= String sigma model in a spacetime metric
{title2=$S=-\frac T2\int\sqrt{-h}\,h^{ab}g_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu$}
Replacing the flat target metric in the <Polyakov action> by a <spacetime metric> $g_{\mu\nu}(X)$ gives the displayed string sigma model. The embedding fields $X^\mu$ map the <worldsheet> into target spacetime, while $h_{ab}$ is an independent <worldsheet metric>. Classical <worldsheet diffeomorphisms> and <Weyl transformations> permit <conformal gauge>; quantum consistency additionally requires vanishing <sigma-model beta functions>.
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