On Critical Dimensions of String Theories
Dimitry A. Leites and Christoph Sachse
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Submission date: 23. Dec. 2006
published in: Mathematical physics : proceedings of the 12th Regional Conference ; Islamabad, Pakistan ; 27 March - 1 April 2006 / M. J. Aslam ... (eds.)
New Jersey [u.a.] : World Scientific, 2007. - P. 31 - 38
Keywords and phrases: stringy Lie superalgebra, critical dimension
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Exactly 10 of all simple Lie superalgebras of vector fields on 1|N-dimensional supercircles (superstrings) preserving a structure (either nothing, or a volume element, or a contact form) have nontrivial central extensions. The values of the central charges in (projective) spinor-oscillator representations of these stringy superalgebras associated with the adjoint module can be interpreted as critical dimensions of respective superstrings.
Apart from the well-known values 26, 10, 2 and 0 corresponding to N=0, 1, 2 and >2, respectively, for the contact type stringy superalgebras (Neveu-Schwarz and Ramond types alike), there are two more non-zero values of critical dimension: For the general and divergence-free algebras, both for N=2. These dimensions, found here, are -1 and -2, respectively. We also mention related problems.