The Hamiltonian including the boundary interaction is still of Sugawara form, simply by shifting all of the Fourier modes of J 1 by a constant, and the partition function is still given by a sum over U (1) Kac-Moody modules, but with weights shifted by a constant due to the boundary interaction, The band structure of the R = ∞ spectrum was discussed in Refs. [11, 13] for the case in which both ends of the string interact with the tachyonic background with equal strength. Here we will derive the corresponding results for the case of interest in our string-cutoff CP $^{N}$− 1 model, where only one end of the string interacts with the tachyon background. By open string/closed string duality, the partion function for the open string can be rewritten as a partition function for a closed string propagating from one boundary to the other, Here | N 〉 is the CFT boundary state associated with Neumann boundary conditions at the string end, and | B 〉 is the same state “rotated” by the boundary perturbation. q is the conjugate nome related to w by a Jacobi transformation, By writing the boundary states in terms of closed string left- and right-moving SU (2) Kac-Moody modules and observing that the boundary perturbation (16) is a global SU (2) rotation acting on the left-moving states, we find Here χ V ir j ( q ) = q − 1 24 f ( q ) − $^{1}$( q j 2 − q ( j +1) $^{2}$) is the discrete Virasoro character of spin j , and the J$_{i}$ ’s are the SU (2) generators in the closed string channel. The net rotation angle β is given in terms Z R $^{=}$∞ BN = 1 √$_{2}$ ∑ j =0 , 1 $_{2}$, 1 ,... χ V ir j ( q $^{2}$) ∫ π − π dp 2 π j ∑ m $^{=}$− $_{j}$〈 j, m | e ipJ 3 e iπgJ 1 e ipJ 3 | j, m 〉 = 1 √$_{2}$ ∑ j =0 , 1 $_{2}$, 1 ,... χ V ir j ( q $^{2}$) ∫ π − π dp 2 π j ∑ m $^{=}$− j sin(2 j + 1) β/ 2 sin β/ 2 (26) log q = 2 π 2 log w (25) Z R $^{=}$∞ BN = 〈 B | q L 0 + ¯ L 0 | N 〉 R $^{=}$∞ (24) Z R 0 BN = w − 1 24 f ( w ) ∞ ∑ Q $^{=}$−∞$^{w}$$^{(}$$^{Q}$$^{+}$$^{g}$ $_{2}$) 2 (23) H 0 + H int =: J $^{1}$J 1 : +const . , (22) J 1 n ≡ J 1 n + 1 2 g (21) 12