In order to derive the classical string action from the worldsheet, it is necessary to take string theory off shell. This can be done by a prescription of Tseytlin, who proposed taking the worldsheet sphere QFT partition function and *differentiating* it with the log of the UV cutoff. I will explain why this strange prescription always gives the correct answers, for both the S-matrix and equations of motion, to all orders in perturbation theory. I will also compare the Susskind-Uglum off-shell method of calculating black hole entropy, to the more popular (but also more dubious) orbifold method. Based on work with Amr Ahmadain (arXiv:2211.08607 and arXiv:2211.16448).
