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    Algebra and Twisted Algebra in Toroidal Target Space

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    KU-AMP 95015hep-th/9601079January 1996revised in March 1996Algebra and Twisted Algebra in Toroidal Target Space ?Shogo Tanimura ?Department of Applied Mathematics and Physics,Faculty of Engineering,Kyoto University,Kyoto 606-01,Japan Abstract Target space duality is reconsidered from the viewpoint of quanti-zation in a space with nontrivial topology.An algebra of operators for the toroidal bosonic string is de?ned and its representations are constructed.It is shown that there exist an in?nite number of

    inequivalent quantizations,which are parametrized by two param-eters 0≤s,t <1.The spectrum exhibits the duality only when s =t or ?t (mod 1).A deformation of the algebra by a central extension is also introduced.It leads to a kind of twisted rela-tion between the zero mode quantum number and the topological winding number.