\subsection{Motivation}

As seen in the previous sections, the study of viable phenomenological models in string theory involves the analysis of the properties of systems of D-branes.
The inclusion of the physical requirements deeply constrains the possible scenarios.
In particular the chiral spectrum of the \sm acts as a strong restriction on the possible setup.
In this section we study models based on \emph{intersecting branes}, which represent a relevant class of such models with interacting chiral matter.
We focus on the development of technical tools for the computation of Yukawa interactions for D-branes at angles~\cite{Chamoun:2004:FermionMassesMixing,Cremades:2003:YukawaCouplingsIntersecting,Cvetic:2010:BranesInstantonsIntersecting,Abel:2007:RealisticYukawaCouplings,Chen:2008:RealisticWorldIntersecting,Chen:2008:RealisticYukawaTextures,Abel:2005:OneloopYukawasIntersecting}.
The fermion--boson couplings and the study of flavour changing neutral currents~\cite{Abel:2003:FlavourChangingNeutral} are keys to the validity of the models.
These and many other computations require the ability to calculate correlation functions of (excited) twist and (excited) spin fields.
The goal of the section is therefore to address such challenges in specific scenarios.

The computation of the correlation functions of Abelian twist fields can be found in literature and plays a role in many scenarios, such as magnetic branes with commuting magnetic fluxes~\cite{Angelantonj:2000:TypeIStringsMagnetised,Bertolini:2006:BraneWorldEffective,Bianchi:2005:OpenStoryMagnetic,Pesando:2010:OpenClosedString,Forste:2018:YukawaCouplingsMagnetized}, strings in gravitational wave background~\cite{Kiritsis:1994:StringPropagationGravitational,DAppollonio:2003:StringInteractionsGravitational,Berkooz:2004:ClosedStringsMisner,DAppollonio:2005:DbranesBCFTHppwave}, bound states of D-branes~\cite{Gava:1997:BoundStatesBranes,Duo:2007:NewTwistField,David:2000:TachyonCondensationD0} and tachyon condensation in Superstring Field Theory~\cite{David:2000:TachyonCondensationD0,David:2001:TachyonCondensationUsing,David:2002:ClosedStringTachyon,Hashimoto:2003:RecombinationIntersectingDbranes}.
A similar analysis can be extended to excited twist fields even though they are more subtle to treat and hide more delicate aspects~\cite{Burwick:1991:GeneralYukawaCouplings,Stieberger:1992:YukawaCouplingsBosonic,Erler:1993:HigherTwistedSector,Anastasopoulos:2011:ClosedstringTwistfieldCorrelators,Anastasopoulos:2012:LightStringyStates,Anastasopoulos:2013:ThreeFourpointCorrelators}.
Results were however found starting from dual models up to modern interpretations of string theory~\cite{Sciuto:1969:GeneralVertexFunction,DellaSelva:1970:SimpleExpressionSciuto}.
Correlation functions involving arbitrary numbers of plain and excited twist fields were more recently studied~\cite{Pesando:2014:CorrelatorsArbitraryUntwisted,Pesando:2012:GreenFunctionsTwist,Pesando:2011:GeneratingFunctionAmplitudes} blending the CFT techniques with the path integral approach and the canonical quantization~\cite{Pesando:2008:MultibranesBoundaryStates,DiVecchia:2007:WrappedMagnetizedBranes,Pesando:2011:StringsArbitraryConstant,DiVecchia:2011:OpenStringsSystem,Pesando:2013:LightConeQuantization}.

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.
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.
}
\begin{equation}
  \left\langle
    \finiteprod{t}{1}{N_B}
    \sigma_{\rM_{(t)}}(x_{(t)})
  \right\rangle
  =
  \cN
  \left(
    \left\lbrace x_{(t)},\, \rM_{(t)} \right\rbrace_{1 \le t \le N_B}
  \right)\,
  e^{-S_E\left( \left\lbrace x_{(t)}, \rM_{(t)} \right\rbrace_{1 \le t \le N_B} \right)},
\end{equation}
where $\rM_{(t)}$ (for $1 \le t \le N_B$) are the monodromies induced by the twist fields, $N_B$ is the number of D-branes and $x_{(t)}$ are the intersection points on the worldsheet.
Even though quantum corrections are crucial to the complete determination of the normalisation of the correlator, the classical contribution of the Euclidean action represents the leading term of the Yukawa couplings.
We focus on its contribution to better address the differences from the usual factorised case and generalise the results to non Abelian rotations of the D-branes.
We do not consider the quantum corrections as they cannot be computed with the actual techniques.
Their calculations requires the correlator of four twist fields which in turn requires knowledge of the connection formula for Heun functions which is not known.

We therefore study the boundary conditions for the open string describing the D-branes embedded in $\R^4$.
In particular we first address the issue connected to the global description of the embedding of the D-branes.
In conformal coordinates we rephrase such problem into the study of the monodromies acquired by the string coordinates.
These additional phase factors can then be specialised to \SO{4}, which can be studied in spinor representation as a double copy of \SU{2}.
We thus recast the issue of finding the solution as $4$-dimensional real vector to a tensor product of two solutions in the fundamental representation of \SU{2}.
We then see that these solutions are well represented by hypergeometric functions, up to integer factors.
Physical requirements finally restrict the number of possible solutions.


\subsection{D-brane Configuration and Boundary Conditions}

We focus on the bosonic string embedded in $\ccM^{1, d + 4}$.
The relevant configuration of the D-branes is seen as two-dimensional Euclidean planes in $\R^4$.
We specifically concentrate on the Euclidean solution for the classical bosonic string in this scenario.
The mathematical analysis is however more general and can be applied to any Dp-brane embedded in a generic Euclidean space $\R^q$.
The classical solution can in principle be defined in this case provided the ability to write the explicit form of the basis of functions with the proper boundary and monodromy conditions.
This is possible in the case of three intersecting D-branes but in general it is an open mathematical issue.
In the case of three D-branes with generic embedding we can in fact connect a local basis around one intersection point to another, the third depending on the first two intersections, by means of Mellin-Barnes integrals.
This way the solution can be explicitly and globally constructed.
With more than three D-branes the situation is by far more difficult since the explicit form of the connection formulas is not known.


\subsubsection{Intersecting D-branes at Angles}

Let $N_B$ be the total number of D-branes and $t = 1,\, 2,\, \dots,\, N_B$ be
an index defined modulo $N_B$ to label them.
We describe one of the D-branes in a well adapted system of coordinates
$X_{(t)}^I$, where $I = 1,\, 2,\, 3,\, 4$, as:
\begin{equation}
  X_{(t)}^3 = X_{(t)}^4 = 0.
  \label{eq:well-adapt-embed}
\end{equation}
We thus choose $X_{(t)}^1$ and $X_{(t)}^2$ to be the coordinates parallel to the D-brane $D_{(t)}$ while $X_{(t)}^3$ and $X_{(t)}^4$ are the coordinates orthogonal to it.

\begin{figure}[tbp]
  \centering
  \begin{subfigure}[b]{0.45\linewidth}
    \centering
    \def\svgwidth{\linewidth}
    \import{img/}{branesangles.pdf_tex}
    \caption{%
      D-branes as lines on $\R^2$.
    }
  \end{subfigure}
  \hfill
  \begin{subfigure}[b]{0.45\linewidth}
    \centering
    \def\svgwidth{\linewidth}
    \import{img/}{welladapted.pdf_tex}
    \caption{Well adapted system of coordinates.}
  \end{subfigure}
  \caption{%
    Geometry of D-branes at angles.
  }
  \label{fig:branes_at_angles}
\end{figure}

The well adapted reference coordinates system is connected to the global $\R^{4}$ coordinates $X^I$ by a transformation:
\begin{equation}
  \tensor{(X_{(t)})}{^I}
  =
  \tensor{(R_{(t)})}{^I_J}\, \tensor{X}{^J}
  -
  \tensor{(g_{(t)})}{^I},
  \qquad
  I,\, J = 1,\, 2,\, 3,\, 4,
  \label{eq:brane_rotation}
\end{equation}
where $R_{(t)}$ represents the rotation of the D-brane $D_{(t)}$ and $g_{(t)} \in \R^4$ its translation with respect to the origin of the global set of coordinates (see \Cref{fig:branes_at_angles} for a two-dimensional example).

While we could naively consider $R_{(t)} \in \SO{4}$, rotating separately the subset of coordinates parallel and orthogonal to the D-brane does not affect the embedding.
In fact it just amounts to a trivial redefinition of the initial well adapted coordinates.
The rotation $R_{(t)}$ is actually defined in the Grassmannian:
\begin{equation}
  R_{(t)}
  \in
  \mathrm{Gr}(2, 4)
  =
  \frac{\SO{4}}{\rS\left( \OO{2} \times \OO{2} \right)},
\end{equation}
that is we just need to consider the left coset where $R_{(t)}$ is a representative of an equivalence class
\begin{equation}
  \left[ R_{(t)} \right]
  =
  \left\lbrace R_{(t)} \sim \cO_{(t)} R_{(t)} \right\rbrace,
\end{equation}
where $\cO_{(t)} = \rS\left( \OO{2} \times \OO{2} \right)$ is defined as
\begin{equation}
  \cO_{(t)}
  =
  \mqty( \dmat{\cO^{\parallel}_{(t)}, \cO^{\perp}_{(t)}} )
\end{equation}
with $\cO^{\parallel}_{t} \in \OO{2}$, $\cO^{\perp}_{t} \in \OO{2}$ and $\det \cO_{(t)} = 1$.
The superscript $\parallel$ represents any of the coordinates parallel to the D-brane, while $\perp$ any of the orthogonal.
Notice that the additional $\Z_2$ factor in $\rS\left( \OO{2} \times \OO{2} \right)$ with respect to $\SO{2} \times \SO{2}$ can be used to set $g_{(t)}^{\perp} \ge 0$.


\subsubsection{Boundary Conditions for Branes at Angles}

The peculiar embedding of the D-branes has natural consequences on the boundary conditions of the open strings.
Let $\tau_E = i \tau$ be the Wick rotated time direction.
We define the usual upper plane coordinates:
\begin{eqnarray}
  u
  =
  x + i y
  =
  e^{\tau_E + i \sigma}
  & \in &
  \ccH \cup \left\lbrace z \in \C \mid \Im z = 0 \right\rbrace,
  \\
  \bu
  =
  x - i y
  =
  e^{\tau_E - i \sigma}
  & \in &
  \overline{\ccH} \cup \left\lbrace z \in \C \mid \Im z = 0 \right\rbrace,
\end{eqnarray}
where $\ccH = \left\lbrace z \in \C \mid \Im z > 0 \right\rbrace$ is the upper complex plane and $\overline{\ccH} = \left\lbrace z \in \C \mid \Im z < 0 \right\rbrace$ is the lower complex plane.
In conformal coordinates $u$ and $\bu$, D-branes at $\sigma = 0$ and $\sigma = \pi$ are mapped onto the real axis $\Im z = 0$.
We use the symbol $D_{(t)}$ to label both the brane and the interval representing it on the real axis of the upper half plane:
\begin{equation}
  D_{(t)} = \left[ x_{(t)}, x_{(t-1)} \right],
  \qquad
  t = 2,\, 3,\, \dots,\, N_B,
  \qquad
  x_{(t)} < x_{(t-1)}.
\end{equation}
The points $x_{(t)}$ and $x_{(t-1)}$ represent the worldsheet intersection points of the brane $D_{(t)}$ with the branes $D_{(t+1)}$ and $D_{(t-1)}$ respectively.
The choice of the intervals must be carefully considered: since the D-branes are defined modulo $N_B$, the shorthand for the interval $D_{(1)} = \left[ x_{(1)}, x_{(N_B)} \right]$ should actually be:
\begin{equation}
  D_{(1)}
  =
  \left[ x_{(1)}, +\infty \right) \cup \left( -\infty, x_{(N_B)} \right].
\end{equation}

In the global coordinates system associated to the subspace $\R^4 \subset \ccM^{1, d+4}$ where D-branes intersect, the relevant part of the action in conformal gauge is:
\begin{equation}
  \begin{split}
    S_{\R^4}
    & =
    \frac{1}{2 \pi \ap}
    \iint\limits_{\ccH}
    \dd{u} \dd{\bu}\,
    \ipd{u} X^I\, \ipd{\bu} X^J\,
    \eta_{IJ}
    \\
    & =
    \frac{1}{4 \pi \ap}
    \iint\limits_{\R \times \R^+}
    \dd{x}\dd{y}\,
    \left(
      \ipd{x} X^I\, \ipd{x} X^J
      +
      \ipd{y} X^I\, \ipd{y} X^J
    \right)\,
    \eta_{IJ},
  \end{split}
  \label{eq:string_action}
\end{equation}
where $2\, \ipd{u} = \ipd{x} - i\, \ipd{y}$ and $2\, \ipd{\bu} = \ipd{x} + i\, \ipd{y}$.
The \eom in these coordinates are:
\begin{equation}
  \ipd{u} \ipd{\bu} X^I( u, \bu )
  =
  \frac{1}{4}
  \left( \ipd{x}^2 + \ipd{y}^2 \right) X^I( x+iy, x-iy )
  =
  0.
  \label{eq:string_equation_of_motion}
\end{equation}
Their solution factorises as usual in holomorphic and anti-holomorphic components $X^I( u, \bu ) = X^I( u ) + \bX^I( \bu )$.

In the well adapted frame~\eqref{eq:well-adapt-embed} we describe an open string with one of the endpoints on $D_{(t)}$ through the relations:
\begin{eqnarray}
  \eval{\ipd{\sigma} X^i_{(t)}( \tau, \sigma )}_{\sigma = 0}
  =
  \eval{\ipd{y} X^i_{(t)}( u, \bu )}_{y = 0}
  & = &
  0,
  \qquad
  i = 1,\, 2,
  \label{eq:neumann_bc}
  \\
  X^m_{(t)}( \tau, 0 )
  =
  X^m_{(t)}( x, x )
  & = &
  0,
  \qquad
  m = 3,\, 4,
  \label{eq:dirichlet_bc}
\end{eqnarray}
where $x \in D_{(t)} = \left[ x_t, x_{t-1} \right]$ and the index $i$ labels the Neumann boundary conditions while $m$ labels the Dirichlet coordinates associated to the direction orthogonal to the D-branes.

As the presence of $g_{(t)}^m$ in \eqref{eq:brane_rotation} and \eqref{eq:dirichlet_bc} may complicate the analysis, we consider the derivative along the boundary direction of \eqref{eq:dirichlet_bc} to remove the dependence on the translation vector.
This procedure produces simpler boundary conditions which are nevertheless not equivalent to the original~\eqref{eq:neumann_bc} and \eqref{eq:dirichlet_bc}: they will be recovered later by adding further constraints.
The simpler boundary conditions we consider in the global coordinates are:
\begin{eqnarray}
  \tensor{\left( R_{(t)} \right)}{^i_J}
  \eval{\ipd{\sigma} X^J( \tau, \sigma )}_{\sigma = 0}
  & = &
  i\, \tensor{\left( R_{(t)} \right)}{^i_J}
  \left(
    \ipd{u} X^J( x + i\, 0^+ ) - \ipd{\bu} \bX^J( x - i\, 0^+ )
  \right)
  =
  0,
  \\
  \tensor{\left( R_{(t)} \right)}{^m_J}
  \eval{\ipd{\tau} X^J( \tau, \sigma )}_{\sigma = 0}
  & = &
  i\, \tensor{\left( R_{(t)} \right)}{^m_J}
  \left(
    \ipd{u} X^J( x + i\, 0^+ ) + \ipd{\bu} \bX^J( x - i\, 0^+ )
  \right)
  =
  0,
\end{eqnarray}
where $i = 1,\, 2$, $m = 3,\, 4$ and $x \in D_{(t)}$.

With the introduction of the target space embedding of the worldsheet interaction point between D-branes $D_{(t)}$ and $D_{(t+1)}$, $f_{(t)}$, we recover the full boundary conditions in terms of discontinuities on the real axis:
\begin{equation}
  \begin{cases}
    \ipd{u} X^I( x + i\, 0^+ )
    & =
    \tensor{\left( U_{(t)} \right)}{^I_J}
    \ipd{\bu} \bX^J( x - i\, 0^+ ),
    \qquad
    x \in D_{(t)}
    \\
    X^I( x_{(t)}, x_{(t)} )
    & =
    f_{(t)}
  \end{cases}.
  \label{eq:discontinuity_bc}
\end{equation}
In the last expression we introduced the matrix
\begin{equation}
  U_{(t)}
  =
  \left( R_{(t)} \right)^{-1}\,
  \cS\,
  R_{(t)}
  \in
  \frac{\SO{4}}{\rS\left( \OO{2} \times \OO{2} \right)},
  \label{eq:Umatrices}
\end{equation}
where
\begin{equation}
  \cS
  =
  \mqty( \dmat{ 1, 1, -1, -1 } )
  \label{eq:reflection_S}
\end{equation}
embeds the difference between Neumann and Dirichlet conditions.
Given its definition $U_{(t)}$ is such that $U_{(t)} = \left( U_{(t)} \right)^{-1} = \left( U_{(t)} \right)^T$.

The target space vector $f_{(t)}$ recovers the apparent loss of information suffered when losing $g_{(t)}$.
Consider for instance the embedding equations~\eqref{eq:dirichlet_bc} for any two intersecting D-branes $D_{(t)}$ and $D_{(t+1)}$.
Introducing the auxiliary quantities
\begin{eqnarray}
  \cR_{(t,\, t+1)}
  =
  \mqty( R_{(t)}^m \\ R_{(t+1)}^n )
  & \in &
  \GL{4}{\R},
  \qquad
  m, n = 3, 4,
  \\
  \cG_{(t,\, t+1)}
  =
  \mqty( g_{(t)}^m \\ g_{(t+1)}^n )
  & \in &
  \R^4,
  \qquad
  m, n = 3, 4,
\end{eqnarray}
we can compute the intersection point as:
\begin{equation}
  f_{(t)}
  =
  \left( \cR_{(t,\, t+1)} \right)^{-1}\,
  \cG_{(t,\, t+1)}.
\end{equation}
Information on $g_{(t)}$ is thus recovered through the global boundary conditions in the second equation in \eqref{eq:discontinuity_bc}.


\subsubsection{Doubling Trick and Branch Cut Structure}

In conformal coordinates we thus introduced the discontinuities~\eqref{eq:discontinuity_bc} across each D-brane which define a non trivial cut structure on the plane.
One way to deal with them is to introduce the \emph{doubling trick} by gluing the relations along an arbitrary but fixed D-brane $D_{(\bt)}$:
\begin{equation}
  \ipd{z} \cX(z) =
  \begin{cases}
    \ipd{u} X(u)
    &
    \qif
    z = u \qand \Im z > 0 \qor z \in D_{(\bt)}
    \\
    U_{(\bt)}\,
    \ipd{\bu} \bX(\bu)
    & \qif z = \bu \qand \Im z < 0 \qor z \in D_{(\bt)}
  \end{cases}.
  \label{eq:real_doubling_trick}
\end{equation}
Let then $\cU_{(t,\, t+1)} = U_{(t+1)}\, U_{(t)}$ and $\widetilde{\cU}_{(t,\, t+1)} = U_{(\bt)}\, U_{(t)}\, U_{(t+1)}\, U_{(\bt)}$.
The boundary conditions in terms of the doubling field are:
\begin{eqnarray}
  \ipd{z} \cX( x_t + e^{2 \pi i}( \eta + i\, 0^+ ) )
  & = &
  \cU_{(t,\, t+1)}
  \ipd{z} \cX( x_t + \eta + i\, 0^+ ),
  \label{eq:top_monodromy}
  \\
  \partial \cX( x_t + e^{2 \pi i}( \eta - i\, 0^+ ) )
  & = &
  \widetilde{\cU}_{(t,\, t+1)}
  \ipd{z} \cX( x_t + \eta - i\, 0^+ ),
  \label{eq:bottom_monodromy}
\end{eqnarray}
for $0 < \eta < \min\left( \abs{x_{(t-1)} - x_{(t)}}, \abs{x_{(t)} - x_{(t+1)}} \right)$ in order to consider only the two adjacent D-branes $D_{(t)}$ and $D_{(t+1)}$.
Matrices $\cU_{(t,\, t+1)}$ and $\widetilde{\cU}_{(t,\, t+1)}$ are the non trivial monodromies arising from the rotation of the D-branes.

Since the relative rotations between consecutive D-branes are non Abelian, for each interaction point there are two monodromies \cU and $\widetilde{\cU}$ depending on the location of the base point of the closed loop: one for paths starting in the upper plane \ccH and one for paths starting in $\overline{\ccH}$.
As a consequence of the geometry of the rotations of the D-branes, a path on the complex plane enclosing all of them does not present a monodromy:
\begin{equation}
  \finiteprod{t}{1}{N_B}\,
  \cU_{(\bt - t, \bt + 1 - t)}
  =
  \finiteprod{t}{1}{N_B}\,
  \widetilde{\cU}_{(\bt + t, \bt + 1 + t)}
  =
  \1_4.
  \label{eq:homotopy_rep}
\end{equation}
The complex plane has therefore branch cuts running between the D-branes at finite as shown in \Cref{fig:finite_cuts}.
We thus translated the rotations of the D-branes encoded in the matrices $R_{(t)}$ in terms of $\cU_{(t,\, t+1)}$ and $\widetilde{\cU}_{(t,\, t+1)}$ which are matrix representations of the homotopy group of the complex plane with the described branch cut structure.

\begin{figure}[tbp]
  \centering
  \def\svgwidth{0.5\textwidth}
  \import{img/}{branchcuts.pdf_tex}
  \caption{%
    Branch cut structure of the complex plane with $N_B = 4$.
    Cuts are pictured as solid coloured blocks running from one intersection point to another at finite.
  }
  \label{fig:finite_cuts}
\end{figure}

As a consistency check, the action~\eqref{eq:string_action} can be computed in terms of the doubling field \cX.
The map
\begin{equation}
  x_{(t)} + \eta \pm i\, 0^+
  \quad
  \mapsto
  \quad
  x_{(t)} + e^{2 \pi i}( \eta \pm i\, 0^+)
\end{equation}
must leave the action untouched since it does not depend on the branch cut structure.
In fact we can show that
\begin{equation}
  S_{\R^4}
  =
  \frac{1}{4 \pi \ap}
  \iint\limits_{\C}
  \dd{z} \dd{\bz}\,
  \ipd{z} \cX^T(z)\,
  U_{(\bt)}\,
  \ipd{\bz} \cX(\bz).
\end{equation}
As a matter of fact the action does not depend on the branch structure of the complex plane.


\subsection{D-branes at Angles in Spinor Representation}

In the previous section we showed that it is possible to map the information on the rotations of the D-branes to non trivial monodromies of the doubling field.
We thus recast the issue of solving the \eom of the string in the presence of rotated boundary conditions to the search for an explicit solution $\ipd{z} \cX( z )$ reproducing the non trivial monodromies in~\eqref{eq:top_monodromy} and \eqref{eq:bottom_monodromy}.

The field $\ipd{z} \cX( z )$ is technically a $4$-dimensional \emph{real} vector which has $N_B$ non trivial monodromy factors represented by $4
\times 4$ real matrices, one for each interaction point $x_{(t)}$.
A solution of the \eom is encoded in four linearly independent functions with $N_B$ branch points.
In principle we could try to write it as a solution to fourth order differential equations with $N_B$ finite Fuchsian points.
This is however an open mathematical debate.
In fact the basis of such functions around each branch point are usually complicated and defined up to several free parameters.
Moreover the explicit connection formulae between any two of them is an unsolved mathematical problem.
Using contour integrals and writing the functions as Mellin--Barnes integrals it might be possible to solve the issue in the case $N_B = 3$ but it is certainly not the best course of action.

On the other hand $N_B = 3$ is exactly the case we are investigating.
In what follows we use the isomorphism
\begin{equation}
  \SO{4}
  \cong
  \frac{\SU{2} \times \SU{2}}{\Z_2}
  \label{eq:su2isomorphism}
\end{equation}
to map the problem of finding a $4$-dimensional real solution to the \eom to a quest for a $2 \times 2$ complex matrix.
Such matrix is a linear superposition of tensor products of vectors in the fundamental representation of two different \SU{2} groups.
These vectors are solutions to second order differential equations with three Fuchsian points, possibly the hypergeometric equation.
The task is then to find the parameters of the hypergeometric functions producing the spinor representation of the monodromies in~\eqref{eq:top_monodromy} and \eqref{eq:bottom_monodromy}.


\subsubsection{Doubling Trick and Rotations in Spinor Representation}

We recall some of the properties of the isomorphism~\eqref{eq:su2isomorphism} in~\Cref{sec:isomorphism}.
We define the spinor representation of $X$ as:
\begin{equation}
  X_{(s)}( u, \bu ) = X^I( u, \bu )\, \tau_I,
\end{equation}
where $\tau = \left( i\, \1_2,\, \vb{\sigma} \right)$ and $\vb{\sigma}$ is the vector of the Pauli matrices.
Consider then:
\begin{equation}
  \ipd{z} \cX_{(s)}( z )
  =
  \begin{cases}
    \ipd{u} X_{(s)}(u)
    & \qif
    z \in \ccH \qor z \in D_{(\bt)}
    \\
    U_{L}(\vb{n}_{(\bt)})\,
    \ipd{\bu} X_{(s)}(\bu)\,
    U_{R}^{\dagger}(\vb{m}_{(\bt)})
    & \qif z \in \overline{\ccH} \qor z \in D_{(\bt)}
  \end{cases}.
  \label{eq:spinor_doubling_trick}
\end{equation}

As in the real representation the discontinuities on the D-branes can be cast into monodromy factors with respect to the D-brane $D_{(\bt)}$. Branch cut structure and considerations on the homotopy group are left unchanged as long as we consider both left and right sectors of $\SU{2}_L \times \SU{2}_R$ at the same time.
Let $0 < \eta < \min\left( \abs{x_{(t)} - x_{(t-1)}}, \abs{x_{(t+1)} - x_{(t)}} \right)$.
We find:
\begin{eqnarray}
  \ipd{z} \cX_{(s)}( x_t + e^{2 \pi i}( \eta + i\, 0^+) )
  & = &
  \cL_{(t,\, t+1)} \ipd{z}\,
  \cX_{(s)}( x_t + \eta + i\, 0^+ )\,
  \cR_{(t,\, t+1)}^{\dagger},
  \label{eq:top_spinor_monodromy}
  \\
  \ipd{z} \cX_{(s)}( x_t + e^{2 \pi i}( \eta - i\, 0^+) )
  & = &
  \widetilde{\cL}_{(t,\, t+1)}\,
  \ipd{z} \cX_{(s)}( x_t + \eta - i\, 0^+ )\,
  \widetilde{\cR}_{(t,\, t+1)}^{\dagger},
  \label{eq:bottom_spinor_monodromy}
\end{eqnarray}
  where:
\begin{eqnarray}
  \cL_{(t,\, t+1)}
  & = &
  U_{L}(\vb{n}_{(t+1)})\,
  U_{L}^{\dagger}(\vb{n}_{(t)}),
  \\
  \widetilde{\cL}_{(t,\, t+1)}
  & = &
  U_{L}(\vb{n}_{(\bt)})\,
  U_{L}^{\dagger}(\vb{n}_{(t)})\,
  U_{L}(\vb{n}_{(t+1)})\,
  U_{L}^{\dagger}(\vb{n}_{(\bt)}),
  \\
  \cR_{(t,\, t+1)}
  & = &
  U_{R}(\vb{m}_{(t+1)})\,
  U_{R}^{\dagger}(\vb{m}_{(t)}),
  \\
  \widetilde{\cR}_{(t,\, t+1)}
  & = &
  U_{R}(\vb{m}_{(\bt)})\,
  U_{R}^{\dagger}(\vb{m}_{(t)})\,
  U_{R}(\vb{m}_{(t+1)})\,
  U_{R}^{\dagger}(\vb{m}_{(\bt)}).
\end{eqnarray}

In spinor representation the action~\eqref{eq:string_action} becomes
\begin{equation}
  \begin{split}
    S_{\R^4}
    & =
    \frac{1}{4 \pi \ap}
    \iint\limits_{\ccH}
    \dd{u} \dd{\bu}\,
    \tr(\ipd{u} X_{(s)}(u, \bu) \cdot \ipd{\bu} X^{\dagger}_{(s)}(u, \bu))
    \\
    & =
    \frac{1}{8 \pi \ap}
    \iint\limits_{\C}
    \dd{z} \dd{\bz}\,
    \tr(
      U_{L}(\vb{n}_{(\bt)})\,
      \ipd{z} \cX_{(s)}(z, \bz)\,
      U_{R}^{\dagger}(\vb{m}_{(\bt)})\,
      \ipd{\bz} \cX_{(s)}^{\dagger}(z, \bz)
    ).
  \end{split}
  \label{eq:action_doubling_fields_spinor_representation}
\end{equation}
It is possible to show that the closed loop $x_t + \eta \pm i\, 0^+ \mapsto x_t + e^{2 \pi i}( \eta \pm i\, 0^+ )$ does not generate additional contributions in the action.


\subsubsection{Special Form of Matrices for D-Branes at Angles} 
\label{sec:special_SO4}

The \SU{2} matrices involved in this scenario with D-branes intersecting at angles have a particular form.
In the left sector (i.e.\ $\SU{2}_L$ matrices) we have:
\begin{equation}
  \cL_{(t,\, t+1)}
  =
  U_{L}(\vb{n}_{(t+1)})\,
  U_{L}^{\dagger}(\vb{n}_{(t)})\,
  =
  -\vb{n}_{(t+1)} \cdot \vb{n}_{(t)}
  +
  i\, (\vb{n}_{(t+1)} \times \vb{n}_{(t)}) \cdot \vb{\sigma} ,
\end{equation}
with $\vb{n}_{(t)}^2 = 1$.
This is a consequence of the peculiar properties of the \SO{4} matrices $U_{(t)}$ defined in \eqref{eq:Umatrices}.
Hence the corresponding $\SU{2}_L \times \SU{2}_R$ element $(U_{L}(\vb{n}_{(t)}),\, U_{R}(\vb{m}_{(t)}))$ reflects such characteristics.
In particular for the left part we have
\begin{equation}
  U_{L}(\vb{n}_{(t)})
  =
  i\, \vb{n}_{(t)} \cdot \vb{\sigma},
  \qquad
  \vb{n}_{(t)}^2 = 1,
  \label{eq:special_UL_brane_t}
\end{equation}
since $U_{(t)}^2 = \1_4$ implies that $U_{L}^2 = \pm \1_2$.
The right sector clearly follows the same discussion.

In fact \cS in~\eqref{eq:reflection_S} can be represented as $U_{L} = U_{R} = i\, \sigma_1$.
Then any matrix $U_{L}(\vb{n}_{(t)})$ is of the form $U_{L}(\vb{n}_{(t)}) = i\, U(\vb{r}_{(t)}) \cdot \sigma_1 \cdot U^\dagger(\vb{r}_{(t)})$, for some $\vb{r}_{(t)}$ as follows from~\eqref{eq:Umatrices}.
Such matrix has vanishing trace and squares to $-\1_2$ hence the term  proportional to two-dimensional unit matrix in the expression of the generic \SU{2} element given in \Cref{sec:isomorphism} vanishes.
As a consequence $n_{(t)} = \frac{1}{4}$ such that~\eqref{eq:special_UL_brane_t} follows.


\subsection{The Classical Solution}

In the previous sections we defined the principal tools to study the non Abelian embedding of the D-branes.
In what follows we start the investigation of the relation between the hypergeometric solutions and the monodromies due to the geometry of the D-branes.


\subsubsection{The Choice of Hypergeometric Functions}

We build the spinorial representation with \SU{2} matrices and solutions of Fuchsian equations with $N_B$ regular singular points.
We are specifically interested in a solution with $N_B = 3$.
We fix the usual \SL{2}{\R} invariance by mapping the three intersection points $x_{(\bt-1)}$, $x_{(\bt+1)}$ and $x_{(\bt)}$ to $\omega_{\bt-1} = \omega_{x_{(\bt-1)}} = 0$, $\omega_{\bt+1} = \omega_{x_{(\bt+1)}} = 1$ and $\omega_{\bt} = \omega_{x_{(\bt)}} = \infty$ respectively through:
\begin{equation}
  \omega_{u}
  =
  \frac{u - x_{(\bt-1)}}{u - x_{(\bt)}}
  \cdot
  \frac{x_{(\bt+1)} - x_{(\bt-1)}}{x_{(\bt+1)} - x_{(\bt)}}
  \label{eq:def_omega}
\end{equation}
The cut structure for this choice is presented in~\Cref{fig:hypergeometric_cuts}.
The map also defines $\arg(\omega_t - \omega_z) \in \left[ 0,\, 2\pi \right)$ for $t = \bt-1,\, \bt+1$.
We choose $\bt = 1$ in what follows.

\begin{figure}[tbp]
  \centering
  \def\svgwidth{0.35\linewidth}
  \import{img/}{threebranes_plane.pdf_tex}
  \caption{%
    Fixing the \SL{2}{\R} invariance for $N_B = 3$ and $\bt = 1$ leads to a cut structure with all the cuts defined on the real axis towards $\omega_{\bt} = \infty$.}
  \label{fig:hypergeometric_cuts}
\end{figure}

The map~\eqref{eq:def_omega} moves the generic Fuchsian singularities to known points on the complex plane.
The functions reproducing the necessary monodromies are basis of hypergeometric functions.
We define:
\begin{equation}
  \hyp{a}{b}{c}{z}
  =
  \zeroinfsum{k}\,
  \frac{\poch{a}{k}\poch{b}{k}}{\gfun{c+k}}~
  \frac{z^k}{k!}
  =
  \frac{1}{\gfun{c}}~
  \tensor[_2]{F}{_1}(a,\, b;\, c;\, z),
\end{equation}
where $\tensor[_2]{F}{_1}(a,\, b;\, c;\, z)$ is the Gauss hypergeometric function and $\gfun{s}$ is the Euler Gamma function.
The function \hyp{a}{b}{c}{z} is well defined for any value of its parameters.\footnotemark{}
\footnotetext{%
  It is not necessary to require $c \in \Z_+$ as in the definition of the Gauss hypergeometric function.
}
We define a vector of independent hypergeometric functions:
\begin{equation}
  B_{\vb{0}}(z)
  =
  \mqty(
    \hyp{a}{b}{c}{z}
    \\
    (-z)^{1-c}~\hyp{a+1-c}{b+1-c}{2-c}{z}
  )
  \label{eq:basis_0}
\end{equation}
as basis of functions around $z = 0$ with a branch cut on the interval $\left[ 0, +\infty \right)$.
The choice of the branch cuts follows from the cut on $\left[ 1, +\infty \right)$ coming from $\hyp{a}{b}{c}{z}$ which has a singularity at $z = 1$ and the cut on $\left[ 0, +\infty \right)$ from $(-z)^{1-c}$.

As argued in~\eqref{eq:homotopy_rep}, the homotopy group of the complex plane with the branch cut structure of~\Cref{fig:hypergeometric_cuts} is such that a closed loop around all the singularities is homotopically trivial.
The corresponding product of the monodromy matrices~\eqref{eq:homotopy_rep} is the unit matrix.
Let for instance $\cM_{\omega_z}^{\pm}$ be the monodromy matrix which represents a closed loop around $\omega_z$ (the $+$ sign represents a path starting in $\ccH$, while $-$ is a path with base point in $\overline{\ccH}$).
The triviality property is realised through:
\begin{equation}
  \cM_{\vb{0}}^+\,
  \cM_{\vb{1}}^+\,
  \cM_{\vb{\infty}}^+
  =
  \cM_{\vb{\infty}}^-\,
  \cM_{\vb{1}}^-\,
  \cM_{\vb{0}}^-
  =
  \1_2
  \label{eq:monodromy_relations}
\end{equation}
The monodromy matrix $\omega_{\bt+1} = 1$ can thus be recovered as a product of monodromies around $0$ and $\infty$ given the properties
\begin{equation}
  \begin{split}
    \cM_{\vb{0}}^+
    & =
    \cM_{\vb{0}}^-
    =
    \cM_{\vb{0}},
    \\
    \cM_{\vb{\infty}}^+
    & =
    \cM_{\vb{\infty}}^-
    =
    \cM_{\vb{\infty}},
  \end{split}
\end{equation}
which encode the peculiar branch cut structure due to the doubling trick gluing the intervals on one arbitrary D-brane.
These matrices are an abstract representation of the monodromy group since they are in an arbitrary basis.

Using the basis in $z = 0$~\eqref{eq:basis_0} it is straightforward to find the explicit representation $\rM_{\vb{0}}$ of the abstract monodromy $\cM_{\vb{0}}$:
\begin{equation}
  \rM_{\vb{0}}( c ) = \mqty( \dmat{1, e^{-2\pi i c}} ).
  \label{eq:monodromy_zero}
\end{equation}
The computation of the monodromy matrix $\rM_{\vb{\infty}}$ representing the monodromy in $\omega_z  =\infty$ in the basis \eqref{eq:basis_0} requires to first compute the monodromy representation $\widetilde{\rM}_{\vb{\infty}}$ of the abstract monodromy $\cM_{\vb{\infty}}$ in the basis of hypergeometric functions around $z = \infty$:
\begin{equation}
  B_{\vb{\infty}}(z)
  =
  \mqty(
    (-z)^{-a}~\hyp{a}{a+1-c}{a+1-b}{z^{-1}}
    \\
    (-z)^{-b}~\hyp{b}{b+1-c}{b+1-a}{z^{-1}}
  ).
\end{equation}
This basis is connected to~\eqref{eq:basis_0} through the transition matrix
\begin{equation}
  \cC( a,\, b,\, c )
  =
  \frac{\pi}{\sin(\pi(a-b))}
  \mqty(
    \frac{1}{\gfun{b}\gfun{c-a}}
    &
    -\frac{1}{\gfun{a} \gfun{c-b}}
    \\
    \frac{1}{\gfun{1-a}\gfun{b+1-c}}
    &
    -\frac{1}{\gfun{1-b}\gfun{a+1-c}}
  ),
  \label{eq:transition_matrix}
\end{equation}
as $B_{\vb{0}}(z) = \cC(a,\, b,\, c)~B_{\vb{\infty}}(z)$.
Through the loop $z \mapsto z e^{-2\pi i}$ we find:
\begin{equation}
  \widetilde{\rM}_{\vb{\infty}}( a,\, b )
  =
  \mqty( \dmat{e^{2\pi i a}, e^{2\pi i b}} ).
\end{equation}
Finally we can build the desired monodromy:
\begin{equation}
  \rM_{\vb{\infty}}
  =
  \cC(a,\, b,\, c)\,
  \widetilde{\rM}_{\vb{\infty}}(a,\, b)\,
  \cC^{-1}(a,\, b,\, c).
  \label{eq:monodromy_infty}
\end{equation}


\subsubsection{The Monodromy Factors}

With the previous definitions we reproduce the monodromies of the doubling field in its spinor representation~\eqref{eq:top_spinor_monodromy}.\footnotemark{}
\footnotetext{%
  In general we do not need to consider~\eqref{eq:bottom_spinor_monodromy} since they are the same monodromies.
}
These monodromies are tensor products of two basis of hypergeometric functions: the first basis reproduces the monodromies defined as $\cL$ and the second one those defined as $\cR$ in~\eqref{eq:top_spinor_monodromy}.
Since in principle there can be several combinations of parameters of the hypergeometric function yielding the same monodromy, we consider the full solution to be a linear superposition of all possible contributions:
\begin{equation}
  \ipd{z} \cX(z)
  =
  \pdv{\omega_z}{z}\,
  \sum\limits_{l,\, r}
  c_{lr}\,
  \ipd{z} \cX_{l,r}(\omega_z),
  \label{eq:formal_solution}
\end{equation}
where we drop the index representing the spinorial representation to lighten the notation.
We write any possible solution in a factorised form as
\begin{equation}
  \ipd{z} \cX_{l,\,r}(\omega_z)
  =
  (-\omega_z)^{A_{lr}}\,
  (1-\omega_z)^{B_{lr}}\,
  \cB_{\vb{0},\, l}^{(L)}(\omega_z)
  \left( \cB_{\vb{0},\, r}^{(R)}(\omega_z) \right)^T,
  \label{eq:formal_solution_lr}
\end{equation}
where  $l$ and $r$ label the parameters associates with the left and right sectors of the hypergeometric function.
We introduce the left basis element
\begin{equation}
  \begin{split}
    \cB_{\vb{0},\, l}^{(L)}(\omega_z)
    & =
    D^{(L)}_l~
    B_{\vb{0},\,l}^{(L)}(\omega_z)
    \\
    & =
    \mqty( 1 & 0 \\ 0 & K_l^{(L)} )\,
    \mqty(
      \hyp{a_l}{b_l}{c_l}{\omega_z}
      \\
      (-z)^{(1-c_l)}\,
      \hyp{a_l+1-c_l}{b_l+1-c_l}{2-c_l}{\omega_z}
    )
  \end{split}
\end{equation}
where $D_l^{(L)} \in \GL{2}{\C}$ is a relative normalisation matrix weighting differently the components of the basis.\footnotemark{}
\footnotetext{%
  In general they can be different for each solution.
}
The right sector follows in a similar way.
Notice that the matrices $D^{(L)}_l$ do not fix the absolute normalisation contained in $c_{lr}$.


\subsubsection{Parameters of the Trivial Monodromy}

Using the previous relations we can determine the possible $\ipd{z} \cX_{l,r}(\omega_z)$ with the desired monodromies.
In this section we study the case of the most general \SU{2} matrices despite the fact that in~\Cref{sec:special_SO4} we argued that they have a specific form.

First of all consider the matrices in \eqref{eq:monodromy_zero} and \eqref{eq:monodromy_infty}.
We impose:
\begin{eqnarray}
  &&\begin{cases}
    D^{(L)}\,
    \rM_{\vb{0}}^{(L)}\,
    \left( D^{(L)} \right)^{-1}
    =
    e^{-2\pi i \delta_{\vb{0}}^{(L)}}\,
    \cL(\vb{n}_{\vb{0}})
    \\
    D^{(R)}\,
    \rM_{\vb{0}}^{(R)}\,
    \left( D^{(R)} \right)^{-1}
    =
    e^{-2\pi i \delta_{\vb{0}}^{(R)}}\,
    \cR^*(\vb{m}_{\vb{0}})
    =
    e^{-2\pi i \delta_{\vb{0}}^{(R)}}\,
    \cR(\widetilde{\vb{m}}_{\vb{0}})
    \\
    e^{2\pi i ( A_{lr} - \delta_{\vb{0}}^{(L)} -
    \delta_{\vb{0}}^{(R)} )}
    =
    1
    \end{cases},
  \label{eq:parameters_equality_zero}
  \\
  &&\begin{cases}
    D^{(L)},
    \rM_{\vb{\infty}}^{(L)}\,
    \left( D^{(L)} \right)^{-1}
    =
    e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\,
    \cL(\vb{n}_{\vb{\infty}})
    \\
    D^{(R)}\,
    \rM_{\vb{\infty}}^{(R)}\,
    \left( D^{(R)} \right)^{-1}
    =
    e^{-2\pi i \delta_{\vb{\infty}}^{(R)}}\,
    \cR^*(\vb{m}_{\vb{\infty}})
    =
    e^{-2\pi i \delta_{\vb{\infty}}^{(R)}}\,
    \cR(\widetilde{\vb{m}}_{\vb{\infty}})
    \\
    e^{2\pi i ( A_{lr} + B_{lr} - \delta_{\vb{\infty}}^{(L)} -
    \delta_{\vb{\infty}}^{(R)} )}
    =
    1
  \end{cases},
  \label{eq:parameters_equality_infty}
\end{eqnarray}
where we defined
\begin{eqnarray}
  \cL(\vb{n}_{\vb{0}})
  & = &
  \cL_{(\bt-1,\,\bt)}
  =
  U_L(\vb{n}_{(\bt)})\,
  U_L^{\dagger}(\vb{n}_{(\bt-1)}),
  \\
  \cL(\vb{n}_{\vb{\infty}})
  & = &
  \cL_{(\bt,\, \bt+1)}
  =
  U_L(\vb{n}_{(\bt+1)})
  U_L^{\dagger}(\vb{n}_{(\bt)}),
  \\
  \cR(\vb{m}_{\vb{0}})
  & = &
  \cR_{(\bt-1,\, \bt)}
  =
  U_R(\vb{n}_{(\bt)})
  U_R^{\dagger}(\vb{n}_{(\bt-1)}),
  \\
  \cR(\vb{m}_{\vb{\infty}})
  & = &
  \cR_{(\bt,\, \bt+1)}
  =
  U_R(\vb{n}_{(\bt+1)})
  U_R^{\dagger}(\vb{n}_{(\bt)}).
\end{eqnarray}
The range of $\delta_{\vb{0}}^{(L)}$ is
\begin{equation}
  \alpha \le \delta_{\vb{0}}^{(L)} \le  \alpha + \frac{1}{2},
\end{equation}
that is the width of the range is only $\frac{1}{2}$ and not $1$ as one would naively expect.
This is a consequence of the fact that $e^{- 4 \pi i \delta_{\vb{0}}^{(L)}}$ is the determinant of the right hand side of the first equation in \eqref{eq:parameters_equality_zero}.
We then choose $\alpha = 0$ for simplicity.
The same considerations hold true for all the other additional parameters $\delta_{\vb{0}}^{(R)}$ and $\delta_{\vb{\infty}}^{(L,\,R)}$.

Since we are interested in relative rotations of the D-branes, we choose the
rotation in $\omega_{\bt-1} = 0$ in the maximal torus of $\SU{2}_L \times \SU{2}_R$ without loss of generality: as we have two independent groups, we can in fact fix the orientation of both vectors $\vb{n}_{\vb{0}}$ and $\vb{m}_{\vb{0}}$.
In particular we set:
\begin{eqnarray}
  \vb{n}_{\vb{0}}
  =
  ( 0,\, 0,\, n_{\vb{0}}^3 ) \in \R^3,
  & \qquad &
  0 < n_{\vb{0}}^3 < \frac{1}{2},
  \label{eq:maximal_torus_left}
  \\
  \widetilde{\vb{m}}_{\vb{0}}
  =
  ( 0,\, 0,\, -m_{\vb{0}}^3 ) \in \R^3,
  & \qquad &
  0 < m_{\vb{0}}^3 < \frac{1}{2},
  \label{eq:maximal_torus_right}
\end{eqnarray}
where $n_{\vb{0}}^3 = 0$ is excluded to avoid considering a trivial rotation.
We then define the parameters of the rotation in $\omega_{\bt} = \infty$ to be the most general
\begin{equation}
  \begin{split}
    \vb{n}_{\vb{\infty}}
    & =
    ( n_{\vb{\infty}}^1,\, n_{\vb{\infty}}^2,\, n_{\vb{\infty}}^3 ),
    \\
    \widetilde{\vb{m}}_{\vb{\infty}}
    & =
    ( -m_{\vb{\infty}}^1,\, m_{\vb{\infty}}^2,\, -m_{\vb{\infty}}^3 ),
\end{split}
\end{equation}
We could actually set $n_{\vb{\infty}}^2 = m_{\vb{\infty}}^2 = 0$ since the choice of the ``gauge''~\eqref{eq:maximal_torus_left} and~\eqref{eq:maximal_torus_right} is preserved by \U{1} rotations mixing $n_{\vb{\infty}}^1$ and $n_{\vb{\infty}}^2$.
We nevertheless keep the general expression in order to check the computations.

Solving~\eqref{eq:parameters_equality_zero} and~\eqref{eq:parameters_equality_infty} connects the parameters of the hypergeometric function to the parameter of the rotations (see \Cref{sec:parameters}) thus reproducing the boundary conditions of the intersecting D-branes through the non trivial monodromies of the basis of hypergeometric functions.
We find:
\begin{eqnarray}
  a_l^{(L)}
  =
  n_{\vb{0}}
  +
  (-1)^{f^{(L)}}\, n_{\vb{1}}
  +
  n_{\vb{\infty}}
  +
  \ffa^{(L)}_l,
  & \qquad &
  \ffa^{(L)}_l \in \Z,
  \\
  b_l^{(L)}
  =
  n_{\vb{0}}
  +
  (-1)^{f^{(L)}}\, n_{\vb{1}}
  -
  n_{\vb{\infty}}
  +
  \ffb^{(L)}_l,
  & \qquad &
  \ffb^{(L)}_l \in \Z,
  \\
  c_l^{(L)}
  =
  2\, n_{\vb{0}}
  +
  \ffc^{(L)}_l,
  & \qquad &
  \ffc^{(L)}_l \in \Z,
  \\
  \delta_{\vb{0}}^{(L)}
  =
  n_{\vb{0}},
  \\
  \delta_{\vb{\infty}}^{(L)}
  =
  -
  n_{\vb{0}}
  -
  (-1)^{f^{(L)}}\, n_{\vb{1}},
  \\
  K^{(L)}_l
  =
  -\frac{1}{2 \pi^2}\,
  \cG(a_l^{(L)},\, b_l^{(L)},\, c_l^{(L)})\,
  \cF(a_l^{(L)},\, b_l^{(L)},\, c_l^{(L)})\,
  \frac{n^1_{\vb{\infty}}+ i\, n^2_{\vb{\infty}}}{n_{\vb{\infty}}},
\label{eq:K_factor_value}
\end{eqnarray}
where $f^{(L)} \in \left\lbrace 0,\, 1 \right\rbrace$.
For the sake of brevity we defined two auxiliary functions, namely $\cG(a,\, b,\, c) = \gfun{1-a}\, \gfun{1-b}\, \gfun{a+1-c}\, \gfun{b+1-c}$ and $\cF(a,\, b,\, c) = \sin(\pi c)\, \sin(\pi(a-b))$.
We also introduced the norm $n_{\vb{1}} = \norm{\vb{n}_{\vb{1}}}$ of the rotation vector around $\omega_{\bt+1} = 1$.
Its dependence on the other parameters is encoded in~\eqref{eq:monodromy_relations}, where $\rM^+_{\vb{1}} = \rM^{-1}_{\vb{0}}\, \rM^{-1}_{\vb{\infty}}$, and the composition rule~\eqref{eq:product_in_SU2}:
\begin{equation}
  \cos(2\pi n_{\vb{1}})
  =
  \cos(2\pi n_{\vb{0}})\,
  \cos(2\pi n_{\vb{\infty}})
  -
  \sin(2\pi n_{\vb{0}})\,
  \sin(2\pi n_{\vb{\infty}})\,
  \frac{n_{\vb{\infty}}^3}{n_{\vb{\infty}}}.
  \label{eq:dependent_monodromy_main_text}
\end{equation}
Relations for the right sector follow under the interchange of $(L)$ with $(R)$ and $\vb{n} \leftrightarrow \vb{m}$.

Parameters $A_{lr}$ and $B_{lr}$ follow the previous results and equations~\eqref{eq:parameters_equality_zero} and \eqref{eq:parameters_equality_infty}:
\begin{eqnarray}
  A_{lr}
  =
  n_{\vb{0}} + m_{\vb{0}} + \ffA_{lr},
  & \qquad &
  \ffA_{lr} \in \Z,
  \\
  B_{lr}
  (-1)^{f^{(L)}}\, n_{\vb{1}} + (-1)^{f^{(R)}}\, m_{\vb{1}} + \ffB_{lr},
  & \qquad &
  \ffB_{lr} \in \Z.
\end{eqnarray}


% vim: ft=tex
