where Here the sum is over all open string states satisfying the specified boundary conditions, and we consider only the motion of the string coordinates in 2-dimensional Euclidean spacetime. With Neumann boundary conditions and in the absence of background fields, it is straightforward to separate the Virasoro eigenvalues into a p 2 term from translational motion and an m 2 i term from internal motion. For general background fields, such a separation is not possible, but we will consider a background λ ( X ) field that depends on only one spatial coordinate X , and is periodic in that coordinate, λ ( X ) = λ ( X + 2 πR$_{0}$ ), where R$_{0}$ = √$_{2}$$_{α}$$_{′}$ is the self-dual radius. For this case, the open string spectrum exhibits a band structure that can be understood in terms of Bloch wave states, as discussed in the next Section. [In the case of HQCD, the justification for considering solutions which depend on only one spacetime coordinate follows from the observation that, while the D 7 brane, which represents the perturbative vacuum, is non-BPS and unstable to tachyonic decay, the D6 branes which result from lamination along one coordinate direction are BPS and stable against further dimensional collapse.] For the free string with λ = 0, a crucial feature of the sum over discrete string states (13) is that it yields an amplitude with modular properties characterized by an elliptic nome For a single compactified bosonic coordinate at the radius R = R$_{0}$ satisfying Neumann boundary conditions at both ends of the string, the sum over states in (13) gives where f ( w ) = ∏ ∞ n $_{=1}$(1 − w $^{n}$). The factor w − 1 24 reflects the negative mass squared of the tachyon mode. Physically, this mode arises as a Casimir effect from the zero-point energy of the open string oscillators. The presence of a tachyon mode in the string-theoretic completion of the CP $^{N}$− 1 model is a signal that the perturbative vacuum in the cutoff field theory is unstable toward tachyon condensation. Note that the mass of the tachyon is of order the cutoff scale, just like the positive mass-squared states of the string. However, unlike the positive states, the tachyon does not decouple in the continuum limit, but in fact drives the transition from the perturbative vacuum to the physical vacuum. The scenario for this transition follows Sen’s description of the tachyon-induced decay of an unstable D-brane [12]. Boundary conformal field theory for the type IIA string in Z R 0 $_{NN}$( w ) ≡ ∑ i e − τm 2 i = w − 1 24 f ( w ) ∞ ∑ n $^{=}$−∞$^{w}$$^{n}$$^{2}$ (15) w = e − τ/ 4 α ′ (14) Z ( e − τ/ 4 α $^{′}$) ≡ ∞ ∑ i =0 e − τm 2 i (13) 9