- #151

arivero

Gold Member

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- 59

[tex]m_{k} = {R'^2 \over M_0 \alpha'^2} (n_k + {\theta_{k^i} - \theta_{k^f} \over 2 \pi})^2= m_0 (n_k + \lambda_k)^2[/tex]

where [itex]\lambda_k[/itex] is the distance between the two branes and [itex]n_k[/itex] an extra wrapping that the open string can perform. This is a free adaptation of Johnson's primer hep-th/0007170:

and the idea of see-sawing is not justified anywhere but lets say that it is just to test how compatible Koide formula is. Remember that the standard formula is [itex]m_0(1+\mu_k)^2[/itex], with conditions [itex]\sum \mu_k = 0, \sum (\mu_k^2 - 1) =0[/itex]. Here, for the special case of three parallel branes, we have from the construction that [itex]\lambda_3 = \lambda_2+\lambda_1[/itex]. I am not sure of how freely we can change the sign of a distance between branes; the formula apparently allows for it but it would need more discussion, so lets simply to investigate the case for wrapping [itex]n_3=0[/itex], so that we can freely flip [itex]\lambda_3 \to - \lambda_3 [/itex] and grant [itex]\lambda_3 + \lambda_2+\lambda_1 =0[/itex] and lets put the other two wrappings at the same level n=1. With this, Koide formula should be

[tex]

\frac 32 = { (n_1+n_2 + n_3)^2 \over n_1^2 + n_2^2 +n_3^2 + 2 \sum n_k \lambda_k + \sum \lambda_k^2}

={ 2 \over 1 + (\lambda_1+\lambda_2) + \lambda_1^2 + \lambda_2^2 + \lambda_1 \lambda_2}

[/tex]

with the extra condition that the distances are normalized to be all of them less than 1. The parametrization is a lot more inconvenient that the one we are used to, but solutions are not strange.

For instance if [itex] \lambda_1=\lambda_2[/itex] we have

[tex]

\frac 32 = { 2 \over 1 + 2 \lambda_1 + 3 \lambda_1^2 }

[/tex]

and [itex]\lambda_1= -\frac 13 + {\sqrt 2 \over 3} [/itex]

[tex]

m_1=m_0 (1 + \lambda_1)^2= (\frac 23 + \frac {\sqrt 2 }3)^2 \\

m_2=m_0 (1 + \lambda_2)^2= (\frac 23 + \frac {\sqrt 2 }3)^2 \\

m_3=m_0 (0 - \lambda_1-\lambda_2)^2= 4 (-\frac 13 + \frac {\sqrt 2 }3)^2 [/tex]

So it seems doable but, well, the question of the winding numbers and the sign of the non-integer part needs more detail. Of course, without an argument to choose winding of the strings and distance between the branes they are attached to, it is just a change of variables.