Quantum string dynamics in the conformal invariantSL(2,R)WZWN background: Anti–de Sitter space with torsion

Abstract

We consider classical and quantum strings in the conformally invariant background corresponding to the $\mathrm{SL}(2,R)$ WZWN model. This background is locally anti--de Sitter spacetime with non-vanishing torsion. Conformal invariance is expressed as the torsion being parallelizing. The precise effect of the conformal invariance on the dynamics of both circular and generic classical strings is extracted. In particular, the conformal invariance gives rise to a repulsive interaction of the string with the background which precisely cancels the dominant attractive term arising from gravity. We perform both semi-classical and canonical string quantization, in order to see the effect of the conformal invariance of the background on the string mass spectrum. Both approaches yield that the high-mass states are governed by $m\ensuremath{\sim}\mathrm{HN}$ ($N\ensuremath{\in}{N}_{0},$ $N$ ``large''), where $m$ is the string mass and $H$ is the Hubble constant. It follows that the level spacing grows proportionally to ${N:d(m}^{2}{\ensuremath{\alpha}}^{\ensuremath{'}})/dN\ensuremath{\sim}N,$ while the string entropy goes like $S\ensuremath{\sim}\sqrt{m}.$ Moreover, it follows that there is no Hagedorn temperature, so that the partition function is well defined at any positive temperature. All results are compared with the analogue results in anti--de Sitter spacetime, which is a nonconformal invariant background. Conformal invariance simplifies the mathematics of the problem but the physics remains mainly unchanged. Differences between conformal and non-conformal backgrounds only appear in the intermediate region of the string spectrum, but these differences are minor. For low and high masses, the string mass spectra in conformal and non-conformal backgrounds are identical. Interestingly enough, conformal invariance fixes the value of the spacetime curvature to be $\ensuremath{-}69/(26{\ensuremath{\alpha}}^{\ensuremath{'}}).$