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String background fields and Riemann-Cartan geometry

We study classical dynamics of cylindrical membranes wrapped around the extra compact dimension of a $(D+1)$-dimensional Riemann-Cartan spacetime. The world-sheet equations and boundary conditions are obtained from the universally valid conservation equations of the stress-energy and spin tensors. Specifically, we consider membranes made of macroscopic matter with maximally symmetric distribution of spin. In the narrow membrane limit, the dimensionally reduced theory is obtained. It describes how effective strings couple to the effective $D$-dimensional geometry. The striking coincidence with the string theory $σ$-model is observed. In this correspondence, the string background fields $G_{μν}$, $B_{μν}$, $A_μ$ and $Φ$ are related to the metric and torsion of the Riemann-Cartan spacetime.