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1, the coordinate for which we impose Neumann boundary conditions parametrises a circle of radius R1 . The orthogonal circle possesses radius RR1 R2 = R2 . 86). This reasoning can also be used to determine the normalisation κφ of the boundary state, κφ = R1 M = 2R2 cos φ R1 N = 2R2 sin φ R2 . 102) This choice may be confirmed easily through a direct computation of the overlap. 100) allow us to compute the excitation spectrum of a brane at a “rational” angle in the usual manner, Zφ φ (q) = φ; x0 , x0 || q˜L0 − 24 || φ; x0 , x0 = κ2φ c χg1 (k,w) (˜ q )χg2 (k,w) (˜ q) k,w 38 Free field theory with boundaries g2 where we used the U(1) characters χg (˜ q ) = q˜ 2 η −1 (˜ q ) that come with the charges g1 (k, w) = kN + R1 wM , 2R1 g2 (k, w) = − kM + R2 wN .

37) can be carried over to our new framework, by restricting the wave number in the coherent states |(0, k) D to integer values of k. The boundary state is given by || D(x0 ) 1 =√ 2r eikx0 /r |(k, 0) . 86) k∈Z The right-hand side is invariant under shifts of the parameter x0 by integer multiples of 2πr. Hence, we may still interpret x0 as the value of the compactified √ boson X at the boundary z = z¯. The normalisation with the prefactor 1/ 2r is a new feature not present in the uncompactified case and will be justified below.

If the number of components D is even, the superconformal algebra may be extended to N = 2. 28 Free field theory with boundaries Boundary theories for these models are obtained in a straightforward manner from the previous findings. We shall see that under very simple conditions our boundary conditions preserve worldsheet supersymmetry. Free field theory and N = 1 superconformal algebra The simplest free action depending on the fermionic fields ψ μ and the familiar bosonic fields X μ is given by S[ψ, X] = 1 4π d2 z gμν 2 ¯ ν + ψ μ ∂ψ ¯ ν + ψ μ ∂ψ ν √ ∂X μ ∂X α .

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