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We develop a general response theory of gapless Fermi points with nontrivialtopological charges for gauge and nonlinear sigma fields, which asserts thatthe topological character of the Fermi points is embodied as the terms withdiscrete coefficients proportional to the corresponding topological charges.Applying the theory to the effective non-linear sigma models for topologicalFermi points with disorders in the framework of replica approach, we deriverigorously the Wess-Zumino terms with the topological charges being theirlevels in the two complex symmetry classes of A and AIII. Intriguingly, twonontrivial examples of quadratic Fermi points with the topological charge `2'are respectively illustrated for the classes A and AIII. We also address aqualitative connection of topological charges of Fermi points in the realsymmetry classes to the topological terms in the non-linear sigma models, basedon the one-to-one classification correspondence.

General response theory of topologically stable Fermi points and its implications for disordered cases