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Electromagnetic and gravitational responses of two-dimensional non-interacting electrons in background magnetic field

We compute electromagnetic, gravitational and mixed linear response functions of two- dimensional free fermions in external quantizing magnetic field at an integer filling factor. The results are presented in the form of the effective action and as an expansion of currents and stresses in wave-vectors and frequencies of the probing electromagnetic and metric fields. We identify the terms in linear response functions coming from geometric Chern-Simons, Wen-Zee, and gravitational Chern-Simons terms in effective action. We derive the expressions for Hall conductivity, Hall viscosity and find the current and charge density responses to the spatial curvature as well as stresses caused by inhomogeneous electromagnetic fields.