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Sine-Gordon Theory with Higher Spin $N=2$ Supersymmetry and the Massless Limit

The Sine-Gordon theory at $\frac{β^{2}}{8π} = \frac{2}{(2n+3)},\; n= 1,2,3 \cdots $ has a higher spin generalization of the $N=2$ supersymmetry with the central terms which arises from the affine quantum group $U_{q}( \hat{s \ell} (2))$. Observing that the algebraic determination of $S$ matrices $( \approx {\rm quantum~ integrability })$ requires the saturation of the generalized Bogomolny bound, we construct a variant of the Sine-Gordon theory at this value of the coupling in the framework of $S$ matrix theory. The spectrum consists of a doublet of fractionally charged solitons as well as that of anti-solitons in addition to the ordinary breathers. The construction demonstrates the existence of the theory other than the one by the truncation to the breathers considered by Smirnov. The allowed values for the fractional part of the fermion number is also determined. The central charge in the massless limit is found to be $c= 1$ from the TBA calculation for nondiagonal S matrices. The attendant $c=1$ conformal field theory is the gaussian model with ${\bf Z_{2}}$ graded chiral algebra at the radius parameter $r= \sqrt{2n+3}$. In the course of the calculation, we find $4n+2$ zero modes from the (anti-)soliton distributions.