Strings at the Tip of the Cone and Black Hole Entropy From the Worldsheet: Part I

TL;DR

This paper investigates the worldsheet description of closed bosonic strings on a 2D flat cone, calculating the cylinder partition function, entropy, and analyzing IR divergences, revealing insights into black hole entropy from string theory.

Contribution

It introduces a novel calculation of string configurations and entropy on a conical target space, connecting string worldsheet dynamics to black hole entropy without IR divergences.

Findings

The entropy depends linearly on the radial cutoff and is IR finite under a specific renormalization scheme.

The number of string configurations includes line defects that connect boundaries, interpreted as open strings ending on a point.

The entropy exhibits a maximum at a specific radius in each winding sector, remaining finite after summation.

Abstract

We study the nonlinear sigma model (NLSM) worldsheet action describing the motion of closed bosonic strings in the target space of a two-dimensional (2D) flat cone in polar coordinates. We calculate the cylinder partition function. We first place the cylindrical worldsheet on a rectangular lattice before taking the continuum limit. We find an integer number of string configurations on the worldsheet, which we call line defects, that run from one boundary of the cylinder to the other. We insert two sources (conical defects) at each boundary and fix the two ends of the line defect by Dirichlet boundary conditions to a point $r_c$ in target space. In target space, a line defect appears as an Susskind\&Uglum-type open string ending on $r_c$. We compute the semiclassical contribution to the off-shell cylinder amplitude by saddle point approximation. The amplitude has an interesting infrared (IR) divergence structure that depends on the given range of the cone angle. We then compute the entropy by varying the cone angle. In a particular renormalization scheme that relates the ultraviolet (UV) and to the infrared (IR) limits of the modulus integral, we find the entropy to be free of IR divergences but linearly dependent on the radial cutoff. We argue that our calculation provides a well-defined state on a constant Euclidean-time slice directly from the string worldsheet. We also study the 2D flat cone NLSM without discretization. We compute the entropy from the off-shell stationary action and show it is finite in each winding sector $W$ with a maximum at $r_c=\sqrt{\alpha'}/|W|$. After summing over all winding sectors, it still has a finite maximum in the UV limit but for $r_c >0$.