  The synthetic dimension, a research topic of both fundamental significance
and practical applications, is attracting increasing attention in recent years.
In this paper, we propose a theoretical framework to construct arbitrary
synthetic dimensions, or N-boson synthetic lattices, using multiple bosons on
one-dimensional lattices. We show that a one-dimensional lattice hosting N
indistinguishable bosons can be mapped to a single boson on a N-dimensional
lattice with high symmetry. Band structure analyses on this N-dimensional
lattice can then be mathematically performed to predict the existence of exotic
eigenstates and the motion of N-boson wavepackets. As illustrative examples, we
demonstrate the edge states in two-boson Su-Schrieffer-Heeger synthetic
lattices without interactions, interface states in two-boson
Su-Schrieffer-Heeger synthetic lattices with interactions, and weakly-bound
triplon states in three-boson tight-binding synthetic lattices with
interactions. The interface states and weakly-bound triplon states have not
been thoroughly understood in previous literatures. Our proposed theoretical
framework hence provides a novel perspective to explore the multi-boson
dynamics on lattices with boson-boson interactions, and opens up a future
avenue in the fields of multi-boson manipulation in quantum engineering.
