Off-Shell Strings II: Black Hole Entropy

In 1994, Susskind and Uglum argued that it is possible to derive the Bekenstein-Hawking entropy $A/4G_N$ from string theory. In this article we explain the conceptual underpinnings of this argument, while elucidating its relationship to induced gravity and ER=EPR. Following an off-shell calculation by Tseytlin, we explicitly derive the classical closed string effective action from sphere diagrams at leading order in $\alpha^{\prime}$. We then show how to use this to obtain black hole entropy from the RG flow of the NLSM on conical manifolds. (We also briefly discuss the more problematic ``open string picture'' of Susskind and Uglum, in which strings end on the horizon.) We then compare these off-shell results with the rival ``orbifold replica trick'' using the on-shell $\mathbb{C}/Z_{N}$ background, which does not account for the leading order Bekenstein-Hawking entropy -- unless perhaps tachyons are allowed to condense on the orbifold. Possible connections to the ER=EPR conjecture are explored. Finally, we discuss prospects for various extensions, including prospects for deriving holographic entanglement entropy in the bulk of AdS.