Wilson-Polyakov loops for critical strings and superstrings at finite temperature

Abstract

An open string with end-points fixed at spatial separation L is a string theory analogue of the static quark-antiquark system in quenched QCD. Following a review of the quantum mechanics of this system in critical bosonic string theory the partition function at finite β (the inverse temperature) for fixed end-point open strings is discussed. This is related by a conformal transformation (“world-sheet duality”) to the correlation function of two closed strings fixed at distinct spatial points (a string theory analogue of two Wilson-Polyakov loops). Temperature duality ( β → β′ = 4 π 2/ β) relates this correlation function, in turn, to the finite-temperature Green function for a closed strong propagating between initial and final states that are at distinct (euclidean) space-time points. In addition, spatial duality relates the fixed end-point open string to the familiar open string with free end-points. A generalization to fixed end-points superstrings is suggested, in which the superalgebra may be viewed as the spatial dual of the usual open-string superalgebra. At zero temperature world-sheet duality relates the partition function of supersymmetric fixed end-point open strings to the correlation function of point-like closed-string states. These couple to combinations of the scalar and pseudoscalar states of a type-2b superstring superfield. At finite temperature supersymmetry is broken and this correlation function involves the propagation of non-supersymmetric states with non-zero winding numbers (which formally include a tachyon at temperatures above the Hagedorn transition). Temperature duality again relates the partition function to the finite-temperature Green function describing the propagator for point-like closed-string states of the dual theory, in which supersymmetry is broken. The singularity that arises in the critical bosonic theory as L is reduced below L = 2 π√ α′ is absent in the superstring and the static potential is well defined for all L.