The Atick-Witten free energy, closed tachyon condensation and deformed Poincare' symmetry

TL;DR

This paper explains the high-temperature behavior of string theory's free energy through a deformed Poincare' symmetry arising from closed tachyon condensation, leading to non-commutative geometry and spacelike branes.

Contribution

It demonstrates that above the Hagedorn temperature, Poincare' symmetry deforms into a quantum algebra due to vortex condensation, revealing new geometric and symmetry structures in string theory.

Findings

High-temperature free energy scales as T^2 with divergence due to tachyons

Deformed Poincare' symmetry emerges from vortex condensation above Hagedorn temperature

Endpoint of tachyon condensation is an infinite stack of spacelike branes with non-commutative geometry

Abstract

The dependence of the free energy of string theory on the temperature at T much larger than the Hagedorn temperature was found long ago by Atick and Witten and is $F(T)\sim \Lambda T^2$, where $\Lambda$ diverges because of a tachyonic instability. We show that this result can be understood assuming that, above the Hagedorn transition, Poincare' 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 Poincare' 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 spacelike 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 spacetime fermions is broken by world-sheet vortices, and the would-be T-dual variable becomes non-compact.