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Add first part of the defect CFT

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-09-24 16:13:00 +02:00
parent 5fd7638885
commit e363da4b90
5 changed files with 628 additions and 9 deletions
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18
sec/part1/dbranes.tex
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@@ -16,13 +16,11 @@ Correlation functions involving arbitrary numbers of plain and excited twist fie
We consider D6-branes intersecting at angles in the case of non Abelian relative rotations presenting non Abelian twist fields at the intersections.
We try to understand subtleties and technical issues arising from a scenario which has been studied only in the formulation of non Abelian orbifolds \cite{Inoue:1987:NonAbelianOrbifolds,Inoue:1990:StringInteractionsNonAbelian,Gato:1990:VertexOperatorsNonabelian,Frampton:2001:ClassificationConformalityModels} and for D-branes relative \SU{2} rotations~\cite{Pesando:2016:FullyStringyComputation}.
In this configuration we study three D6-branes in $10$-dimensional Minkowski space $\ccM^{1,9}$ with an internal space of the form $\R^4 \times \R^2$ before the compactification.
The D-branes are embedded as lines in $\R^2$ and as two-dimensional surfaces inside $\R^4$.
We focus on the relative rotations which characterise each D-brane in $\R^4$ with respect to the others.
In total generality, they are non commuting \SO{4} matrices.
In this paper we study the classical solution of the bosonic string which dominates the behaviour of the correlator of twist fields.
We study the classical solution of the bosonic string which dominates the behaviour of the correlator of twist fields.
Using the path integral approach we can in fact separate the classical contribution from the quantum fluctuations and write the correlators of $N_B$ twist fields $\sigma_{\rM_{(t)}}( x_{(t)} )$ as:\footnotemark{}
\footnotetext{%
Ultimately $N_B = 3$ in our case.
@@ -2503,7 +2501,7 @@ Now~\eqref{eq:area_tmp} becomes:
\end{split}
\label{eq:action_with_imaginary_part}
\end{equation}
where $g^{(\perp)}_{(t)\, I} = \sum_{m \in \{3, 4\}} \tensor{\left( R_{(t)}^{-1} \right)}{_{mI}}\, g_{(t)\, m}$ is the transverse shift of $D_{(t)}$ in global coordinates:
where $g^{(\perp)}_{(t)\, I} = \sum_{m \in \{3, 4\}} \tensor{\left( R_{(t)}^{-1} \right)}{_{mI}}\, g_{(t)\, m}$ is the transverse shift of $D_{(t)}$ in the global coordinates of the target space:
\begin{equation}
g^{(\perp)}_{(t)\, I}\, (f_{(t-1)} - f_{(t)})^I = 0.
\label{eq:g_perp_Delta_f}
@@ -2555,7 +2553,7 @@ For the \SU{2} case we can use a rotation to map $(f_{(t-1)} - f_{(t)})^i$ to th
Each term of the action can be interpreted again as an area of a triangle where the distance between the interaction points is the base.
\subsubsection{The General Non Abelian Case}
\subsubsection{Generalisation and Summary}
\begin{figure}[tbp]
\centering
@@ -2573,9 +2571,11 @@ Most probably the value of the action is larger than in the holomorphic case sin
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.
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.
\subsection{A Brief Summary}
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.
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.
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.
In the generic case presented so far this is in general no longer true.
The reason can probably be traced back to supersymmetry, even though we only dealt with the bosonic string.
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.
% vim: ft=tex