In their review of string perturbation theory equation 2.149, D’Hoker and Phong start from a surface integral of a Beltrami differential $\mu$ times a quadratic differential $b$ over a Riemann surface, and they convert that to a contour integral of the quadratic differential $b$ alone.
They do not give enough details for me (new to Beltrami differentials) to figure out how to do this kind of thing in practice, though. I do get the impression they’re rather cavalier about something, given that contour integrals of quadratic differentials are not coordinate invariant. I’ve been Googling around, but without much to show for it.
So, can anyone help me find references/reviews either explaining explicitly how to do this (for example, given the Beltrami differential $\mu$ corresponding to the modification of one certain element of the period matrix, how do I compute $\iint b\mu = \oint_? b\ldots$ for an arbitrary holomorfic/meromorfic quadratic differential $b$?), or explaining the concepts they use (quasiconformal vector fields written in terms of functions with unit step)?
Thank you for any help you can give me!