The Atick–Witten free energy, closed tachyon condensation and deformed Poincaré symmetry Original Research Article

The dependence of the free energy of string theory on the temperature at T⪢THag was found long ago by Atick and Witten and is F(T)∼ΛT2, where Λ diverges because of a tachyonic instability. We show that this result can be understood assuming that, above the Hagedorn transition, Poincaré symmetry is deformed into a quantum algebra. Physically this quantum algebra describes a non-commutative spatial geometry and a discrete Euclidean time. We then show that in string theory this deformed Poincaré symmetry indeed emerges above the Hagedorn temperature from the condensation of vortices on the world-sheet. This result indicates that the endpoint of the condensation of closed string tachyons with non-zero winding is an infinite stack of space-like branes with a given non-commutative world-volume geometry. On a more technical side, we also point out that T-duality along a circle with antiperiodic boundary conditions for space–time fermions is broken by world-sheet vortices, and the would-be T-dual variable becomes non-compact.