With weight-shifting operators, \(d\neq 2\) looks increasingly like \(d=2\) in CFT

When working on conformal field theory, your life is very different depending on whether the dimension is two or not. In \(d=2\) you have that infinite-dimensional symmetry algebra called the Virasoro algebra, and in some important cases such as minimal models you can classify your CFTs, and solve them analytically. In \(d\neq 2\), your symmetry algebra is finite-dimensional, and you mostly have to do with numerical results. This not only makes you code a lot, but also incites you to make technical assumptions that are physically restrictve, such as unitarity.

Degenerate fields in \(d=2\) CFT

What makes \(d=2\) CFT solvable in many cases is the existence of degenerate primary fields.