  We develop a general response theory of gapless Fermi points with nontrivial
topological charges for gauge and nonlinear sigma fields, which asserts that
the topological character of the Fermi points is embodied as the terms with
discrete coefficients proportional to the corresponding topological charges.
Applying the theory to the effective non-linear sigma models for topological
Fermi points with disorders in the framework of replica approach, we derive
rigorously the Wess-Zumino terms with the topological charges being their
levels in the two complex symmetry classes of A and AIII. Intriguingly, two
nontrivial examples of quadratic Fermi points with the topological charge `2'
are respectively illustrated for the classes A and AIII. We also address a
qualitative connection of topological charges of Fermi points in the real
symmetry classes to the topological terms in the non-linear sigma models, based
on the one-to-one classification correspondence.
