Confinement and chiral symmetry breaking in holography: a smooth switch-off

A bstract We revisit the holographic description of the thermal first order phase transition of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 SYM compactified on a spatial circle. At the transition, the dominant bulk saddle exchanges between a geometry with a compact spatial circle and one with a compact Euclidean time circle. We construct a one-parameter family of Euclidean geometries that describes the unstable branch of the transition, completing the swallow-tail structure of the free energy. Although these configurations are thermodynamically unstable, they provide a continuous interpolation between the confining soliton and the deconfined black hole phases. Using probe fundamental strings, we show that the theory remains confining along the unstable branch, with a string tension that decreases smoothly and vanishes only in the black hole limit. Introducing fundamental matter via probe D5-branes, we find that chiral symmetry breaking follows the same pattern: the condensate decreases continuously and switches off precisely where confinement disappears. We discuss the implications for the confinement and chiral symmetry breaking mechanisms at large N c .