29 lines
3.3 KiB
TeX
29 lines
3.3 KiB
TeX
We thus showed that the specific geometry of the intersecting D-branes leads to different results when computing the value of the classical action, that is the leading contribution to the Yukawa couplings in string theory.
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In particular in the Abelian case the value of the action is exactly the area formed by the intersecting D-branes in the $\R^2$ plane, i.e.\ the string worldsheet is completely contained in the polygon on the plane.
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The difference between the \SO{4} case and \SU{2} is more subtle as in the latter there are complex coordinates in $\R^4$ for which the classical string solution is holomorphic in the upper half plane.
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In the generic case presented so far this is in general no longer true.
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The reason can probably be traced back to supersymmetry, even though we only dealt with the bosonic string.
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In fact when considering \SU{2} rotated D-branes part of the spacetime supersymmetry is preserved, while this is not the case for \SO{4} rotations.
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In the general case there does not seem to be any possible way of computing the action~\eqref{eq:action_with_imaginary_part} in term of the global data.
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Most probably the value of the action is larger than in the holomorphic case since the string is no longer confined to a plane.
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Given the nature of the rotation its worldsheet has to bend in order to be attached to the D-brane as pictorially shown in~\Cref{fig:brane3d} in the case of a $3$-dimensional space.
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The general case we considered then differs from the known factorized case by an additional contribution in the on-shell action which can be intuitively understood as a small ``bump'' of the string worldsheet in proximity of the boundary.
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We thus showed that the specific geometry of the intersecting D-branes leads to different results when computing the value of the classical action, that is the leading contribution to the Yukawa couplings in string theory.
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In particular in the Abelian case the value of the action is exactly the area formed by the intersecting D-branes in the $\R^2$ plane, i.e.\ the string worldsheet is completely contained in the polygon on the plane.
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The difference between the \SO{4} case and \SU{2} is more subtle as in the latter there are complex coordinates in $\R^4$ for which the classical string solution is holomorphic in the upper half plane.
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|
In the generic case presented so far this is in general no longer true.
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|
The reason can probably be traced back to supersymmetry, even though we only dealt with the bosonic string.
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In fact when considering \SU{2} rotated D-branes part of the spacetime supersymmetry is preserved, while this is not the case for \SO{4} rotations.
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In a technical and direct way we showed the computation of amplitudes involving an arbitrary number of Abelian spin and matter fields.
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The approach we introduced does not generally rely on \cft techniques and can be seen as an alternative to bosonization and old methods based on the Reggeon vertex.
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Starting from this work the future direction may involve the generalisation to non Abelian spin fields and the application to twist fields.
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In this sense this approach might be the only way to compute the amplitudes involving these complicated scenarios.
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This analytical approach may also shed some light on the non existence of a technique similar to bosonisation for twist fields.
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