Update on the Abelian limit of the D-branes
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
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@@ -84,16 +84,14 @@ We thus choose $X_{(t)}^1$ and $X_{(t)}^2$ to be the coordinates parallel to the
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\begin{subfigure}[b]{0.45\linewidth}
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\centering
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\centering
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\def\svgwidth{\linewidth}
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\import{img/}{branesangles.pdf_tex}
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\import{img}{branesangles.pdf_tex}
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\caption{%
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\caption{D-branes as lines on $\R^2$.}
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D-branes as lines on $\R^2$.
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}
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\end{subfigure}
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\end{subfigure}
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\hfill
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\caption{Well adapted system of coordinates.}
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\caption{Well adapted system of coordinates.}
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\end{subfigure}
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\end{subfigure}
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\caption{%
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\caption{%
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@@ -388,7 +386,7 @@ We thus translated the rotations of the D-branes encoded in the matrices $R_{(t)
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\import{img/}{branchcuts.pdf_tex}
|
\import{img}{branchcuts.pdf_tex}
|
||||||
\caption{%
|
\caption{%
|
||||||
Branch cut structure of the complex plane with $N_B = 4$.
|
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.
|
Cuts are pictured as solid coloured blocks running from one intersection point to another at finite.
|
||||||
@@ -606,7 +604,7 @@ We choose $\bt = 1$ in what follows.
|
|||||||
\begin{figure}[tbp]
|
\begin{figure}[tbp]
|
||||||
\centering
|
\centering
|
||||||
\def\svgwidth{0.35\linewidth}
|
\def\svgwidth{0.35\linewidth}
|
||||||
\import{img/}{threebranes_plane.pdf_tex}
|
\import{img}{threebranes_plane.pdf_tex}
|
||||||
\caption{%
|
\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 $\omega_{\bt} = \infty$.}
|
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}
|
\label{fig:hypergeometric_cuts}
|
||||||
@@ -1810,273 +1808,359 @@ by the same symmetry.
|
|||||||
We can then study the two solutions in the two cases.
|
We can then study the two solutions in the two cases.
|
||||||
We first perform the computations common to both cases and then we explicitly specialise the calculations.
|
We first perform the computations common to both cases and then we explicitly specialise the calculations.
|
||||||
Computing the parameters of the hypergeometric functions of the first solution leads to:
|
Computing the parameters of the hypergeometric functions of the first solution leads to:
|
||||||
|
\begin{equation}
|
||||||
|
\begin{cases}
|
||||||
|
a^{(L)} & = n_{\vb{0}} + n_{\vb{1}} + n_{\vb{\infty}} + \ffa^{(L)}
|
||||||
|
\\
|
||||||
%% TODO %%
|
b^{(L)} & = n_{\vb{0}} + n_{\vb{1}} - n_{\vb{\infty}} + \ffb^{(L)}
|
||||||
|
\\
|
||||||
|
c^{(L)} & = 2\, n_{\vb{0}} + \ffc^{(L)}
|
||||||
|
\end{cases},
|
||||||
|
\qquad
|
||||||
|
\begin{cases}
|
||||||
\begin{align}
|
a^{(R)} & = m_{\vb{0}} + m_{\vb{1}} + m_{\vb{\infty}} + \ffa^{(R)}
|
||||||
\left\{\begin{array}{l}
|
\\
|
||||||
a^{(L)}=n_{\vb{0}}+n_{\vb{1}}+n_{\vb{\infty}}+\ffa^{(L)}
|
b^{(R)} & = m_{\vb{0}} + m_{\vb{1}} - m_{\vb{\infty}} + \ffb^{(R)}
|
||||||
\\
|
\\
|
||||||
b^{(L)}=n_{\vb{0}}+n_{\vb{1}}-n_{\vb{\infty}}+\ffb^{(L)}
|
c^{(R)} & = 2\, m_{\vb{0}} + 1 + \ffc^{(R)}
|
||||||
\\
|
\end{cases}.
|
||||||
c^{(L)}=2 n_{\vb{0}}+\ffc^{(L)}
|
\end{equation}
|
||||||
\end{array}
|
The values of the constants are in \Cref{tab:coeffs_k}.
|
||||||
\right.
|
We then derive the factors $K^{(L)}$ and $K^{(R)}$ using~\eqref{eq:K_factor_value}.
|
||||||
,~~~~
|
The first solution reads:
|
||||||
\left\{\begin{array}{l}
|
\begin{equation}
|
||||||
a^{(R)}=m_{\vb{0}}+m_{\vb{1}}+m_{\vb{\infty}}+\ffa^{(R)}
|
\begin{split}
|
||||||
\\
|
\ipd{\omega} \cX_1
|
||||||
b^{(R)}=m_{\vb{0}}+m_{\vb{1}}-m_{\vb{\infty}}+\ffb^{(R)}
|
& =
|
||||||
\\
|
(-\omega)^{n_{\vb{0}} + m_{\vb{0}} - 1}\,
|
||||||
c^{(R)}=2 m_{\vb{0}}+1+\ffc^{(R)}
|
(1-\omega)^{n_{\vb{1}} + m_{\vb{1}} - 1}\,
|
||||||
\end{array}
|
\\
|
||||||
\right.,
|
& \times
|
||||||
\end{align}
|
\mqty(
|
||||||
where the values of the constants can be read from Table~\ref{tab:coeffs_k}.
|
\hyp{a^{(L)}}{b^{(L)}}{c^{(L)}}{\omega}
|
||||||
Then we compute the $K^{(L)}$ and $K^{(R)}$ factors using \eqref{eq:K_factor_value}.
|
\\
|
||||||
Therefore the first solution is:
|
K^{(L)}\, (-\omega)^{1 - c^{(L)}}\,
|
||||||
\begin{align}
|
\hyp{a^{(L)} + 1 - c^{(L)}}{b^{(L)} + 1 - c^{(L)}}{2 - c^{(L)}}{\omega}
|
||||||
\partial_\omega \chi_1 =&
|
)
|
||||||
(-\omega)^{n_{\vb{0}}+m_{\vb{0}}-1 }
|
\\
|
||||||
(1-\omega)^{n_{\vb{1}}+m_{\vb{1}}-1 } \times
|
& \times
|
||||||
\nonumber\\
|
\mqty(
|
||||||
& \times
|
\hyp{a^{(R)}}{b^{(R)}}{c^{(R)}}{\omega}
|
||||||
\mqty(
|
\\
|
||||||
F(a^{(L)}, b^{(L)}; c^{(L)}; \omega) \\
|
K^{(R)}\, (-\omega)^{1 - c^{(R)}}\,
|
||||||
K^{(L)} (-\omega)^{1-c^{(L)}}
|
\hyp{a^{(R)} + 1 - c^{(R)}}{b^{(R)} + 1 - c^{(R)}}{2 - c^{(R)}}{\omega}
|
||||||
F(a^{(L)}+1-c^{(L)}, b^{(L)}+1-c^{(L)}; 2-c^{(L)}; \omega)
|
)
|
||||||
)
|
\end{split}
|
||||||
\nonumber\\
|
\label{eq:first_solution}
|
||||||
&\times
|
\end{equation}
|
||||||
\mqty(
|
|
||||||
F(a^{(R)}, b^{(R)}; c^{(R)}; \omega) \\
|
|
||||||
K^{(R)} (-\omega)^{1-c^{(R)}}
|
|
||||||
F(a^{(R)}+1-c^{(R)}, b^{(R)}+1-c^{(R)}; 2-c^{(R)}; \omega)
|
|
||||||
)^T.
|
|
||||||
\end{align}
|
|
||||||
|
|
||||||
The parameters of the second solution read
|
The parameters of the second solution read
|
||||||
\begin{align}
|
|
||||||
&
|
|
||||||
\left\{\begin{array}{ll}
|
|
||||||
\hat a^{(L)}= n_{\vb{0}}+\hat n_{\vb{1}}+ \hat n_{\vb{\infty}}
|
|
||||||
+\hat {\ffa}^{(L)}
|
|
||||||
&= c^{(L)}-a^{(L)}
|
|
||||||
+\ffa^{(L)}-\ffc^{(L)}+
|
|
||||||
\hat {\ffa}^{(L)}+1
|
|
||||||
\\
|
|
||||||
\hat b^{(L)}=n_{\vb{0}}+\hat n_{\vb{1}}- \hat n_{\vb{\infty}}+
|
|
||||||
\hat {\ffb}^{(L)}
|
|
||||||
&= c^{(L)}-b^{(L)}
|
|
||||||
+\ffb^{(L)}-\ffc^{(L)}+
|
|
||||||
\hat {\ffb}^{(L)}
|
|
||||||
\\
|
|
||||||
\hat c^{(L)}=2 n_{\vb{0}} +\hat {\ffc}^{(L)}
|
|
||||||
&= c^{(L)} -\ffc^{(L)}+\hat {\ffc}^{(L)}
|
|
||||||
\end{array}
|
|
||||||
\right.,
|
|
||||||
\nonumber\\
|
|
||||||
&
|
|
||||||
\left\{\begin{array}{ll}
|
|
||||||
\hat a^{(R)}= m_{\vb{0}}+\hat m_{\vb{1}}+ \hat m_{\vb{\infty}}
|
|
||||||
+\hat {\ffa}^{(R)}
|
|
||||||
&= c^{(R)}-a^{(R)}
|
|
||||||
+\ffa^{(R)}-\ffc^{(R)}+
|
|
||||||
\hat {\ffa}^{(R)}+1
|
|
||||||
\\
|
|
||||||
\hat b^{(R)}=m_{\vb{0}}+\hat m_{\vb{1}}- \hat m_{\vb{\infty}}+
|
|
||||||
\hat {\ffb}^{(R)}
|
|
||||||
&= c^{(R)}-b^{(R)}
|
|
||||||
+\ffb^{(R)}-\ffc^{(R)}+
|
|
||||||
\hat {\ffb}^{(R)}
|
|
||||||
\\
|
|
||||||
\hat c^{(R)}=2 m_{\vb{0}} +\hat {\ffc}^{(R)}
|
|
||||||
&= c^{(R)} -\ffc^{(R)}+\hat {\ffc}^{(R)}
|
|
||||||
\end{array}
|
|
||||||
\right.
|
|
||||||
.
|
|
||||||
\end{align}
|
|
||||||
We see that the two cases differ only for the constants and not for
|
|
||||||
the structure.
|
|
||||||
|
|
||||||
|
|
||||||
\subsubsection{Case 1}
|
|
||||||
\label{sec:case1}
|
|
||||||
We start with the case $n_{\vb{0}}>m_{\vb{0}}$, $n_{\vb{1}}>m_{\vb{1}}$ and
|
|
||||||
$n_{\vb{\infty}}> m_{\vb{\infty}}$
|
|
||||||
for which the second solution is
|
|
||||||
$n_{\vb{0}}>m_{\vb{0}}$, $\hat n_{\vb{1}}< \hat m_{\vb{1}}$ and
|
|
||||||
$\hat n_{\vb{\infty}}< \hat m_{\vb{\infty}}$
|
|
||||||
The parameters for the second are explicitly
|
|
||||||
\begin{align}
|
|
||||||
&
|
|
||||||
\left\{\begin{array}{l}
|
|
||||||
\hat a^{(L)}= c^{(L)}-a^{(L)}
|
|
||||||
\\
|
|
||||||
\hat b^{(L)}= c^{(L)}-b^{(L)}
|
|
||||||
\\
|
|
||||||
\hat c^{(L)}= c^{(L)}
|
|
||||||
\end{array}
|
|
||||||
\right.
|
|
||||||
,~~~~
|
|
||||||
\left\{\begin{array}{l}
|
|
||||||
\hat a^{(R)}= c^{(R)}-a^{(R)}
|
|
||||||
\\
|
|
||||||
\hat b^{(R)}= c^{(R)}-b^{(R)}+1
|
|
||||||
\\
|
|
||||||
\hat c^{(R)}= c^{(R)} +1
|
|
||||||
\end{array}
|
|
||||||
\right.
|
|
||||||
.
|
|
||||||
\end{align}
|
|
||||||
The $K$ factors are
|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\hat K^{(L)}= K^{(L)},~~~~
|
\begin{split}
|
||||||
\hat K^{(R)}= \frac{K^{(R)}}{a^{(R)} (c^{(R)}-b^{(R)})}
|
&
|
||||||
.
|
\begin{cases}
|
||||||
|
\hat{a}^{(L)}
|
||||||
|
& =
|
||||||
|
n_{\vb{0}} + \hat{n}_{\vb{1}} + \hat{n}_{\vb{\infty}} + \hat{\ffa}^{(L)}
|
||||||
|
=
|
||||||
|
c^{(L)} - a^{(L)} + \ffa^{(L)} - \ffc^{(L)} + \hat{\ffa}^{(L)} + 1
|
||||||
|
\\
|
||||||
|
\hat{b}^{(L)}
|
||||||
|
& =
|
||||||
|
n_{\vb{0}} + \hat{n}_{\vb{1}} - \hat{n}_{\vb{\infty}} + \hat{\ffb}^{(L)}
|
||||||
|
=
|
||||||
|
c^{(L)} - b^{(L)} + \ffb^{(L)} - \ffc^{(L)} + \hat{\ffb}^{(L)}
|
||||||
|
\\
|
||||||
|
\hat{c}^{(L)}
|
||||||
|
& =
|
||||||
|
2\, n_{\vb{0}} + \hat{\ffc}^{(L)}
|
||||||
|
=
|
||||||
|
c^{(L)} - \ffc^{(L)} + \hat{\ffc}^{(L)}
|
||||||
|
\end{cases}
|
||||||
|
\\
|
||||||
|
&
|
||||||
|
\begin{cases}
|
||||||
|
\hat{a}^{(R)}
|
||||||
|
& =
|
||||||
|
m_{\vb{0}} + \hat{n}_{\vb{1}} + \hat{n}_{\vb{\infty}} + \hat{\ffa}^{(R)}
|
||||||
|
=
|
||||||
|
c^{(R)} - a^{(R)} + \ffa^{(R)} - \ffc^{(R)} + \hat{\ffa}^{(R)} + 1
|
||||||
|
\\
|
||||||
|
\hat{b}^{(R)}
|
||||||
|
& =
|
||||||
|
m_{\vb{0}} + \hat{n}_{\vb{1}} - \hat{n}_{\vb{\infty}} + \hat{\ffb}^{(R)}
|
||||||
|
=
|
||||||
|
c^{(R)} - b^{(R)} + \ffb^{(R)} - \ffc^{(R)} + \hat{\ffb}^{(R)}
|
||||||
|
\\
|
||||||
|
\hat{c}^{(R)}
|
||||||
|
& =
|
||||||
|
2\, m_{\vb{0}} + \hat{\ffc}^{(R)}
|
||||||
|
=
|
||||||
|
c^{(R)} - \ffc^{(R)} + \hat{\ffc}^{(R)}
|
||||||
|
\end{cases}
|
||||||
|
\end{split}
|
||||||
|
\end{equation}
|
||||||
|
The two cases differ only for constant factors and not in structure.
|
||||||
|
|
||||||
|
|
||||||
|
\paragraph{Case 1}
|
||||||
|
|
||||||
|
Consider $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} > m_{\vb{\infty}}$.
|
||||||
|
The associated second solution is $n_{\vb{0}} > m_{\vb{0}}$, $\hat{n}_{\vb{1}} < \hat{m}_{\vb{1}}$ and $\hat{n}_{\vb{\infty}} < \hat{m}_{\vb{\infty}}$.
|
||||||
|
Its parameters are:
|
||||||
|
\begin{equation}
|
||||||
|
\begin{cases}
|
||||||
|
\hat{a}^{(L)} & = c^{(L)} - a^{(L)}
|
||||||
|
\\
|
||||||
|
\hat{b}^{(L)} & = c^{(L)} - b^{(L)}
|
||||||
|
\\
|
||||||
|
\hat{c}^{(L)} & = c^{(L)}
|
||||||
|
\end{cases},
|
||||||
|
\qquad
|
||||||
|
\begin{cases}
|
||||||
|
\hat{a}^{(R)} & = c^{(R)} - a^{(R)}
|
||||||
|
\\
|
||||||
|
\hat{b}^{(R)} & = c^{(R)} - b^{(R)} + 1
|
||||||
|
\\
|
||||||
|
\hat{c}^{(R)} & = c^{(R)} + 1
|
||||||
|
\end{cases},
|
||||||
|
\end{equation}
|
||||||
|
The normalisation factors are
|
||||||
|
\begin{equation}
|
||||||
|
\hat{K}^{(L)} = K^{(L)},
|
||||||
|
\qquad
|
||||||
|
\hat{K}^{(R)} = \frac{K^{(R)}}{a^{(R)} (c^{(R)} - b^{(R)})}.
|
||||||
\end{equation}
|
\end{equation}
|
||||||
Using Euler relation
|
Using Euler relation
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
F(a,b;c; \omega) =(1-\omega)^{c-a-b} F(c-a,c-b;c; \omega)
|
\hyp{a}{b}{c}{\omega} = (1-\omega)^{c-a-b}\, \hyp{c-a}{c-b}{c}{\omega},
|
||||||
,
|
|
||||||
\end{equation}
|
\end{equation}
|
||||||
we can finally write the second solution as
|
we can write the second solution as
|
||||||
\begin{align}
|
|
||||||
\partial_\omega \chi_2 =&
|
|
||||||
(-\omega)^{n_{\vb{0}}+m_{\vb{0}}-1}
|
|
||||||
(1-\omega)^{n_{\vb{1}}+m_{\vb{1}}} \times
|
|
||||||
\nonumber\\
|
|
||||||
& \times
|
|
||||||
\mqty(
|
|
||||||
F(a^{(L)}, b^{(L)}; c^{(L)}; \omega) \\
|
|
||||||
K^{(L)} (-\omega)^{1-c^{(L)}}
|
|
||||||
F(a^{(L)}+1-c^{(L)}, b^{(L)}+1-c^{(L)}; 2-c^{(L)}; \omega)
|
|
||||||
)
|
|
||||||
\nonumber\\
|
|
||||||
&\times
|
|
||||||
\mqty(
|
|
||||||
F(a^{(R)}+1, b^{(R)}; c^{(R)}+1; \omega) \\
|
|
||||||
\hat K^{(R)}
|
|
||||||
% \frac{K^{(R)}}{a^{(R)} (c^{(R)}-b^{(R)})}
|
|
||||||
(-\omega)^{-c^{(R)}}
|
|
||||||
F(a^{(R)}+1-c^{(R)}, b^{(R)}-c^{(R)}; 1-c^{(R)}; \omega)
|
|
||||||
)^T
|
|
||||||
,
|
|
||||||
\end{align}
|
|
||||||
in which the left basis is exactly equal to the first solution while
|
|
||||||
the right basis differs for $a^{(R)}\rightarrow a^{(R)}+1$ and
|
|
||||||
$c^{(R)}\rightarrow c^{(R)}+1$.
|
|
||||||
|
|
||||||
\subsubsection{Case 2}
|
|
||||||
\label{sec:case2}
|
|
||||||
|
|
||||||
Consider now the second case $n_{\vb{0}}>m_{\vb{0}}$, $n_{\vb{1}}>m_{\vb{1}}$ and
|
|
||||||
$n_{\vb{\infty}}< m_{\vb{\infty}}$.
|
|
||||||
For the second solution we have
|
|
||||||
$n_{\vb{0}}> m_{\vb{0}}$, $\hat n_{\vb{1}}< \hat m_{\vb{1}}$ and
|
|
||||||
$\hat n_{\vb{\infty}}> \hat m_{\vb{\infty}}$ and the parameters are explicitly
|
|
||||||
\begin{align}
|
|
||||||
&
|
|
||||||
\left\{\begin{array}{l}
|
|
||||||
\hat a^{(L)}= c^{(L)}-a^{(L)}-1
|
|
||||||
\\
|
|
||||||
\hat b^{(L)}= c^{(L)}-b^{(L)}+1
|
|
||||||
\\
|
|
||||||
\hat c^{(L)}= c^{(L)}
|
|
||||||
\end{array}
|
|
||||||
\right.
|
|
||||||
,~~~~
|
|
||||||
\left\{\begin{array}{l}
|
|
||||||
\hat a^{(R)}= c^{(R)}-a^{(R)}
|
|
||||||
\\
|
|
||||||
\hat b^{(R)}= c^{(R)}-b^{(R)}
|
|
||||||
\\
|
|
||||||
\hat c^{(R)}= c^{(R)}
|
|
||||||
\end{array}
|
|
||||||
\right.
|
|
||||||
.
|
|
||||||
\end{align}
|
|
||||||
The $K$ factors are
|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\hat K^{(L)}= K^{(L)}\frac{(b^{(L)}-1)(c^{(L)}-a^{(L)}-1)}{a^{(L)}(c^{(L)}-b^{(L)})},~~~~
|
\begin{split}
|
||||||
\hat K^{(R)}= K^{(R)}
|
\ipd{\omega} \cX_2
|
||||||
.
|
& =
|
||||||
|
(-\omega)^{n_{\vb{0}} + m_{\vb{0}} - 1}\,
|
||||||
|
(1-\omega)^{n_{\vb{1}} + m_{\vb{1}}}\,
|
||||||
|
\\
|
||||||
|
& \times
|
||||||
|
\mqty(
|
||||||
|
\hyp{a^{(L)}}{b^{(L)}}{c^{(L)}}{\omega}
|
||||||
|
\\
|
||||||
|
K^{(L)}\, (-\omega)^{1 - c^{(L)}}\,
|
||||||
|
\hyp{a^{(L)} + 1 - c^{(L)}}{b^{(L)} + 1 - c^{(L)}}{2 - c^{(L)}}{\omega}
|
||||||
|
)
|
||||||
|
\\
|
||||||
|
& \times
|
||||||
|
\mqty(
|
||||||
|
\hyp{a^{(R)} + 1}{b^{(R)}}{c^{(R)} + 1}{\omega}
|
||||||
|
\\
|
||||||
|
\hat{K}^{(R)}\, (-\omega)^{- c^{(R)}}\,
|
||||||
|
\hyp{a^{(R)} + 1 - c^{(R)}}{b^{(R)} - c^{(R)}}{1 - c^{(R)}}{\omega}
|
||||||
|
)
|
||||||
|
\end{split}.
|
||||||
\end{equation}
|
\end{equation}
|
||||||
Using Euler relation we can finally write the second solution for the
|
In this solution the left basis is exactly the same as in the first solution~\eqref{eq:first_solution} while the right basis differs for $a^{(R)} \mapsto a^{(R)} + 1$ and $c^{(R)} \mapsto c^{(R)} + 1$.
|
||||||
second case as
|
|
||||||
\begin{align}
|
|
||||||
\partial_\omega \chi_2 =&
|
|
||||||
(-\omega)^{n_{\vb{0}}+m_{\vb{0}}-1}
|
|
||||||
(1-\omega)^{n_{\vb{1}}+m_{\vb{1}}} \times
|
|
||||||
\nonumber\\
|
|
||||||
& \times
|
|
||||||
\mqty(
|
|
||||||
F(a^{(L)}+1, b^{(L)}-1; c^{(L)}; \omega) \\
|
|
||||||
\hat K^{(L)}
|
|
||||||
%K^{(L)}\frac{(b^{(L)}-1)(c^{(L)}-a^{(L)}-1)}{a^{(L)}(c^{(L)}-b^{(L)})}
|
|
||||||
(-\omega)^{1-c^{(L)}}
|
|
||||||
F(a^{(L)}+2-c^{(L)}, b^{(L)}-c^{(L)}; 2-c^{(L)}; \omega)
|
|
||||||
)
|
|
||||||
\nonumber\\
|
|
||||||
&\times
|
|
||||||
\mqty(
|
|
||||||
F(a^{(R)}, b^{(R)}; c^{(R)}; \omega) \\
|
|
||||||
K^{(R)} (-\omega)^{1-c^{(R)}}
|
|
||||||
F(a^{(R)}+1-c^{(R)}, b^{(R)}+1-c^{(R)}; 2-c^{(R)}; \omega)
|
|
||||||
)^T
|
|
||||||
,
|
|
||||||
\end{align}
|
|
||||||
in which the right basis is exactly equal to the first solution while
|
|
||||||
the left basis differs for $a^{(L)}\rightarrow a^{(L)}+1$ and
|
|
||||||
$b^{(L)}\rightarrow b^{(L)}-1$.
|
|
||||||
|
|
||||||
|
|
||||||
|
\paragraph{Case 2}
|
||||||
|
|
||||||
|
Consider now the second option $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} < m_{\vb{\infty}}$.
|
||||||
\subsection{The Solution}
|
For the second solution we have $n_{\vb{0}} > m_{\vb{0}}$, $\hat{n}_{\vb{1}} < \hat{m}_{\vb{1}}$ and $\hat{n}_{\vb{\infty}} > \hat{m}_{\vb{\infty}}$ and the parameters are explicitly:
|
||||||
In the previous section we have shown that there are two independent
|
\begin{equation}
|
||||||
solutions, therefore the general solution for
|
\begin{cases}
|
||||||
$\partial_\omega \chi$ obviously reads
|
\hat{a}^{(L)} & = c^{(L)} - a^{(L)} - 1
|
||||||
\begin{equation}
|
\\
|
||||||
\partial_\omega \chi= C_1 \partial_\omega \chi_1 + C_2 \partial_\omega \chi_2
|
\hat{b}^{(L)} & = c^{(L)} - b^{(L)} - 1
|
||||||
\label{eq:general_solution}
|
\\
|
||||||
.
|
\hat{c}^{(L)} & = c^{(L)}
|
||||||
|
\end{cases},
|
||||||
|
\qquad
|
||||||
|
\begin{cases}
|
||||||
|
\hat{a}^{(R)} & = c^{(R)} - a^{(R)}
|
||||||
|
\\
|
||||||
|
\hat{b}^{(R)} & = c^{(R)} - b^{(R)}
|
||||||
|
\\
|
||||||
|
\hat{c}^{(R)} & = c^{(R)}
|
||||||
|
\end{cases},
|
||||||
\end{equation}
|
\end{equation}
|
||||||
Therefore the final solution depends now only on two complex
|
The normalisation factors $K$ are:
|
||||||
constants, $C_1$ and $C_2$ which we can fix imposing the global conditions
|
\begin{equation}
|
||||||
in \eqref{eq:discontinuity_bc}, i.e. the second equation for all
|
\hat{K}^{(L)}
|
||||||
$t$'s in the solution \eqref{eq:classical_solution}.
|
|
||||||
Since the three target space intersection
|
|
||||||
points always define a triangle on a 2-dimensional plane, we can
|
|
||||||
impose the boundary conditions knowing two angles formed by the sides (i.e.
|
|
||||||
the branes between two intersections) and the length of one of
|
|
||||||
them.
|
|
||||||
We already fixed the parameters of the rotations, then we need to
|
|
||||||
compute the length of one of the sides.
|
|
||||||
and consider, for instance, the length of the side
|
|
||||||
$X(x_{\bt+1},x_{\bt+1}) - X(x_{\bt-1}, x_{\bt-1})$:
|
|
||||||
Explicitly we impose the four real equations in spinorial formalism
|
|
||||||
\begin{equation}
|
|
||||||
\int_0^1 \dd{\omega} \partial_\omega \cX(\omega)
|
|
||||||
+
|
|
||||||
U_L^{\dagger}(\vb{n}_{{\bt}})
|
|
||||||
~\int_0^1 \dd{\bar\omega} \partial_\omega \cX(\bar\omega)
|
|
||||||
~U_R(\vb{m}_{{\bt}})
|
|
||||||
=
|
=
|
||||||
f_{{\bt+1}\,(s)}-f_{{\bt-1}\,(s)}
|
K^{(L)}\,
|
||||||
,
|
\frac{(b^{(L)} - 1)(c^{(L)} - a^{(L)} - 1)}{a^{(L)} (c^{(L)} - b^{(L)})},
|
||||||
|
\qquad
|
||||||
|
\hat{K}^{(R)}
|
||||||
|
=
|
||||||
|
K^{(R)}.
|
||||||
\end{equation}
|
\end{equation}
|
||||||
where we have used the mapping \eqref{eq:def_omega} to write the
|
Using Euler relation we write the second solution for the second case as
|
||||||
integrals directly in $\omega$ variables.
|
\begin{equation}
|
||||||
This equation has then enough degrees of freedom to fix completely
|
\begin{split}
|
||||||
the two complex parameters $C_1$ and $C_2$,
|
\ipd{\omega} \cX_2
|
||||||
thus completing the determination of the full solution in its general form.
|
& =
|
||||||
|
(-\omega)^{n_{\vb{0}} + m_{\vb{0}} - 1}\,
|
||||||
|
(1-\omega)^{n_{\vb{1}} + m_{\vb{1}}}\,
|
||||||
|
\\
|
||||||
|
& \times
|
||||||
|
\mqty(
|
||||||
|
\hyp{a^{(L)} + 1}{b^{(L)} - 1}{c^{(L)}}{\omega}
|
||||||
|
\\
|
||||||
|
\hat{K}^{(L)}\, (-\omega)^{1 - c^{(L)}}\,
|
||||||
|
\hyp{a^{(L)} + 2 - c^{(L)}}{b^{(L)} - c^{(L)}}{2 - c^{(L)}}{\omega}
|
||||||
|
)
|
||||||
|
\\
|
||||||
|
& \times
|
||||||
|
\mqty(
|
||||||
|
\hyp{a^{(R)}}{b^{(R)}}{c^{(R)}}{\omega}
|
||||||
|
\\
|
||||||
|
\hat{K}^{(R)}\, (-\omega)^{- c^{(R)}}\,
|
||||||
|
\hyp{a^{(R)} + 1 - c^{(R)}}{b^{(R)} + 1 - c^{(R)}}{2 - c^{(R)}}{\omega}
|
||||||
|
)
|
||||||
|
\end{split}.
|
||||||
|
\end{equation}
|
||||||
|
The right basis is the same as in the first solution while the left basis differs for $a^{(L)} \mapsto a^{(L)} + 1$ and $b^{(L)} \mapsto b^{(L)} - 1$.
|
||||||
|
|
||||||
|
|
||||||
|
\subsubsection{The Solution}
|
||||||
|
|
||||||
|
We showed that there are two independent solutions.
|
||||||
|
The general solution for $\ipd{\omega} \cX$ is therefore:
|
||||||
|
\begin{equation}
|
||||||
|
\ipd{\omega} \cX
|
||||||
|
=
|
||||||
|
C_1\, \ipd{\omega} \cX_1 + C_2\, \ipd{\omega} \cX_2.
|
||||||
|
\label{eq:general_solution}
|
||||||
|
\end{equation}
|
||||||
|
The final solution depends only on two complex constants, $C_1$ and $C_2$, which we can fix imposing the global conditions in \eqref{eq:discontinuity_bc}, that is the second equation in the solution \eqref{eq:classical_solution}.
|
||||||
|
As the three intersection points in target space always define a triangle on a 2-dimensional plane, we impose the boundary conditions knowing two angles formed by the sides of the triangle (i.e.\ the branes between two intersections) and the length of one of them.
|
||||||
|
Since we already fixed the parameters associated to the rotations, we need to compute the length of one of the sides.
|
||||||
|
Consider for instance the length of $X(x_{\bt+1},\, x_{\bt+1}) - X(x_{\bt-1},\, x_{\bt-1})$.
|
||||||
|
Explicitly we impose the four real equations in spinorial formalism
|
||||||
|
\begin{equation}
|
||||||
|
\finiteint{\omega}{0}{1}
|
||||||
|
\ipd{\omega} \cX(\omega)
|
||||||
|
+
|
||||||
|
U_L^{\dagger}(\vb{n}_{{\bt}})
|
||||||
|
\left[
|
||||||
|
\finiteint{\bomega}{0}{1} \ipd{\bomega} \cX(\bomega)
|
||||||
|
\right]
|
||||||
|
U_R(\vb{m}_{{\bt}})
|
||||||
|
=
|
||||||
|
f_{{\bt+1}\, (s)} - f_{{\bt-1}\, (s)},
|
||||||
|
\end{equation}
|
||||||
|
where we used the mapping~\eqref{eq:def_omega} to write the integrals in the $\omega$ variables.
|
||||||
|
This equation has enough degrees of freedom to fix completely the two complex parameters $C_1$ and $C_2$.
|
||||||
|
The final generic solution is thus uniquely determined.
|
||||||
|
|
||||||
|
|
||||||
|
\subsection{Recovering the \texorpdfstring{\SU{2}}{SU(2)} and the Abelian Solution}
|
||||||
|
|
||||||
|
In this section we show how this general procedure includes both the solution with pure \SU{2} rotation matrices and the solution with Abelian rotations of the D-branes.
|
||||||
|
The Abelian solution emerges from this construction as a limit and produces the known result for Abelian $\SO{2} \times \SO{2} \subset \SO{4}$ rotations in the case of a factorised space $\R^4 = \R^2 \times \R^2$.
|
||||||
|
|
||||||
|
\subsubsection{Abelian Limit of the \texorpdfstring{\SU{2}}{SU(2)} Monodromies}
|
||||||
|
|
||||||
|
Here we compute the parameter $\vb{n}_{\vb{1}}$ given two Abelian rotation in $\upomega = 0$ and $\upomega = \infty$ using the standard expression for two \SU{2} element multiplication given in~\eqref{eq:product_in_SU2} in~\Cref{sec:isomorphism}.
|
||||||
|
Results are shown in~\Cref{tab:Abelian_composition}.
|
||||||
|
\begin{table}[tbp]
|
||||||
|
\centering
|
||||||
|
\begin{tabular}{@{}rr|cc|cr|c@{}}
|
||||||
|
\toprule
|
||||||
|
$\vb{n}_{\vb{0}}$ &
|
||||||
|
$\vb{n}_{\vb{\infty}}$ &
|
||||||
|
\multicolumn{2}{c|}{relations} &
|
||||||
|
$n_{\vb{1}}$ &
|
||||||
|
$\vb{n}_{\vb{1}}$ &
|
||||||
|
$\sum\limits_{t} \vb{n}_{\vb{t}}$
|
||||||
|
\\
|
||||||
|
\midrule
|
||||||
|
$n_{\vb{0}}\, \vb{k}$ &
|
||||||
|
$n_{\vb{\infty}}\, \vb{k}$ &
|
||||||
|
$n_{\vb{0}} + n_{\vb{\infty}} < \frac{1}{2}$ &
|
||||||
|
$n_{\vb{0}} \lessgtr n_{\vb{\infty}}$ &
|
||||||
|
$n_{\vb{0}} + n_{\vb{\infty}}$ &
|
||||||
|
$-n_{\vb{1}}\, \vb{k}$ &
|
||||||
|
$\vb{0}$
|
||||||
|
\\
|
||||||
|
$n_{\vb{0}}\, \vb{k}$ &
|
||||||
|
$n_{\vb{\infty}}\, \vb{k}$ &
|
||||||
|
$n_{\vb{0}} + n_{\vb{\infty}} > \frac{1}{2}$ &
|
||||||
|
$n_{\vb{0}} \lessgtr n_{\vb{\infty}}$ &
|
||||||
|
$1 - (n_{\vb{0}} + n_{\vb{\infty}})$ &
|
||||||
|
$n_{\vb{1}}\, \vb{k}$ &
|
||||||
|
$\vb{k}$
|
||||||
|
\\
|
||||||
|
$n_{\vb{0}}\, \vb{k}$ &
|
||||||
|
$-n_{\vb{\infty}}\, \vb{k}$ &
|
||||||
|
$n_{\vb{0}} + n_{\vb{\infty}} \lessgtr \frac{1}{2}$ &
|
||||||
|
$n_{\vb{0}} > n_{\vb{\infty}}$ &
|
||||||
|
$n_{\vb{0}} - n_{\vb{\infty}}$ &
|
||||||
|
$-n_{\vb{1}}\, \vb{k}$ &
|
||||||
|
$\vb{0}$
|
||||||
|
\\
|
||||||
|
$n_{\vb{0}}\, \vb{k}$ &
|
||||||
|
$-n_{\vb{\infty}}\, \vb{k}$ &
|
||||||
|
$n_{\vb{0}} + n_{\vb{\infty}} \lessgtr \frac{1}{2}$ &
|
||||||
|
$n_{\vb{0}} < n_{\vb{\infty}}$ &
|
||||||
|
$-n_{\vb{0}} + n_{\vb{\infty}}$ &
|
||||||
|
$n_{\vb{1}}\, \vb{k}$ &
|
||||||
|
$\vb{0}$
|
||||||
|
\\
|
||||||
|
\bottomrule
|
||||||
|
\end{tabular}
|
||||||
|
\caption{Abelian limit of \SU{2} monodromies}
|
||||||
|
\label{tab:Abelian_composition}
|
||||||
|
\end{table}
|
||||||
|
Under the parity transformation $P_2$ the previous four cases are grouped
|
||||||
|
into two sets $\{ n_{\vb{1}} = n_{\vb{0}} + n_{\vb{\infty}},\, \hat{n}_{\vb{1}} = -n_{\vb{0}} + \hat{n}_{\vb{\infty}} \}$ and $\{ n_{\vb{1}} = 1 - (n_{\vb{0}} + n_{\vb{\infty}}),\, \hat{n}_{\vb{1}} = n_{\vb{0}} - \hat{n}_{\vb{\infty}} \}$.
|
||||||
|
Geometrically the first group corresponds to the same geometry which is depicted in~\Cref{fig:Abelian_angles_1} while the second in~\Cref{fig:Abelian_angles_2}.
|
||||||
|
We can in fact arbitrarily fix the orientation of $D_{(3)}$ to obtain these geometrical interpretations.
|
||||||
|
Since $n^3_{\vb{0}} > 0$ we can then fix the orientation of $D_{{1}}$.
|
||||||
|
$D_{{2}}$ is then fixed relatively to $D_{{1}}$ by the sign of $n^3_{\vb{\infty}}$.
|
||||||
|
The sign of $n^3_{\vb{1}}$ then follows.
|
||||||
|
|
||||||
|
Differently from the usual geometric Abelian case, this group analytical approach distinguishes between the possible orientations of the D-branes.
|
||||||
|
In fact we can compare all possible D-brane orientation and the group parameter $n^3$ with the angles in the Abelian configuration in~\Cref{fig:usual_Abelian_angles}.
|
||||||
|
The relation between the usual Abelian paramter $\epsilon_{\vb{t}}$ and $n_{\vb{t}}^3$ is
|
||||||
|
\begin{equation}
|
||||||
|
\varepsilon_{\vb{t}}
|
||||||
|
=
|
||||||
|
n_{\vb{t}}^3 + \theta(-n^3_{\vb{t}})
|
||||||
|
\label{eq:Abelian_vs_n_simple_case},
|
||||||
|
\end{equation}
|
||||||
|
when all $m = 0$.
|
||||||
|
|
||||||
|
\begin{figure}[tbp]
|
||||||
|
\centering
|
||||||
|
\def\svgwidth{0.8\textwidth}
|
||||||
|
\import{img}{abelian_angles_case1.pdf_tex}
|
||||||
|
\caption{%
|
||||||
|
The Abelian limit when the triangle has all acute angles.
|
||||||
|
This corresponds to the cases $n_{\vb{0}} + n_{\vb{\infty}}< \frac{1}{2}$ and $n_{\vb{0}}< n_{\vb{\infty}}$ which are exchanged under the parity $P_2$.}
|
||||||
|
\label{fig:Abelian_angles_1}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[tbp]
|
||||||
|
\centering
|
||||||
|
\def\svgwidth{0.8\textwidth}
|
||||||
|
\import{img}{abelian_angles_case2.pdf_tex}
|
||||||
|
\caption{%
|
||||||
|
The Abelian limit when the triangle has one obtuse angle.
|
||||||
|
This corresponds to the cases $n_{\vb{0}} + n_{\vb{\infty}}> \frac{1}{2}$ and $n_{\vb{0}}> n_{\vb{\infty}}$ which are exchanged under the parity $P_2$.}
|
||||||
|
\label{fig:Abelian_angles_2}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\begin{figure}[tbp]
|
||||||
|
\centering
|
||||||
|
\def\svgwidth{0.8\textwidth}
|
||||||
|
\import{img}{usual_abelian_angles.pdf_tex}
|
||||||
|
\caption{%
|
||||||
|
The geometrical angles used in the usual geometrical approach to the Abelian configuration do not distinguish among the possible branes orientations.
|
||||||
|
In fact we have $0 \le \alpha < 1$ and $0 < \upvarepsilon < 1$.
|
||||||
|
}
|
||||||
|
\label{fig:usual_Abelian_angles}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
% vim: ft=tex
|
% vim: ft=tex
|
||||||
|
|||||||
Reference in New Issue
Block a user