Currently I am reading the Di Francesco-Mathieu-Sénéchal textbook on conformal field theory. Above the equation 5.52, the author argues that the EM tensor should have scaling dimension 2 and spin 2. But why did the author then conclude that both the holomorphic and anti-holomorphic dimension of the tensor are 2? Isn't it that the holomorphic dimension is 2 while the anti-holomorphic one is 0?
Another question: why does the EM tensor transform like a primary field?