  BPS spectrum with finite number of states are found for higher rank four
dimensional N=2 theory engineered from six dimensional A_{N-1} (2,0) theory on
a Riemann surface with various kinds of defects. The wall crossing formula is
interpreted as the quantum dilogarithm identity. Various methods including
quiver representation theory, maximal green mutation, and cluster algebra are
used extensively. The spectral generator and its refined version for the higher
rank theory are written down using the explicit spectrum information. The
finite chamber has an interesting N^3 behavior in the large N limit.

High Energy Physics - Theory;Algebraic Geometry;Combinatorics;Quantum Algebra