  Geometrical dimensionality plays a fundamentally important role in the
topological effects arising in discrete lattices. While direct experiments are
limited by three spatial dimensions, the research topic of synthetic dimensions
implemented by the frequency degree of freedom in photonics is rapidly
advancing. The manipulation of light in such artificial lattices is typically
realized through electro-optic modulation, yet their operating bandwidth
imposes practical constraints on the range of interactions between different
frequency components. Here we propose and experimentally realize all-optical
synthetic dimensions involving specially tailored simultaneous short- and
long-range interactions between discrete spectral lines mediated by frequency
conversion in a nonlinear waveguide. We realize triangular chiral-tube lattices
in three-dimensional space and explore their four-dimensional generalization.
We implement a synthetic gauge field with nonzero magnetic flux and observe the
associated multidimensional dynamics of frequency combs, all within one
physical spatial port. We anticipate that our method will provide a new means
for the fundamental study of high-dimensional physics and act as an important
step towards using topological effects in optical devices operating in the time
and frequency domains.
