We discuss the transition between black strings and fundamental strings in the presence of a compact dimension, \( {\mathbbm{S}}_z^1 \) . In particular, we study the Horowitz-Polchinski effective field theory in \( {\mathbb{R}}^d\times {\mathbbm{S}}_z^1 \) , with a reduction on the Euclidean time circle \( {\mathbbm{S}}_{\tau}^1 \) . 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 < 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 \( {\mathbb{R}}^d\times {\mathbbm{S}}_{\tau}^1\times {\mathbbm{S}}_z^1 \) Euclidean spacetime.
