in (2 + 1)-Dimensional Relativistic Hydrodynamics

We develop a hydrostatic and first-order constitutive description of a mixed anomaly involving \( U(1)_Q \), \( U(1)_F \), charge conjugation (acting only on \( U(1)_Q \)), and parity in 2+1 dimensions, characterized by a mutual Chern-Simons level \( kQF=12 \,\,(mod\ Z) \). Using the established multi-charge anomaly sector, we make explicit that this response is odd under C and \( P \) separately but even under \( CP \), specify a global inflow representative, and derive the equilibrium cross-charge response, including \( ∂ n_Q/∂ B_F=∂ n_F/∂ B_Q=1/4π 1/2π \). An analytic neutral-fluid example shows that this response is nonzero while the source-free hydrodynamic poles are unaffected.