Duality in statistical mechanics and string theory

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Abstract
A closed (bosonic) string propagating in a flat space-time where one of the dimensions is curled up into a circle has a Kramers-Wannier duality symmetry identical to that possessed by the Villain model. This is also similar to duality in string theory (in the sense of modular invariance). One is also led naturally to extend the space-time coordinate system by introducing dual coordinates. One of the consequences is that the radius of the internal manifold does not have a coordinate-independent meaning and one cannot physically distinguish between a radius R and a radius α’/R.