Topological pump of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mi>Q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> quantum chain and Diophantine equation

A topological pump of the $SU(Q)$ quantum chain is proposed associated with a current due to local $[U(1)]^{\otimes Q}$ gauge invariance colored fermions. The invariant dimer phases are characterized by $Z_Q$ Berry as order parameter $d$-dimensional twist space ($d=Q-1$) synthetic Brillouin zone. By inclusion symmetry breaking perturbation specified rational $\Phi=P/Q$, pump, that encloses around phase boundary, $Q$ Chern numbers currents uniform infinitesimal twists. analysis systems under open/periodic/twisted boundary conditions clarifies bulk-edge correspondence where large transformation generated center mass (CoM) plays central role. An explicit formula for number given using Diophanine equation. Numerical demonstration exact diagonalization and DMRG finite ($Q=3,4$ $5$) have been presented confirm general discussions low energy spectra, edge states, CoM's, correspondence. modified Lieb-Schultz-Mattis type argument also mentioned.