Summary
The equations of motion of the general background of curved space-time, Einstein’s equations, are derived simply by demanding the renormalized energy-momentum tensor of a bosonic string propagating in this background to be traceless. The energy-momentum tensor of such a string is then separable into a holomorphic and an antiholomorphic part as a consequence of the conformal invariance of the theory regained at the quantum level.
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Marrakchi, A.L., Singh, L.P. String dynamics in curved space-time revisited. Nuov Cim B 107, 903–909 (1992). https://doi.org/10.1007/BF02899292
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DOI: https://doi.org/10.1007/BF02899292