# Variation of the auxiliary worldsheet metric

## Main Question or Discussion Point

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

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Ben Niehoff
Gold Member
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?

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

Ben Niehoff
Gold Member
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

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