  Recent developments of experimental techniques have given us unprecedented
opportunities of studying topological insulators in high dimensions, while some
of the dimensions are "synthetic", in the sense that the effective lattice
momenta along these synthetic dimensions are controllable periodic tuning
parameters. In this work, we study interaction effects on topological
insulators with synthetic dimensions. We show that although the free fermion
band structure of high dimensional topological insulators can be precisely
simulated with the "synthetic techniques", the generic interactions in these
effective synthetic topological insulators are qualitatively different from the
local interactions in ordinary condensed matter systems. And we show that these
special but generic interactions have unexpected effects on topological
insulators, namely they would change (or reduce) the classification of
topological insulators differently from the previously extensively studied
local interactions.
