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I have several question regarding axion. Could anyone give me some brief introduction to what is a axion string, axion field and how is this related to fermion zero mode and chiral zero mode?

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Axionic strings are special cosmic strings defined by the condition that if you look at the axion $a(\phi)$ as a function of the cylindrical coordinate around the cosmic string (which I choose to be along the $z$-axis), it winds around the origin, so $$a(\phi) = a_0 \exp(i\phi) $$ There is a nontrivial "monodromy" around the axionic string, we say. At the very center of the string, $a$ (a complex number) continuously goes to zero. So such an object is protected (at least temporarily, in the case of a gauge $U(1)$ symmetry) by a topological "knot" or "twist". Note that $a$ is a complex scalar field and it is charged under a $U(1)$ group which explains its lightness; the lightness is necessary for it to play its standard role in QCD.

The axionic string nevertheless behaves as an object and it has "collective coordinates" - fields that only exist near the locus of the string and are only functions of the coordinate along the length of the string (and time). In particular, the part of the degrees of freedom that doesn't depend on the position along the string are "zero modes", see

What are zero modes?

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