Finite-temperature string theory and twisted tori

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Abstract
We show that the introduction of a finite temperature into string theory is mathematically equivalent to compactifying the timelike coordinate on a twisted torus of radius β/2π=1/2πkT, the reciprocal of the temperature. We derive a general formula for the one-loop vacuum amplitude Λ(β) of a string theory compactified on a twisted torus that is manifestly modular invariant and is almost as easy to apply as in the compactification of a string theory on an ordinary torus. The connection with statistical mechanics is made through the formula lnZ(β)=-βVΛ(β) where Z(β) is the partition function and V the volume of the system. We discuss how the function σ(E) which counts the number of string states of total energy E and plays a central role in the microcanonical ensemble can be computed by taking the inverse Laplace transform of Z(β) and we give estimates of σ(E) for low and high energy densities by this method. We study the effect of finite temperature on the effective potentials for different parameters which determine a string theory’s compactification. As supersymmetry is broken at finite temperature these potentials are nontrivial. We study the effect of finite temperature on the Wilson loop mechanism for gauge symmetry breaking and find that for a supersymmetric theory compactified on a torus finite-temperature effects favor the trivial vacuum. Also the effective potential for the radius R of the torus becomes modified so as to be minimum at R=0 or R=∞. The introduction of interactions and curvature effects into our formalism is discussed.