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it behaves the same as in the compactified theory. For subcritical boundary perturbations, g < 1 $_{2}$, propagating states and momentum dependence for the energy eigenvalues within a band arise from mixing between the states on neighboring unit cells. On the other hand, for g ≈ 0, the NFS view of the band structure starts with the continuous relativistic dispersion relation of the free string at R = ∞ . The boundary perturbation injects discrete units of momentum which are integer multiples of R − 1 0 , causing left- and right-propagating string states near the edges of the Brillouin zones to mix, opening up band gaps at the zone boundaries. 

To obtain the spectrum of the compactified R = R$_{0}$ theory, define the SU (2) Kac-Moody currents constructed from the holomorphic (left-moving) part of the coordinate X$_{L}$ ( z ) (where z = τ + iσ ), 

<img src='content_image/348988.jpg'>

There is also a set of right-moving Kac-Moody currents, but by standard arguments, the open string Hamiltonian can be written entirely in terms of the left-moving fields defined on the doubled line segment − l < σ < l with periodic boundary conditions. The free open string Hamiltonian density is given by a Sugawara construction which, by SU (2) symmetry, can be written in several ways 

<img src='content_image/348987.jpg'>

In terms of the Kac-Moody currents, the marginal boundary perturbation (16) is simply 

<img src='content_image/348986.jpg'>

For the free string in the : J $^{3}$J3 : form, the partition function is easily calculated by observing that the states fall into U (1) Kac-Moody modules, one for each allowed value of the U (1) charge Q = ∫ dσ/ (2 π ) J $^{3}$( σ ). The module with conformal weight Q 2 contributes w − 1 / $^{24}$f ( w ) − $^{1}$w Q 2 to the partition function. The Kac-Moody charges are equal to the allowed translational zero mode momenta of the string, which for the R = R$_{0}$ compactified theory are integers. The sum over modules gives the free string partition function, 

<img src='content_image/348985.jpg'>

By SU (2) symmetry, the same partition function is obtained from the : J $^{1}$J1 : form of the Sugawara Hamiltonian. In this latter form, the marginal boundary perturbation can be included 

