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Phase Transitions in Separated $D_{p-1}$ and anti-$D_{p-1}$ Branes at Finite Temperature

We consider a pair of parallel ${\rm D}_{p-1}$ and anti-${\rm D}_{p-1}$ branes in flat space, with a finite separation $d$ along some perpendicular spatial direction and at finite temperature. If this spatial direction is compactified on a circle then by T-duality, the system is equivalent to a ${\rm D}_{p}$-anti ${\rm D}_{p} $ pair wrapped around the dual circle with a constant Wilson line $A \approx d $ on one of the branes. We focus in particular on the $p=9$ case and compute the free energy of this system and study the occurrence of second order phase transitions as both the temperature and Wilson line (brane-antibrane separation) are varied. In the limit of vanishing Wilson line we recover the previous results obtained in the literature, whereby the open string vacuum at the origin of the tachyon field T=0 is stabilized at sufficiently high temperature at which a second order phase transition occurs. For sufficiently large Wilson line, we find new second order phase transitions corresponding to the existence of two minima in the tachyon effective potential at finite temperature and tachyon field value. Entropic arguments suggest that as the system cools, the tachyon is likely to find itself in the minimum that approaches infinity as the temperature vanishes (i.e. the one corresponding to the closed string vacuum), rather than the minimum at T=0 (corresponding to the open string vacuum).