34 real, so we can choose the matrices R and S to be orthogonal) have to be diagonal. Since in general it is not possible to diagonalize four independent matrices using just two rotations, we will need to impose a set of consistency conditions on the wave-functions f a L,R j and g a L,R j and the constants c a $_{j}$, that will determine the wave functions uniquely. Let’s define: the conditions that we need to impose on the KK modes are then: We want to reduce the set of eqs. (A5) to a more explicit form. As a first thing, consider the integral appearing in the left-hand side of eq. (A5b). It can be rewritten If the eigenfunctions ˜ f a L,R j satisfy the equation of motion: where we leave the eigenvalue m$_{j}$ for now unspecified, then the integral in eq. (A6) can be further simplified to − m 2 k ∫ πR 0 ˜ f a L j ˜ f a L k dy + b ( πR ) ˜ f a $_{L j}$∂$_{y}$ ˜ f a L k ∣ ∣$^{πR}$ 0 + ( L → R ) . (A8) ˆ Df a L,R j = − m 2 $_{j}$f a $_{L,R j}$, (A7) ∫ πR 0 b ( y ) ∂$_{y}$ ˜ f a $_{L j}$∂$_{y}$ ˜ f a L k dy + ( L → R ) = − ∫ πR 0 ˜ f a $_{L j}$∂$_{y}$ ( b ( y ) ∂$_{y}$ ˜ f a $_{Lk}$) dy + b ( πR ) ˜ f a $_{Lj}$∂$_{y}$ ˜ f a L k ∣ ∣$^{πR}$ 0 + ( L → R ) . (A6) a f a L,R j = R$_{jk}$ ˜ f a $_{L,R k}$; g a L,R j = S$_{jk}$ ˜ g a $_{L,R k}$, c a j = S$_{jk}$ ˜ c a $_{k}$; (A4) 1 g 2 5 ∫ πR 0 dy ( ˜ f a L j ˜ f a L k + ˜ f a R j ˜ f a R k ) + 1 ˜ g 2 ˜ f a L j ˜ f a L k ∣ πR + 1 ˜ g ′ 2 ˜ f 3 R j ˜ f 3 R k ∣ πR = a$_{j}$δ$_{jk}$ ; (A5a) 1 g 2 5 ∫ πR 0 dy b ( y ) ( ∂$_{y}$ ˜ f a $_{L j}$∂$_{y}$ ˜ f a L k + ∂$_{y}$ ˜ f a $_{R j}$∂$_{y}$ ˜ f a R k ) + ˜ v 2 4 b ( πR ) ( ˜ f a L j ˜ f a L k + ˜ f 3 R j ˜ f 3 R k − 2 ˜ f 3 L j ˜ f 3 R k ∣ πR = b$_{j}$δ$_{jk}$ ; (A5b) 1 g 2 5 ∫ πR 0 dy b y ) ( ˜ g a $_{Lj}$˜ g a Lk + ˜ g a $_{R j}$˜ g a R k ) + ˜ c a $_{j}$˜ c a k = c$_{j}$δ$_{jk}$ ; (A5c) 1 g 2 5 πR 0 dy b ( y ) ∂$_{y}$ ˜ f a $_{L j}$˜ g a L k + ∂$_{y}$ ˜ f a $_{R j}$˜ g a R k ) − ˜ v 2 ( ˜ f a $_{L j}$˜ c a k − ˜ f 3 $_{R j}$˜ c 3 k )∣ ∣ πR = d$_{j}$δ$_{jk}$. (A5d)