From black strings to fundamental strings: non-uniformity and phase transitions

A bstract We discuss the transition between black strings and fundamental strings in the presence of a compact dimension, $$ {\mathbbm{S}}_z^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> . In particular, we study the Horowitz-Polchinski effective field theory in $$ {\mathbb{R}}^d\times {\mathbbm{S}}_z^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> , with a reduction on the Euclidean time circle $$ {\mathbbm{S}}_{\tau}^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>τ</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> . The classical solution of this theory describes a bound state of self-gravitating strings, known as a “string star”, in Lorentzian spacetime. By analyzing non-uniform perturbations to the uniform solution, we identify the critical mass at which the string star becomes unstable towards non-uniformity along the spatial circle (i.e., Gregory-Laflamme instability) and determine the order of the associated phase transition. For 3 ≤ d &lt; 4, we argue that at the critical mass, the uniform string star can transition into a localized black hole. More generally, we describe the sequence of transitions from a large uniform black string as its mass decreases, depending on the value of d . Additionally, using the SL(2) k / U(1) model in string theory, we show that for sufficiently large d , the uniform black string is stable against non-uniformity before transitioning into fundamental strings. We also present a novel solution that exhibits double winding symmetry breaking in the asymptotically $$ {\mathbb{R}}^d\times {\mathbbm{S}}_{\tau}^1\times {\mathbbm{S}}_z^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>τ</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> Euclidean spacetime.