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I have a question about deriving variation of metric under Weyl and coordinate transformations in Polchinski's string theory (3.3.16).

Under transformation $$\zeta: g \rightarrow g^{\zeta}, \,\,\, g_{ab}^{\zeta}(\sigma')=\exp[ 2 \omega (\sigma) ] \frac{ \partial \sigma^c }{\partial \sigma'^a} \frac{ \partial \sigma^d}{\partial \sigma'^b} g_{cd}(\sigma) \tag{3.3.10} $$

how to show $$ \delta g_{ab} = 2 \delta \omega g_{ab} - \nabla_a \delta \sigma_b-\nabla_b \delta \sigma_a ? \tag{3.3.16}$$ The first term in (3.3.16) comes from Weyl transformation. I am unable to derive the second and third terms.

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1 Answer 1

up vote 1 down vote accepted

It goes a little something like this:

\begin{align*} \delta g_{ab}(\sigma)&=g_{ab}^{\zeta}(\sigma)-g_{ab}(\sigma)\\ &=\exp(2\omega(\sigma-\delta\sigma))\frac{\partial (\sigma^c-\delta \sigma^c)}{\partial \sigma^a}\frac{\partial( \sigma^d-\delta \sigma^d)}{\partial \sigma^b}g_{cd}(\sigma-\delta \sigma)-g_{ab}(\sigma)\\ &\approx (1+2\omega )({\delta^c}_a-\partial_a\delta \sigma^c)({\delta^d}_b-\partial_b\delta \sigma^d)(g_{cd}(\sigma)-\delta\sigma^e\partial_eg_{cd}(\sigma))-g_{ab}(\sigma)\\ &\approx 2\omega g_{ab}(\sigma)-\partial_a\delta\sigma_b-\partial_b\delta\sigma_a-\delta\sigma^e\partial_eg_{ab}(\sigma). \end{align*} The last expression we recognize as the Lie derivative of the metric along the vector field $\delta\sigma^a$. What you wrote down is an equivalent form using the covariant derivative.

P.S. you made it to the hardest chapter :)

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