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The area of the flat metric of a quadratic differential is ${\int }_X|q|$. Multiplying $q$ by a scalar $a$ multiplies this area by $|a|$ and lengths by $|a{|}^{1/2}$. It is preserved by the SL2R action on differentials.

A section of the square of the canonical bundle, locally $q=\varphi (z)d{z}^2$; its coefficient changes by the square of the coordinate derivative. A holomorphic quadratic differential has holomorphic coefficient, and its flat metric of a quadratic differential is $|q{|}^{1/2}$.