Some caution must be used in writing down the completeness and orthogonality re- lations for the f 1 , 2 Ln mode functions. The differential operator ˆ D (16) is in fact not hermitian with respect to the ordinary scalar product when evaluated on functions obeying BCs of the kind (18), due to the presence of terms explicitly containing the eigenvalues m$_{L n}$ which are induced by πR -localized terms in the action. To obtain the correct completeness and orthogonality properties of this function set, a generalized scalar product must be used which takes into account such terms. This is given by where L$_{m}$ sets the normalization. Since the scalar product ( · , · )$_{˜}$ g is dimensionless, we will set: L$_{m}$ ≡ 1. This will ensure that the kinetic terms of the bosons of this sector are canonically normalized. From this definition we deduce the completeness relation: • A right charged sector coming from the expansion of ( W 1 , 2 R )$_{M}$. The explicit form of the expansion for this sector is The wave-functions of the vector fields satisfy a similar equation of motion: and the set of BCs: The scalar profiles are given by: g 1 , 2 R n = 1 m$_{R n}$ ∂$_{y}$f 1 , 2 $_{R n}$. (26) ∂$_{y}$f 1 , 2 R n = 0 at y = 0 , (24) f 1 , 2 R n = 0 at y = πR. (25) ˆ Df 1 , 2 R n = − m 2 $_{R n}$f 1 , 2 $_{R n}$, (23) W 1 , 2 $_{R µ}$( x, y ) = ∞ ∑ n =0 f 1 , 2 $_{R n}$( y ) W 1 , 2( n ) Rµ ( x ) , W 1 , 2 R $_{5}$( x, y ) = ∞ ∑ n =0 g 1 , 2 $_{R n}$( y ) G 1 , 2 ( n ) R ( x ) . (22) 1 g 2 5 ∑ k f 1 , 2 $_{L k}$( y ) f 1 , 2 $_{Lk}$( z ) + 1 ˜ g $_{2}$δ ( z − πR ) ∑ k f 1 , 2 $_{L k}$( y ) f 1 , 2 $_{Lk}$( πR ) = δ ( y − z ); (21) ( f 1 , 2 $_{L n}$, f 1 , 2 Lm ) ˜ g = L 2 $_{m}$δ$_{mn}$ , ( f, h )$_{˜}$ g = 1 g 2 5 ∫ πR 0 dy fh + 1 ˜ g 2 fh | $_{πR}$, (20) 11