  Nonlinear optical frequency conversion, observed more than half a century
ago, is a corner stone in modern applications of nonlinear and quantum optics.
It is well known that frequency conversion processes are constrained by
conservation laws, such as momentum conservation that requires phase matching
conditions for efficient conversion. However, conservation laws alone could not
fully capture the features of nonlinear frequency conversion. Here it is shown
that topology can provide additional constraints in nonlinear multi-frequency
conversion processes. Unlike conservation laws, a topological constraint
concerns with the conserved properties under continuous deformation, and can be
regarded as a new indispensable degree of freedom to describe multi-frequency
processes. We illustrate such a paradigm by considering sum frequency
generation under a multi-frequency pump wave, showing that, akin topological
phases in topological insulators, topological phase transitions can be observed
in the frequency conversion process both at classical and quantum level.
