The quasiopen string

The quasiopen string in D dimensions is defined by the Nambu-Goto action and the boundary conditions (x’+x)(${\sigma }_{\alpha }$,τ)= $V_{\alpha }$(x’-x)(${\sigma }_{\alpha }$,τ), where σ=${\sigma }_1$ and ${\sigma }_2$ denote the ends of the string, x’≡∂x/∂σ, and the $V_{\alpha }$ (α=1,2) are real symmetric orthogonal matrices. (The usual open string corresponds to $V_1$=$V_2$=-1.) We impose Poincaré invariance in d dimensions, d<D. Classically this requires ($V_{\alpha }$ ${)}_{\mathrm{jk}}$=-${\delta }_{\mathrm{jk}}$ for j,k∈ Poincaré sector and ($V_{\alpha }$ ${)}_{\mathrm{jk}}$=0 if only one of j,k belongs to the Poincaré sector. Further quantization gives D=26 and a mass spectrum with a ground-state mass squared $M_G$ ${}^2$= -(1- ${\mathcal{J}}_i$‖${\mathrm{theta}}_i$‖(1-‖${\mathrm{theTa}}_I$‖)/4)/α’, where ‖${\mathrm{theta}}_i$‖≤(1/2), exp(2iπ${\mathrm{theta}}_i$) are the eigenvalues of $V_2$ $V_1$, and α’ is the slope parameter in the string action. A choice of ${\mathrm{theta}}_i$ giving a tachyon-free spectrum is thus possible if d≤10.