Variation of the auxiliary worldsheet metric

[Jack2013]

Can somebody clarify how the formula for variation of the auxilliary worldsheet metric is obtained due to reparametrization of the worldsheet in string theory??

[Ben Niehoff]

I could probably explain if I knew which formula you are talking about. Could you type it out, or maybe give a reference to one of the string theory books?

[Jack2013]

In D-branes Clifford J. Johnson page 30 Equation 2.26

[Ben Niehoff]

OK, Clifford is using \zeta^1, \zeta^2 to refer to the worldsheet coordinates \sigma, \tau. So the embedding functions are

X^\mu(\zeta^1, \zeta^2)
and the worldsheet metric with "up" indices is

\gamma^{ab}(\zeta^1, \zeta^2) \frac{\partial}{\partial \zeta^a} \otimes \frac{\partial}{\partial \zeta^b}
So, all you need to do is look at the infinitesimal variations of these expressions when you change coordinates

\zeta^a = \zeta^a(\xi^1, \xi^2)