Off-shell strings II: Black hole entropy

Abstract

In 1994, Susskind and Uglum argued that it is possible to derive the Bekenstein-Hawking entropy A/4G_N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>/</mml:mi> <mml:mn>4</mml:mn> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> </mml:math> 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} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>α</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> . 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} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>ℂ</mml:mi> <mml:mi>/</mml:mi> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> </mml:math> 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.