“Some very pretty 19th-century mathematics now comes into play. A two-manifold whose metric is given up to a Weyl transformation is called a Riemann surface. As in the 1D case, a Riemann surface can be characterized up to diffeomorphism by finitely many parameters. There are two big differences: The parameters are now complex rather than real, and their range is restricted in a way that leaves no room for an ultraviolet divergence. I will return to that last point later.”

I think I am behind on my 19th century mathematics.