Stretched horizon, replica trick and off-shell winding condensate, and all that

A bstract α ′ corrections to near horizon dynamics of a Schwarzschild black hole in a large number of spacetime dimensions D are governed by the worldsheet theory composed of the cigar CFT and the classical sigma model on the sphere at the horizon, along with a timelike-Liouville theory of central charge 26 – D . At leading order in weak string coupling, black hole thermodynamics is insensitive to the details of timelike Liouville theory. In this limit, we use the Lewkowycz-Maldacena-trick motivated infinitesimally off-shell closed string worldsheet formalism in [arxiv: 2310.02313] to calculate thermal entropy exactly in α ′ . The leading term in the α ′ → 0 limit and the first stingy correction of our result are in precise agreement with the target space Callan-Myers-Perry formula. Also, we point out a remarkable simplification at an ultra-low temperature of order 1/√αD 1 / α ′ D and show that for a certain special set of vertex operators involving modes of the D dimensional spacetime and the time-like Liouville theory, the resulting worldsheet theory is related to Virasoro minimal string.