[Paper Review] Thermal Duality Transformations and the Canonical Ensemble: The Long String Phase Transition

Shyamoli Chaudhuri|arXiv (Cornell University)|Sep 29, 2004

Black Holes and Theoretical PhysicsPhysics and Astronomy23 references2 citations

TL;DR

This paper formulates string statistical mechanics in the canonical ensemble using Euclidean timelike T-duality, showing that all six supersymmetric string theories exhibit T² free energy growth at high temperatures. It rules out exponential divergence in the one-loop free energy above a critical temperature and identifies a Wilson-Polyakov-Susskind loop as an order parameter for a string-scale thermal phase transition in type I theories, with dual descriptions enabling precise computation across the phase boundary.

ABSTRACT

We give a first principles formulation of the equilibrium statistical mechanics of strings in the canonical ensemble, compatible with the Euclidean timelike T-duality transformations that link the six supersymmetric string theories in pairs. We demonstrate that each exhibits a T^2 growth in the free energy at high temperatures far above the string scale. We verify that the low energy field theory limit of our expression for the string free energy reproduces the expected T^{10} growth when the contribution from massive string modes is suppressed. In every case, heterotic, type I, and type II, we can definitively rule out the occurrence of an exponential divergence in the one-loop string free energy above some critical temperature. Finally, we identify a macroscopic loop amplitude in the type I string theories which yields the expectation value of a single Wilson-Polyakov-Susskind loop in the low energy finite temperature supersymmetric gauge theory limit, an order parameter for a thermal phase transition at a string scale temperature. We point out that precise computations can nevertheless be carried out on either side of the phase boundary by using the low energy finite temperature supersymmetric gauge theory limits of the pair of thermal dual string theories, type IB and type I'. Note Added (Sep 2005).

Motivation & Objective

• To develop a first-principles formulation of equilibrium statistical mechanics for strings in the canonical ensemble.
• To incorporate Euclidean timelike T-duality transformations that link the six supersymmetric string theories in dual pairs.
• To analyze the high-temperature behavior of string free energy and verify consistency with low-energy field theory limits.
• To determine whether exponential divergence occurs in the one-loop free energy above a critical temperature.
• To identify an order parameter for a thermal phase transition at the string scale in type I string theories.

Proposed method

• Formulate the canonical ensemble for strings using T-duality-invariant boundary conditions in Euclidean time.
• Apply the duality transformations to relate the free energy of dual string theories, particularly type IB and type I'.
• Compute the one-loop string free energy in the high-temperature regime, focusing on T² scaling behavior.
• Suppress contributions from massive string modes to recover the low-energy field theory limit.
• Identify a macroscopic loop amplitude in type I theories that corresponds to the Wilson-Polyakov-Susskind loop in the finite-temperature supersymmetric gauge theory limit.
• Use dual descriptions on either side of the phase boundary to perform precise computations via the low-energy gauge theory limits.

Experimental results

Research questions

• RQ1 Does the one-loop string free energy exhibit exponential divergence above a critical temperature in the canonical ensemble?
• RQ2 How does the free energy scale at high temperatures in the absence of massive string mode contributions?
• RQ3 What is the role of T-duality in connecting the thermodynamic behavior of dual string theories like type IB and type I'?
• RQ4 Can a Wilson-Polyakov-Susskind loop serve as a reliable order parameter for a thermal phase transition at the string scale?
• RQ5 How can precise computations be carried out across the phase boundary using dual low-energy gauge theory limits?

Key findings

• The one-loop string free energy exhibits T² growth at high temperatures in all string theories—heterotic, type I, and type II—far above the string scale.
• The low-energy field theory limit of the string free energy correctly reproduces the expected T¹⁰ scaling when massive string modes are suppressed.
• The paper definitively rules out exponential divergence in the one-loop string free energy above any critical temperature.
• A macroscopic loop amplitude in type I string theories yields the expectation value of a single Wilson-Polyakov-Susskind loop in the low-energy finite-temperature supersymmetric gauge theory limit.
• The phase transition at the string scale is accessible via dual descriptions: type IB and type I' theories allow precise computation on either side of the phase boundary.
• The T-duality framework ensures consistency between the high-temperature behavior of dual string theories and their low-energy gauge theory limits.