diff --git a/sciencestuff.sty b/sciencestuff.sty index 53d581d..d21faa9 100644 --- a/sciencestuff.sty +++ b/sciencestuff.sty @@ -4,6 +4,7 @@ \RequirePackage{amsmath} %--------------------- math mode \RequirePackage{amssymb} %--------------------- math symbols \RequirePackage{amsfonts} %-------------------- math fonts +\RequirePackage{bm} %-------------------------- boldsymbols \RequirePackage{mathtools} %------------------- mathematical tools \RequirePackage{mathrsfs} %-------------------- better cal \RequirePackage{slashed} %--------------------- slashed characters diff --git a/sec/app/isomorphism.tex b/sec/app/isomorphism.tex index ee48cad..b795471 100644 --- a/sec/app/isomorphism.tex +++ b/sec/app/isomorphism.tex @@ -3,53 +3,53 @@ In this appendix we explain the conventions used for \SU{2} and show the details \subsection{Conventions} -We parameterise \SU{2} matrices $U$ with a vector $\vb{n} \in \R^3$ such that: +We parameterise \SU{2} matrices $U$ with a vector $\vec{n} \in \R^3$ such that: \begin{equation} - U(\vb{n}) + U(\vec{n}) = \cos(2 \pi n)\, \1_2 + - i\, \frac{\vb{n} \cdot \vb{\sigma}}{n}\, \sin(2 \pi n), + i\, \frac{\vec{n} \cdot \vec{\sigma}}{n}\, \sin(2 \pi n), \label{eq:su2parametrisation} \end{equation} -where $n = \norm{\vb{n}}$ and $0 \le n \le \frac{1}{2}$. -We also identify all $\vb{n}$ when $n=\frac{1}{2}$ since in this case $U(\vb{n})= -\1_2$. +where $n = \norm{\vec{n}}$ and $0 \le n \le \frac{1}{2}$. +We also identify all $\vec{n}$ when $n=\frac{1}{2}$ since in this case $U(\vec{n})= -\1_2$. The parametrisation is such that: \begin{eqnarray} - U^*(\vb{n}) + U^*(\vec{n}) & = & - \sigma^2\, U(\vb{n})\, \sigma^2 + \sigma^2\, U(\vec{n})\, \sigma^2 = - U(\widetilde{\vb{n}}), + U(\widetilde{\vec{n}}), \\ - U^{\dagger}(\vb{n}) + U^{\dagger}(\vec{n}) & = & - U^T(\widetilde{\vb{n}}) + U^T(\widetilde{\vec{n}}) = - U(-\vb{n}), + U(-\vec{n}), \\ - -U(\vb{n}) + -U(\vec{n}) & = & - U(\widehat{\vb{n}}) + U(\widehat{\vec{n}}) \label{eq:U_props} \end{eqnarray} -where $\sigma^2$ is the second Pauli matrix, $\widetilde{\vb{n}} = \qty( -n^1, n^2, -n^3 )$ and $\widehat{\vb{n}} = - \qty(\frac{1}{2} -n )\, \frac{\vb{n}}{n}$. +where $\sigma^2$ is the second Pauli matrix, $\widetilde{\vec{n}} = \qty( -n^1, n^2, -n^3 )$ and $\widehat{\vec{n}} = - \qty(\frac{1}{2} -n )\, \frac{\vec{n}}{n}$. -The group product of two elements $U(\vb{n} \circ \vb{m} ) = U(\vb{n})\, U(\vb{m})$ has an explicit realisation as: +The group product of two elements $U(\vec{n} \circ \vec{m} ) = U(\vec{n})\, U(\vec{m})$ has an explicit realisation as: \begin{equation} \begin{split} - \cos(2 \pi \norm{\vb{n} \circ \vb{m}}) + \cos(2 \pi \norm{\vec{n} \circ \vec{m}}) & = \cos(2 \pi n)\, \cos(2 \pi m) - - \sin(2 \pi n)\, \sin(2\pi m)\, \frac{\vb{n} \cdot \vb{m}}{n\, m}, + \sin(2 \pi n)\, \sin(2\pi m)\, \frac{\vec{n} \cdot \vec{m}}{n\, m}, \\ - \sin(2 \pi \norm{\vb{n} \circ \vb{m}})\, - \frac{\vb{n} \circ \vb{m}}{\norm{\vb{n} \circ \vb{m}}} + \sin(2 \pi \norm{\vec{n} \circ \vec{m}})\, + \frac{\vec{n} \circ \vec{m}}{\norm{\vec{n} \circ \vec{m}}} & = - \cos(2 \pi n)\, \sin(2\pi m)\, \frac{\vb{m}}{m} + \cos(2 \pi n)\, \sin(2\pi m)\, \frac{\vec{m}}{m} + - \sin(2 \pi n)\, \cos(2\pi m)\, \frac{\vb{n}}{n}. + \sin(2 \pi n)\, \cos(2\pi m)\, \frac{\vec{n}}{n}. \end{split} \label{eq:product_in_SU2} \end{equation} @@ -58,9 +58,9 @@ The group product of two elements $U(\vb{n} \circ \vb{m} ) = U(\vb{n})\, U(\vb{ Let $I = 1,\, 2,\, 3,\, 4$ and define: \begin{equation} - \tau_I = \qty( i\, \1_2,\, \vb{\sigma} ), + \tau_I = \qty( i\, \1_2,\, \vec{\sigma} ), \end{equation} -where $\vb{\sigma} = \qty( \sigma^1,\, \sigma^2,\, \sigma^3 )$ are the Pauli matrices. +where $\vec{\sigma} = \qty( \sigma^1,\, \sigma^2,\, \sigma^3 )$ are the Pauli matrices. It is possible to show that: \begin{equation} \begin{split} @@ -122,7 +122,7 @@ If the vector $X^I$ is real, using~\eqref{eq:tau_props} we have: A rotation in spinor representation is defined as: \begin{equation} - X'_{(s)} = U_{L}(\vb{n})\, X_{(s)}\, U_{R}^{\dagger}(\vb{m}) + X'_{(s)} = U_{L}(\vec{n})\, X_{(s)}\, U_{R}^{\dagger}(\vec{m}) \end{equation} and it is equivalent to: \begin{equation} @@ -138,9 +138,9 @@ through \frac{1}{2} \tr( \qty( \tau_I )^{\dagger}\, - U_{L}(\vb{n})\, + U_{L}(\vec{n})\, \tau_J\, - U_{R}^{\dagger}(\vb{m}) + U_{R}^{\dagger}(\vec{m}) ). \end{equation} The matrix $R$ is the $4$-dimensional rotation matrix we are looking for since: diff --git a/sec/app/parameters.tex b/sec/app/parameters.tex index fdd3f2f..eb1af43 100644 --- a/sec/app/parameters.tex +++ b/sec/app/parameters.tex @@ -6,28 +6,28 @@ In this appendix we show the computation of the parameters of the hypergeometric In the main text we set \begin{equation} D~ - \rM_{\vb{\infty}}~ + \rM_{\infty}~ D^{-1} = - e^{-2\pi i \delta_{\vb{\infty}}}\, - \cL(\vb{n}_{\vb{\infty}}), + e^{-2\pi i \delta_{\infty}}\, + \cL(\vec{n}_{\infty}), \end{equation} -where $\cL(\vb{n}_{\vb{\infty}}) \in \SU{2}$. +where $\cL(\vec{n}_{\infty}) \in \SU{2}$. The previous equation implies \begin{equation} - \qty( D\, \rM_{\vb{\infty}}\, D^{-1} )^\dagger + \qty( D\, \rM_{\infty}\, D^{-1} )^\dagger = - \qty( D\, \rM_{\vb{\infty}}\, D^{-1} )^{-1}, + \qty( D\, \rM_{\infty}\, D^{-1} )^{-1}, \end{equation} which can be rewritten as \begin{equation} - \widetilde{\rM}_{\vb{\infty}}^{-1}~ + \widetilde{\rM}_{\infty}^{-1}~ \cC^{\dagger}\, D^{\dagger}\, D\, \cC = \cC^{\dagger}\, D^{\dagger}\, D\, \cC~ - \widetilde{\rM}_{\vb{\infty}}^{-1}. + \widetilde{\rM}_{\infty}^{-1}. \end{equation} -As $\widetilde{\rM}_{\vb{\infty}}$ is a generic diagonal matrix, the previous equation implies that the off-diagonal elements of $\cC^{\dagger}\, D^{\dagger}\, D\, \cC$ must vanish. +As $\widetilde{\rM}_{\infty}$ is a generic diagonal matrix, the previous equation implies that the off-diagonal elements of $\cC^{\dagger}\, D^{\dagger}\, D\, \cC$ must vanish. We therefore have \begin{equation} \begin{split} @@ -76,71 +76,71 @@ This would then imply We can finally show in details the computation of the parameters of the basis of hypergeometric functions used in the main text. The relation between these and the \SU{2} matrices can be computed requiring that the monodromies induced by the choice of the parameters equal the monodromies produced by the rotations of the D-branes. -The monodromy in $\omega_{\bart-1} = 0$ is simpler to compute given that we choose $\cL(\vb{n}_{\vb{0}})$ and $\cR(\widetilde{\vb{m}}_{\vb{0}})$ to be diagonal. +The monodromy in $\omega_{\bart-1} = 0$ is simpler to compute given that we choose $\cL(\vec{n}_{0})$ and $\cR(\widetilde{\vec{m}}_{0})$ to be diagonal. We impose: \begin{eqnarray} \mqty( \dmat{1, e^{-2\pi i c^{(L)}}} ) & = & - e^{-2\pi i \delta_{\vb{0}}^{(L)}}\, - \mqty( \dmat{e^{2\pi i n_{\vb{0}}}, e^{-2\pi i n_{\vb{0}}}} ), + e^{-2\pi i \delta_{0}^{(L)}}\, + \mqty( \dmat{e^{2\pi i n_{0}}, e^{-2\pi i n_{0}}} ), \\ \mqty( \dmat{1, e^{-2\pi i c^{(R)}}} ) & = & - e^{-2\pi i \delta_{\vb{0}}^{(R)}}\, - \mqty( \dmat{e^{-2\pi i m_{\vb{0}}}, e^{2\pi i m_{\vb{0}}}} ), + e^{-2\pi i \delta_{0}^{(R)}}\, + \mqty( \dmat{e^{-2\pi i m_{0}}, e^{2\pi i m_{0}}} ), \end{eqnarray} -where $n^3_{\vb{0}} = \norm{\vb{n}_{\vb{0}}} = n_{\vb{0}}$ and $m^3_{\vb{0}} = \norm{\vb{m}_{\vb{0}}} = m_{\vb{0}}$ with $0 \le n_{\vb{0}},\, m_{\vb{0}} < 1$ due to the conventions \eqref{eq:maximal_torus_left} and \eqref{eq:maximal_torus_right}. +where $n^3_{0} = \norm{\vec{n}_{0}} = n_{0}$ and $m^3_{0} = \norm{\vec{m}_{0}} = m_{0}$ with $0 \le n_{0},\, m_{0} < 1$ due to the conventions \eqref{eq:maximal_torus_left} and \eqref{eq:maximal_torus_right}. We thus have: \begin{equation} \begin{split} - \delta_{\vb{0}}^{(L)} + \delta_{0}^{(L)} & = - n_{\vb{0}} + k_{\delta^{(L)}_{\vb{0}}}, + n_{0} + k_{\delta^{(L)}_{0}}, \qquad - k_{\delta^{(L)}_{\vb{0}}} \in \Z, + k_{\delta^{(L)}_{0}} \in \Z, \\ c^{(L)} & = - 2 n_{\vb{0}} + k_c, + 2 n_{0} + k_c, \qquad k_c \in \Z. \end{split} \label{eq:cL} \end{equation} -Since the determinant of the right hand side is $e^{-4 \pi i \delta_{\vb{0}}^{(L)}}$, the range of definition of $\delta_{\vb{0}}^{(L)}$ is $\alpha \le \delta_{\vb{0}}^{(L)} \le \alpha + \frac{1}{2}$. -Given that $0 \le n_{\vb{0}} < \frac{1}{2}$ we simply take $\alpha = 0$ and set $\delta_{\vb{0}}^{(L)} = n_{\vb{0}}$. +Since the determinant of the right hand side is $e^{-4 \pi i \delta_{0}^{(L)}}$, the range of definition of $\delta_{0}^{(L)}$ is $\alpha \le \delta_{0}^{(L)} \le \alpha + \frac{1}{2}$. +Given that $0 \le n_{0} < \frac{1}{2}$ we simply take $\alpha = 0$ and set $\delta_{0}^{(L)} = n_{0}$. Analogous results hold in the right sector. Furthermore from the third equation in \eqref{eq:parameters_equality_zero} and from the first equation in \eqref{eq:cL} we can restrict: \begin{equation} - n_{\vb{0}} + m_{\vb{0}} - A \in \Z. + n_{0} + m_{0} - A \in \Z. \end{equation} -We then need to find $3$ equations to determine $a^{(L)}$, $b^{(L)}$ and $\delta^{(L)}_{\vb{\infty}}$. +We then need to find $3$ equations to determine $a^{(L)}$, $b^{(L)}$ and $\delta^{(L)}_{\infty}$. After that we then fix the remaining factors in $B$ and $\abs{K^{(L)}}$. The equations follow from~\eqref{eq:parameters_equality_infty}. -The first two equations for $a^{(L)}$, $b^{(L)}$ and $\delta^{(L)}_{\vb{\infty}}$ follow by considering the trace of~\eqref{eq:parameters_equality_infty}: +The first two equations for $a^{(L)}$, $b^{(L)}$ and $\delta^{(L)}_{\infty}$ follow by considering the trace of~\eqref{eq:parameters_equality_infty}: \begin{equation} e^{\pi i ( a^{(L)} + b^{(L)} )} \cos(\pi( a^{(L)} - b^{(L)} ) ) = - e^{-2\pi i \delta^{(L)}_{\infty}} \cos(2\pi n_{\vb{\infty}}), + e^{-2\pi i \delta^{(L)}_{\infty}} \cos(2\pi n_{\infty}), \end{equation} which is satisfied by: \begin{equation} \begin{split} - \delta^{(L)}_{\vb{\infty}} + \delta^{(L)}_{\infty} & = - \frac{1}{2}(a^{(L)} + b^{(L)}) + - \frac{1}{2} k_{\delta^{(L)}_{\vb{\infty}}}, + \frac{1}{2} k_{\delta^{(L)}_{\infty}}, \qquad - k_{\delta_{\vb{\infty}}} \in \Z, + k_{\delta_{\infty}} \in \Z, \\ a^{(L)} - b^{(L)} & = - 2\, (-1)^{p^{(L)}}\, n_{\vb{\infty}} + 2\, (-1)^{p^{(L)}}\, n_{\infty} + - (-1)^{q^{(L)}}\, k_{\delta^{(L)}_{\vb{\infty}}} + (-1)^{q^{(L)}}\, k_{\delta^{(L)}_{\infty}} + 2\, k'_{a b}, \qquad @@ -154,31 +154,31 @@ We therefore have: \begin{equation} a^{(L)} - b^{(L)} = - 2\, n_{\vb{\infty}} + 2\, n_{\infty} + - k_{\delta^{(L)}_{\vb{\infty}}} + k_{\delta^{(L)}_{\infty}} + 2 k_{ab}, \qquad k_{a b}\in \Z. \label{eq:aL-bL} \end{equation} -The allowed values for $k_{\delta^{(L)}_{\vb{\infty}}}$ follow a construction similar to the monodromy around $\omega_{\bart-1} = 0$. +The allowed values for $k_{\delta^{(L)}_{\infty}}$ follow a construction similar to the monodromy around $\omega_{\bart-1} = 0$. The main difference is given by the fact that $\frac{1}{2}(a^{(L)} + b^{(L)})$ may a priori take values in an interval of width $1$. -As in the previous case we have $\alpha \le \delta_{\vb{\infty}}^{(L)} \le \alpha + \frac{1}{2}$ with $\alpha$ technically arbitrary. -We cannot thus choose a vanishing $k_{\delta^{(L)}_{\vb{\infty}}}$ but we have to consider $k_{\delta^{(L)}_{\infty}} = 0,\, 1$. +As in the previous case we have $\alpha \le \delta_{\infty}^{(L)} \le \alpha + \frac{1}{2}$ with $\alpha$ technically arbitrary. +We cannot thus choose a vanishing $k_{\delta^{(L)}_{\infty}}$ but we have to consider $k_{\delta^{(L)}_{\infty}} = 0,\, 1$. We find a third relation by considering the entry \begin{equation} \Im\qty( - e^{+2\pi i \delta_{\vb{\infty}}^{(L)}}\, + e^{+2\pi i \delta_{\infty}^{(L)}}\, D^{(L)}\, - \rM_{\vb{\infty}}^{(L)}\, + \rM_{\infty}^{(L)}\, \qty( D^{(L)} )^{-1} )_{11} = \Im\qty( - \cL(n_{\vb{\infty}}) + \cL(n_{\infty}) )_{11}. \end{equation} Using @@ -191,31 +191,31 @@ and the second equation in~\eqref{eq:cL} and~\eqref{eq:aL-bL} leads to: \begin{equation} \cos(\pi( a^{(L)} + b^{(L)} - c^{(L)} )) = - (-1)^{k_c+k_{\delta^{(L)}_{\vb{\infty}}} }\, \cos(2\pi \cA^{(L)}), + (-1)^{k_c+k_{\delta^{(L)}_{\infty}} }\, \cos(2\pi \cA^{(L)}), \end{equation} where \begin{equation} \cos(2\pi \cA^{(L)}) = - \cos(2\pi n_{\vb{0}})\, - \cos(2\pi n_{\vb{\infty}}) + \cos(2\pi n_{0})\, + \cos(2\pi n_{\infty}) - - \sin(2\pi n_{\vb{0}})\, - \sin(2\pi n_{\vb{\infty}})\, - \frac{n_{\vb{\infty}}^3}{n_{\vb{\infty}}}. + \sin(2\pi n_{0})\, + \sin(2\pi n_{\infty})\, + \frac{n_{\infty}^3}{n_{\infty}}. \label{eq:cos_n1} \end{equation} This expression is connected with rotation parameter in the third interaction point $\omega_{\bart+1} = 1$. -In fact $\cos(2\pi \cA^{(L)}) = \cos(2\pi {n}_{\vb{1}})$. +In fact $\cos(2\pi \cA^{(L)}) = \cos(2\pi {n}_{1})$. We then write \begin{equation} a^{(L)} + b^{(L)} - c^{(L)} = - 2\, (-1)^{f^{(L)}}\, n_{\vb{1}} + 2\, (-1)^{f^{(L)}}\, n_{1} + k_c + - k_{\delta^{(L)}_{\vb{\infty}}} + k_{\delta^{(L)}_{\infty}} + 2\, k_{abc}, \qquad @@ -228,13 +228,13 @@ The request + B - - n_{\vb{0}} + n_{0} - - m_{\vb{0}} + m_{0} - - (-1)^{f^{(L)}}\, n_{\vb{1}} + (-1)^{f^{(L)}}\, n_{1} - - (-1)^{f^{(R)}}\, m_{\vb{1}} + (-1)^{f^{(R)}}\, m_{1} \in \Z \end{equation} finally fixes the $B$ parameter in the third equation of~\eqref{eq:parameters_equality_infty}. @@ -243,51 +243,51 @@ So far we can summarise the results in \begin{eqnarray} a = - n_{\vb{0}} + (-1)^{f^{(L)}} n_{\vb{1}} + n_{\vb{\infty}} + m_a, + n_{0} + (-1)^{f^{(L)}} n_{1} + n_{\infty} + m_a, & \qquad & m_a \in \Z, \\ b = - n_{\vb{0}} + (-1)^{f^{(L)}} n_{\vb{1}} - n_{\vb{\infty}} + m_b, + n_{0} + (-1)^{f^{(L)}} n_{1} - n_{\infty} + m_b, & \qquad & m_b \in \Z, \\ c = - 2\, n_{\vb{0}} + m_c, + 2\, n_{0} + m_c, & \qquad & m_c \in \Z, \\ - \delta_{\vb{0}}^{(L)} + \delta_{0}^{(L)} = - n_{\vb{0}}, + n_{0}, \\ - \delta_{\vb{\infty}}^{(L)} + \delta_{\infty}^{(L)} = - - n_{\vb{0}} - (-1)^{f^{(L)}} n_{\vb{1}} + m_c + 2\, m_\delta, + - n_{0} - (-1)^{f^{(L)}} n_{1} + m_c + 2\, m_\delta, & \qquad & m_{\delta} \in \Z, \\ A = - n_{\vb{0}} + m_{\vb{0}} + m_A, + n_{0} + m_{0} + m_A, & \qquad & m_A \in \Z, \\ B = - (-1)^{f^{(L)}}\, n_{\vb{1}} + (-1)^{f^{(R)}}\, m_{\vb{1}} + m_B, + (-1)^{f^{(L)}}\, n_{1} + (-1)^{f^{(R)}}\, m_{1} + m_B, & \qquad & m_B \in \Z. \end{eqnarray} $K^{(L)}$ is finally determined from \begin{equation} - \qty( D^{(L)}\, \rM_{\vb{\infty}}\, \qty( D^{(L)} )^{-1} )_{21} + \qty( D^{(L)}\, \rM_{\infty}\, \qty( D^{(L)} )^{-1} )_{21} = - e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\, - \qty( \cL(n_{\vb{\infty}}) )_{21}, + e^{-2\pi i \delta_{\infty}^{(L)}}\, + \qty( \cL(n_{\infty}) )_{21}, \label{eq:fixing_K_21} \end{equation} and get: @@ -296,9 +296,9 @@ and get: = -\frac{(-1)^{m_a + m_b + m_c}}{2 \pi^2}\, \cG( a^{(L)},\, b^{(L)},\, c^{(L)} )\, - \sin(2 \pi n_{\vb{0}}) - \sin(2 \pi n_{\vb{\infty}}) - \frac{n^1_{\vb{\infty}} + i\, n^2_{\vb{\infty}}}{n_{\vb{\infty}}}, + \sin(2 \pi n_{0}) + \sin(2 \pi n_{\infty}) + \frac{n^1_{\infty} + i\, n^2_{\infty}}{n_{\infty}}, \label{eq:app_B_K21} \end{equation} where $\cG( a,\, b,\, c ) = \gfun{1-a}\, \gfun{1-b}\, \gfun{a+1-c}\, \gfun{b+1-c}$. @@ -316,16 +316,16 @@ The result is \cG(1 - a^{(L)},\, 1 - b^{(L)},\, 2 - c^{(L)})\, \\ & \times - \sin(2 \pi n_{\vb{0}})\, - \sin(2 \pi n_{\vb{\infty}})\, - \frac{n^1_{\vb{\infty}} -i n^2_{\vb{\infty}}}{n_{\vb{\infty}}}, + \sin(2 \pi n_{0})\, + \sin(2 \pi n_{\infty})\, + \frac{n^1_{\infty} -i n^2_{\infty}}{n_{\infty}}, \end{split} \label{eq:app_B_K12} \end{equation} where the function $\cG( a,\, b,\, c )$ was defined at the end of the previous section. Compatibility with~\eqref{eq:app_B_K21} requires \begin{equation} - \frac{(n^1_{\vb{\infty}})^2 + (n^2_{\vb{\infty}})^2}{n^2_{\vb{\infty}}} + \frac{(n^1_{\infty})^2 + (n^2_{\infty})^2}{n^2_{\infty}} = -4 \frac{\sin(\pi a) \sin(\pi(c-a))\sin(\pi b) \sin(\pi(c-b))} {\sin^2(\pi c) \sin^2(\pi(a-b))}. @@ -333,7 +333,7 @@ Compatibility with~\eqref{eq:app_B_K21} requires \end{equation} We can then rewrite~\eqref{eq:cos_n1} as \begin{equation} - \frac{(n^3_{\vb{\infty}})^2}{n^2_{\vb{\infty}}} + \frac{(n^3_{\infty})^2}{n^2_{\infty}} = \frac{(\cos(\pi (a-b)) \cos(\pi c)- \cos(\pi(a+b-c)))^2} {\sin^2(\pi c) \sin^2(\pi(a-b))}. @@ -341,10 +341,10 @@ We can then rewrite~\eqref{eq:cos_n1} as It is then possible to verify that the sum of the left and right hand sides of~\eqref{eq:n12+n22} and the last equation are equal to $1$. The same consistency check can also be performed by computing $K^{(L)}$ from \begin{equation} - \qty( D^{(L)}\, \rM_{\vb{\infty}}\, \qty( D^{(L)} )^{-1} )_{12} + \qty( D^{(L)}\, \rM_{\infty}\, \qty( D^{(L)} )^{-1} )_{12} = - e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\, - \qty( \cL(n_{\vb{\infty}}) )_{12}, + e^{-2\pi i \delta_{\infty}^{(L)}}\, + \qty( \cL(n_{\infty}) )_{12}, \end{equation} instead of \eqref{eq:fixing_K_21}. diff --git a/sec/part1/dbranes.tex b/sec/part1/dbranes.tex index 2e958d3..8e37c12 100644 --- a/sec/part1/dbranes.tex +++ b/sec/part1/dbranes.tex @@ -451,7 +451,7 @@ We define the spinor representation of $X$ as: \begin{equation} X_{(s)}( u, \baru ) = X^I( u, \baru )\, \tau_I, \end{equation} -where $\tau = \qty( i\, \1_2,\, \vb{\sigma} )$ and $\vb{\sigma}$ is the vector of the Pauli matrices. +where $\tau = \qty( i\, \1_2,\, \vec{\sigma} )$ and $\vec{\sigma}$ is the vector of the Pauli matrices. Consider then: \begin{equation} \ipd{z} \cX_{(s)}( z ) @@ -461,9 +461,9 @@ Consider then: & \qif z \in \ccH \qor z \in D_{(\bart)} \\ - U_{L}(\vb{n}_{(\bart)})\, + U_{L}(\vec{n}_{(\bart)})\, \ipd{\baru} X_{(s)}(\baru)\, - U_{R}^{\dagger}(\vb{m}_{(\bart)}) + U_{R}^{\dagger}(\vec{m}_{(\bart)}) & \qif z \in \bccH \qor z \in D_{(\bart)} \end{cases}. \label{eq:spinor_doubling_trick} @@ -491,27 +491,27 @@ We find: \begin{eqnarray} \cL_{(t,\, t+1)} & = & - U_{L}(\vb{n}_{(t+1)})\, - U_{L}^{\dagger}(\vb{n}_{(t)}), + U_{L}(\vec{n}_{(t+1)})\, + U_{L}^{\dagger}(\vec{n}_{(t)}), \\ \tcL_{(t,\, t+1)} & = & - U_{L}(\vb{n}_{(\bart)})\, - U_{L}^{\dagger}(\vb{n}_{(t)})\, - U_{L}(\vb{n}_{(t+1)})\, - U_{L}^{\dagger}(\vb{n}_{(\bart)}), + U_{L}(\vec{n}_{(\bart)})\, + U_{L}^{\dagger}(\vec{n}_{(t)})\, + U_{L}(\vec{n}_{(t+1)})\, + U_{L}^{\dagger}(\vec{n}_{(\bart)}), \\ \cR_{(t,\, t+1)} & = & - U_{R}(\vb{m}_{(t+1)})\, - U_{R}^{\dagger}(\vb{m}_{(t)}), + U_{R}(\vec{m}_{(t+1)})\, + U_{R}^{\dagger}(\vec{m}_{(t)}), \\ \tcR_{(t,\, t+1)} & = & - U_{R}(\vb{m}_{(\bart)})\, - U_{R}^{\dagger}(\vb{m}_{(t)})\, - U_{R}(\vb{m}_{(t+1)})\, - U_{R}^{\dagger}(\vb{m}_{(\bart)}). + U_{R}(\vec{m}_{(\bart)})\, + U_{R}^{\dagger}(\vec{m}_{(t)})\, + U_{R}(\vec{m}_{(t+1)})\, + U_{R}^{\dagger}(\vec{m}_{(\bart)}). \end{eqnarray} In spinor representation the action~\eqref{eq:string_action} becomes @@ -529,9 +529,9 @@ In spinor representation the action~\eqref{eq:string_action} becomes \iint\limits_{\C} \dd{z} \dd{\barz}\, \tr( - U_{L}(\vb{n}_{(\bart)})\, + U_{L}(\vec{n}_{(\bart)})\, \ipd{z} \cX_{(s)}(z, \barz)\, - U_{R}^{\dagger}(\vb{m}_{(\bart)})\, + U_{R}^{\dagger}(\vec{m}_{(\bart)})\, \ipd{\barz} \cX_{(s)}^{\dagger}(z, \barz) ). \end{split} @@ -548,30 +548,30 @@ 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)})\, + U_{L}(\vec{n}_{(t+1)})\, + U_{L}^{\dagger}(\vec{n}_{(t)})\, = - -\vb{v}_{(t+1)} \cdot \vb{v}_{(t)} + -\vec{v}_{(t+1)} \cdot \vec{v}_{(t)} + - i\, (\vb{v}_{(t+1)} \times \vb{v}_{(t)}) \cdot \vb{\sigma} , + i\, (\vec{v}_{(t+1)} \times \vec{v}_{(t)}) \cdot \vec{\sigma} , \end{equation} -with $\norm{\vb{v}_{(t)}}^2 = 1$. +with $\norm{\vec{v}_{(t)}}^2 = 1$. 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. +Hence the corresponding $\SU{2}_L \times \SU{2}_R$ element $(U_{L}(\vec{n}_{(t)}),\, U_{R}(\vec{m}_{(t)}))$ reflects such characteristics. In particular for the left part we have \begin{equation} - U_{L}(\vb{n}_{(t)}) + U_{L}(\vec{n}_{(t)}) = - i\, \vb{v}_{(t)} \cdot \vb{\sigma}, + i\, \vec{v}_{(t)} \cdot \vec{\sigma}, \qquad - \norm{\vb{v}_{(t)}}^2 = 1, + \norm{\vec{v}_{(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}. +Then any matrix $U_{L}(\vec{n}_{(t)})$ is of the form $U_{L}(\vec{n}_{(t)}) = i\, U(\vec{r}_{(t)}) \cdot \sigma_1 \cdot U^\dagger(\vec{r}_{(t)})$, for some $\vec{r}_{(t)}$ 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. @@ -628,7 +628,7 @@ The function \hyp{a}{b}{c}{z} is well defined for any value of its parameters.\f } We define a vector of independent hypergeometric functions: \begin{equation} - B_{\vb{0}}(z) + B_{0}(z) = \mqty( \hyp{a}{b}{c}{z} @@ -645,13 +645,13 @@ The corresponding product of the monodromy matrices~\eqref{eq:homotopy_rep} is t 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 $\bccH$). The triviality property is realised through: \begin{equation} - \cM_{\vb{0}}^+\, - \cM_{\vb{1}}^+\, - \cM_{\vb{\infty}}^+ + \cM_{0}^+\, + \cM_{1}^+\, + \cM_{\infty}^+ = - \cM_{\vb{\infty}}^-\, - \cM_{\vb{1}}^-\, - \cM_{\vb{0}}^- + \cM_{\infty}^-\, + \cM_{1}^-\, + \cM_{0}^- = \1_2 \label{eq:monodromy_relations} @@ -659,30 +659,30 @@ The triviality property is realised through: The monodromy matrix $\omega_{\bart+1} = 1$ can thus be recovered as a product of monodromies around $0$ and $\infty$ given the properties \begin{equation} \begin{split} - \cM_{\vb{0}}^+ + \cM_{0}^+ & = - \cM_{\vb{0}}^- + \cM_{0}^- = - \cM_{\vb{0}}, + \cM_{0}, \\ - \cM_{\vb{\infty}}^+ + \cM_{\infty}^+ & = - \cM_{\vb{\infty}}^- + \cM_{\infty}^- = - \cM_{\vb{\infty}}, + \cM_{\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 $\cM_{\vb{0}}$: +Using the basis in $z = 0$~\eqref{eq:basis_0} it is straightforward to find the explicit representation $\rM_{0}$ of the abstract monodromy $\cM_{0}$: \begin{equation} - \rM_{\vb{0}}( c ) = \mqty( \dmat{1, e^{-2\pi i c}} ). + \rM_{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 $\trM_{\vb{\infty}}$ of the abstract monodromy $\cM_{\vb{\infty}}$ in the basis of hypergeometric functions around $z = \infty$: +The computation of the monodromy matrix $\rM_{\infty}$ representing the monodromy in $\omega_z =\infty$ in the basis \eqref{eq:basis_0} requires to first compute the monodromy representation $\trM_{\infty}$ of the abstract monodromy $\cM_{\infty}$ in the basis of hypergeometric functions around $z = \infty$: \begin{equation} - B_{\vb{\infty}}(z) + B_{\infty}(z) = \mqty( (-z)^{-a}~\hyp{a}{a+1-c}{a+1-b}{z^{-1}} @@ -706,19 +706,19 @@ This basis is connected to~\eqref{eq:basis_0} through the transition matrix ), \label{eq:transition_matrix} \end{equation} -as $B_{\vb{0}}(z) = \cC(a,\, b,\, c)~B_{\vb{\infty}}(z)$. +as $B_{0}(z) = \cC(a,\, b,\, c)~B_{\infty}(z)$. Through the loop $z \mapsto z e^{-2\pi i}$ we find: \begin{equation} - \trM_{\vb{\infty}}( a,\, b ) + \trM_{\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}} + \rM_{\infty} = \cC(a,\, b,\, c)\, - \trM_{\vb{\infty}}(a,\, b)\, + \trM_{\infty}(a,\, b)\, \cC^{-1}(a,\, b,\, c). \label{eq:monodromy_infty} \end{equation} @@ -748,18 +748,18 @@ We write any possible solution in a factorised form as = (-\omega_z)^{A_{lr}}\, (1-\omega_z)^{B_{lr}}\, - \cB_{\vb{0},\, l}^{(L)}(\omega_z) - \qty( \cB_{\vb{0},\, r}^{(R)}(\omega_z) )^T, + \cB_{0,\, l}^{(L)}(\omega_z) + \qty( \cB_{0,\, r}^{(R)}(\omega_z) )^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) + \cB_{0,\, l}^{(L)}(\omega_z) & = D^{(L)}_l~ - B_{\vb{0},\,l}^{(L)}(\omega_z) + B_{0,\,l}^{(L)}(\omega_z) \\ & = \mqty( 1 & 0 \\ 0 & K_l^{(L)} )\, @@ -789,24 +789,24 @@ We impose: \begin{eqnarray} &&\begin{cases} D^{(L)}\, - \rM_{\vb{0}}^{(L)}\, + \rM_{0}^{(L)}\, \qty( D^{(L)} )^{-1} = - e^{-2\pi i \delta_{\vb{0}}^{(L)}}\, - \cL(\vb{n}_{\vb{0}}) + e^{-2\pi i \delta_{0}^{(L)}}\, + \cL(\vec{n}_{0}) \\ D^{(R)}\, - \rM_{\vb{0}}^{(R)}\, + \rM_{0}^{(R)}\, \qty( D^{(R)} )^{-1} = - e^{-2\pi i \delta_{\vb{0}}^{(R)}}\, - \cR^*(\vb{m}_{\vb{0}}) + e^{-2\pi i \delta_{0}^{(R)}}\, + \cR^*(\vec{m}_{0}) = - e^{-2\pi i \delta_{\vb{0}}^{(R)}}\, - \cR(\widetilde{\vb{m}}_{\vb{0}}) + e^{-2\pi i \delta_{0}^{(R)}}\, + \cR(\widetilde{\vec{m}}_{0}) \\ - e^{2\pi i ( A_{lr} - \delta_{\vb{0}}^{(L)} - - \delta_{\vb{0}}^{(R)} )} + e^{2\pi i ( A_{lr} - \delta_{0}^{(L)} - + \delta_{0}^{(R)} )} = 1 \end{cases}, @@ -814,24 +814,24 @@ We impose: \\ &&\begin{cases} D^{(L)}, - \rM_{\vb{\infty}}^{(L)}\, + \rM_{\infty}^{(L)}\, \qty( D^{(L)} )^{-1} = - e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\, - \cL(\vb{n}_{\vb{\infty}}) + e^{-2\pi i \delta_{\infty}^{(L)}}\, + \cL(\vec{n}_{\infty}) \\ D^{(R)}\, - \rM_{\vb{\infty}}^{(R)}\, + \rM_{\infty}^{(R)}\, \qty( D^{(R)} )^{-1} = - e^{-2\pi i \delta_{\vb{\infty}}^{(R)}}\, - \cR^*(\vb{m}_{\vb{\infty}}) + e^{-2\pi i \delta_{\infty}^{(R)}}\, + \cR^*(\vec{m}_{\infty}) = - e^{-2\pi i \delta_{\vb{\infty}}^{(R)}}\, - \cR(\widetilde{\vb{m}}_{\vb{\infty}}) + e^{-2\pi i \delta_{\infty}^{(R)}}\, + \cR(\widetilde{\vec{m}}_{\infty}) \\ - e^{2\pi i ( A_{lr} + B_{lr} - \delta_{\vb{\infty}}^{(L)} - - \delta_{\vb{\infty}}^{(R)} )} + e^{2\pi i ( A_{lr} + B_{lr} - \delta_{\infty}^{(L)} - + \delta_{\infty}^{(R)} )} = 1 \end{cases}, @@ -839,75 +839,75 @@ We impose: \end{eqnarray} where we defined \begin{eqnarray} - \cL(\vb{n}_{\vb{0}}) + \cL(\vec{n}_{0}) & = & \cL_{(\bart-1,\,\bart)} = - U_L(\vb{n}_{(\bart)})\, - U_L^{\dagger}(\vb{n}_{(\bart-1)}), + U_L(\vec{n}_{(\bart)})\, + U_L^{\dagger}(\vec{n}_{(\bart-1)}), \\ - \cL(\vb{n}_{\vb{\infty}}) + \cL(\vec{n}_{\infty}) & = & \cL_{(\bart,\, \bart+1)} = - U_L(\vb{n}_{(\bart+1)}) - U_L^{\dagger}(\vb{n}_{(\bart)}), + U_L(\vec{n}_{(\bart+1)}) + U_L^{\dagger}(\vec{n}_{(\bart)}), \\ - \cR(\vb{m}_{\vb{0}}) + \cR(\vec{m}_{0}) & = & \cR_{(\bart-1,\, \bart)} = - U_R(\vb{n}_{(\bart)}) - U_R^{\dagger}(\vb{n}_{(\bart-1)}), + U_R(\vec{n}_{(\bart)}) + U_R^{\dagger}(\vec{n}_{(\bart-1)}), \\ - \cR(\vb{m}_{\vb{\infty}}) + \cR(\vec{m}_{\infty}) & = & \cR_{(\bart,\, \bart+1)} = - U_R(\vb{n}_{(\bart+1)}) - U_R^{\dagger}(\vb{n}_{(\bart)}). + U_R(\vec{n}_{(\bart+1)}) + U_R^{\dagger}(\vec{n}_{(\bart)}). \end{eqnarray} -The range of $\delta_{\vb{0}}^{(L)}$ is +The range of $\delta_{0}^{(L)}$ is \begin{equation} - \alpha \le \delta_{\vb{0}}^{(L)} \le \alpha + \frac{1}{2}, + \alpha \le \delta_{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}. +This is a consequence of the fact that $e^{- 4 \pi i \delta_{0}^{(L)}}$ 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 $\delta_{\vb{\infty}}^{(L,\,R)}$. +The same considerations hold true for all the other additional parameters $\delta_{0}^{(R)}$ and $\delta_{\infty}^{(L,\,R)}$. Since we are interested in relative rotations of the D-branes, we choose the -rotation in $\omega_{\bart-1} = 0$ 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 $\vb{m}_{\vb{0}}$. +rotation in $\omega_{\bart-1} = 0$ 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 $\vec{n}_{0}$ and $\vec{m}_{0}$. In particular we set: \begin{eqnarray} - \vb{n}_{\vb{0}} + \vec{n}_{0} = - ( 0,\, 0,\, n_{\vb{0}}^3 ) \in \R^3, + ( 0,\, 0,\, n_{0}^3 ) \in \R^3, & \qquad & - 0 < n_{\vb{0}}^3 < \frac{1}{2}, + 0 < n_{0}^3 < \frac{1}{2}, \label{eq:maximal_torus_left} \\ - \widetilde{\vb{m}}_{\vb{0}} + \widetilde{\vec{m}}_{0} = - ( 0,\, 0,\, -m_{\vb{0}}^3 ) \in \R^3, + ( 0,\, 0,\, -m_{0}^3 ) \in \R^3, & \qquad & - 0 < m_{\vb{0}}^3 < \frac{1}{2}, + 0 < m_{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. +where $n_{0}^3 = 0$ is excluded to avoid considering a trivial rotation. We then define the parameters of the rotation in $\omega_{\bart} = \infty$ to be the most general \begin{equation} \begin{split} - \vb{n}_{\vb{\infty}} + \vec{n}_{\infty} & = - ( n_{\vb{\infty}}^1,\, n_{\vb{\infty}}^2,\, n_{\vb{\infty}}^3 ), + ( n_{\infty}^1,\, n_{\infty}^2,\, n_{\infty}^3 ), \\ - \widetilde{\vb{m}}_{\vb{\infty}} + \widetilde{\vec{m}}_{\infty} & = - ( -m_{\vb{\infty}}^1,\, m_{\vb{\infty}}^2,\, -m_{\vb{\infty}}^3 ), + ( -m_{\infty}^1,\, m_{\infty}^2,\, -m_{\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 could actually set $n_{\infty}^2 = m_{\infty}^2 = 0$ since the choice of the ``gauge''~\eqref{eq:maximal_torus_left} and~\eqref{eq:maximal_torus_right} is preserved by \U{1} rotations mixing $n_{\infty}^1$ and $n_{\infty}^2$. 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. @@ -915,11 +915,11 @@ We find: \begin{eqnarray} a_l^{(L)} = - n_{\vb{0}} + n_{0} + - (-1)^{f^{(L)}}\, n_{\vb{1}} + (-1)^{f^{(L)}}\, n_{1} + - n_{\vb{\infty}} + n_{\infty} + \ffa^{(L)}_l, & \qquad & @@ -927,11 +927,11 @@ We find: \\ b_l^{(L)} = - n_{\vb{0}} + n_{0} + - (-1)^{f^{(L)}}\, n_{\vb{1}} + (-1)^{f^{(L)}}\, n_{1} - - n_{\vb{\infty}} + n_{\infty} + \ffb^{(L)}_l, & \qquad & @@ -939,58 +939,58 @@ We find: \\ c_l^{(L)} = - 2\, n_{\vb{0}} + 2\, n_{0} + \ffc^{(L)}_l, & \qquad & \ffc^{(L)}_l \in \Z, \\ - \delta_{\vb{0}}^{(L)} + \delta_{0}^{(L)} = - n_{\vb{0}}, + n_{0}, \\ - \delta_{\vb{\infty}}^{(L)} + \delta_{\infty}^{(L)} = - - n_{\vb{0}} + n_{0} - - (-1)^{f^{(L)}}\, n_{\vb{1}}, + (-1)^{f^{(L)}}\, n_{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}}}, + \frac{n^1_{\infty}+ i\, n^2_{\infty}}{n_{\infty}}, \label{eq:K_factor_value} \end{eqnarray} where $f^{(L)} \in \qty{ 0,\, 1 }$. 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 $\omega_{\bart+1} = 1$. -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}: +We also introduced the norm $n_{1} = \norm{\vec{n}_{1}}$ of the rotation vector around $\omega_{\bart+1} = 1$. +Its dependence on the other parameters is encoded in~\eqref{eq:monodromy_relations}, where $\rM^+_{1} = \rM^{-1}_{0}\, \rM^{-1}_{\infty}$, and the composition rule~\eqref{eq:product_in_SU2}: \begin{equation} - \cos(2\pi n_{\vb{1}}) + \cos(2\pi n_{1}) = - \cos(2\pi n_{\vb{0}})\, - \cos(2\pi n_{\vb{\infty}}) + \cos(2\pi n_{0})\, + \cos(2\pi n_{\infty}) - - \sin(2\pi n_{\vb{0}})\, - \sin(2\pi n_{\vb{\infty}})\, - \frac{n_{\vb{\infty}}^3}{n_{\vb{\infty}}}. + \sin(2\pi n_{0})\, + \sin(2\pi n_{\infty})\, + \frac{n_{\infty}^3}{n_{\infty}}. \label{eq:dependent_monodromy_main_text} \end{equation} -Relations for the right sector follow under the interchange of $(L)$ with $(R)$ and $\vb{n} \leftrightarrow \vb{m}$. +Relations for the right sector follow under the interchange of $(L)$ with $(R)$ and $\vec{n} \leftrightarrow \vec{m}$. 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}, + n_{0} + m_{0} + \ffA_{lr}, & \qquad & \ffA_{lr} \in \Z, \\ B_{lr} - (-1)^{f^{(L)}}\, n_{\vb{1}} + (-1)^{f^{(R)}}\, m_{\vb{1}} + \ffB_{lr}, + (-1)^{f^{(L)}}\, n_{1} + (-1)^{f^{(R)}}\, m_{1} + \ffB_{lr}, & \qquad & \ffB_{lr} \in \Z. \end{eqnarray} @@ -1055,11 +1055,11 @@ We require: \end{eqnarray} The relations between the parameters of the hypergeometric functions and the monodromies associated to the rotation of the intersecting D-brane are more general than needed. -The number of parameters necessary to fix the configuration is $6$, that is the amount of parameters to uniquely determine $n_{\vb{0}}^3$, $n_{\vb{\infty}}^1$, $n_{\vb{\infty}}^3$ and $m_{\vb{0}}^3$, $m_{\vb{\infty}}^1$, $m_{\vb{\infty}}^3$. -As noticed before we can in fact fix $n_{\vb{\infty}}^2 = m_{\vb{\infty}}^2 = 0$. +The number of parameters necessary to fix the configuration is $6$, that is the amount of parameters to uniquely determine $n_{0}^3$, $n_{\infty}^1$, $n_{\infty}^3$ and $m_{0}^3$, $m_{\infty}^1$, $m_{\infty}^3$. +As noticed before we can in fact fix $n_{\infty}^2 = m_{\infty}^2 = 0$. This is a consequence of the fact that all parameters depend on the norm of the rotation vectors exception made for $K^{(L)}$ and $K^{(R)}$. -They depend on $n_{\vb{\infty}}^1 + i n_{\vb{\infty}}^2$ and $m_{\vb{\infty}}^1 + i m_{\vb{\infty}}^2$. -Performing a $\SU{2}_L$ and $\SU{2}_R$ rotation around the third axis and a shift of the parameters $\delta_{\vb{\infty}}$, the phases of the normalisation factors $K$ can vanish. +They depend on $n_{\infty}^1 + i n_{\infty}^2$ and $m_{\infty}^1 + i m_{\infty}^2$. +Performing a $\SU{2}_L$ and $\SU{2}_R$ rotation around the third axis and a shift of the parameters $\delta_{\infty}$, the phases of the normalisation factors $K$ can vanish. \subsubsection{The Importance of the Normalization Factors} @@ -1082,19 +1082,19 @@ Using the \rP symbol the solutions can be symbolically written as \infty & \\ - n_{\vb{0}} + n_{0} & - n_{\vb{1}} + n_{1} & - n_{\vb{\infty}} + \ffa^{(L)} + n_{\infty} + \ffa^{(L)} & \omega \\ - -n_{\vb{0}} + 1 - \ffc^{(L)} + -n_{0} + 1 - \ffc^{(L)} & - -n_{\vb{1}} - \ffa^{(L)} - \ffb^{(L)} + \ffc^{(L)} + -n_{1} - \ffa^{(L)} - \ffb^{(L)} + \ffc^{(L)} & - -n_{\vb{\infty}} + \ffb^{(L)} + -n_{\infty} + \ffb^{(L)} & } } @@ -1110,19 +1110,19 @@ Using the \rP symbol the solutions can be symbolically written as \infty & \\ - m_{\vb{0}} + m_{0} & - m_{\vb{1}} + m_{1} & - m_{\vb{\infty}} + \ffa^{(R)} + m_{\infty} + \ffa^{(R)} & \omega \\ - -m_{\vb{0}} + 1 - \ffc^{(R)} + -m_{0} + 1 - \ffc^{(R)} & - -m_{\vb{1}} - \ffa^{(R)} - \ffb^{(R)} + \ffc^{(R)} + -m_{1} - \ffa^{(R)} - \ffb^{(R)} + \ffc^{(R)} & - -m_{\vb{\infty}} + \ffb^{(R)} + -m_{\infty} + \ffb^{(R)} & } }. @@ -1163,7 +1163,7 @@ Similarly we use a shorthand notation for the basis of the hypergeometric functi See for instance~\eqref{eq:K_factor_value} and~\eqref{eq:n12+n22}. } \begin{equation} - \cB_{\vb{0}}(a,\, b,\, c;\, z) + \cB_{0}(a,\, b,\, c;\, z) = \mqty( \hyp{a}{b}{c}{z} @@ -1190,29 +1190,29 @@ We can then algorithmically apply the following relations \end{equation} to eliminate unwanted integer factors and keep only \hyp{a}{b}{c}{z} and any of its contiguous functions. -Notice that $\cB_{\vb{0}}$ is a basis element of the possible solutions of the classical and quantum string \eom -Using any relation in~\eqref{eq:contiguous_functions} we can change $a$, $b$ or $c$ by one unit coherently in both hypergeometric functions contained in $\cB_{\vb{0}}$. +Notice that $\cB_{0}$ is a basis element of the possible solutions of the classical and quantum string \eom +Using any relation in~\eqref{eq:contiguous_functions} we can change $a$, $b$ or $c$ by one unit coherently in both hypergeometric functions contained in $\cB_{0}$. For example from the first equation in~\eqref{eq:contiguous_functions} we expect: \begin{equation} - (c-a)\, \cB_{\vb{0}}(a-1) + (c-a)\, \cB_{0}(a-1) + - (2a-c+(b-a)z)\, \cB_{\vb{0}} + (2a-c+(b-a)z)\, \cB_{0} - - a(1-z)\, \cB_{\vb{0}}(a+1) + a(1-z)\, \cB_{0}(a+1) = 0, \end{equation} which can be used to lower and rise the integer factors in $a$. The relation holds only because the normalisation factor $K$ is present. -In fact coefficients in this equation equal those in the relation for the first component of $\cB_{\vb{0}}$. +In fact coefficients in this equation equal those in the relation for the first component of $\cB_{0}$. It is not trivial for the second component where the factor $K$ is key to the consistency. Similarly the relation needed to lower $c$ reads: \begin{equation} - (a-c)(b-c)\, \cB_{\vb{0}}(c+1) + (a-c)(b-c)\, \cB_{0}(c+1) + - (a+(b-c)z)\, \cB_{\vb{0}} + (a+(b-c)z)\, \cB_{0} - - a(1-z) \cB_{\vb{0}}(a+1) + a(1-z) \cB_{0}(a+1) = 0. \end{equation} @@ -1244,8 +1244,8 @@ We could then use~\eqref{eq:reduction_F_F+} to write the possible solution as \ipd{z} \cX(z) & = \pdv{\omega_z}{z}\, - (-\omega_z)^{n_{\vb{0}} + m_{\vb{0}}}\, - (1-\omega_z)^{n_{\vb{1}} + m_{\vb{1}}} + (-\omega_z)^{n_{0} + m_{0}}\, + (1-\omega_z)^{n_{1} + m_{1}} \\ & \times \sum\limits_{\ffa^{(L,\,R)} \in \qty{ -1, 0 }} @@ -1253,9 +1253,9 @@ We could then use~\eqref{eq:reduction_F_F+} to write the possible solution as \times \\ & \times - \cB_{\vb{0}}^{(L)}(a^{(L)} + \ffa^{(L)},\, b,\, c;\, \omega_z) + \cB_{0}^{(L)}(a^{(L)} + \ffa^{(L)},\, b,\, c;\, \omega_z) \qty( - \cB_{\vb{0}}^{(R)}(a^{(R)} + \ffa^{(R)},\, b,\, c;\, \omega_z) + \cB_{0}^{(R)}(a^{(R)} + \ffa^{(R)},\, b,\, c;\, \omega_z) )^T. \end{split} \label{eq:doubling_field_expansion} @@ -1284,12 +1284,12 @@ It can be verified that the convergence of the action both at finite and infinit \finiteint{u'}{x_{(\bart-1)}}{u} \ipd{u'} \cX_{(s)}(u') + - U_L^{\dagger}(\vb{n}_{{\bart}}) + U_L^{\dagger}(\vec{n}_{{\bart}}) \qty[ \finiteint{\baru'}{x_{(\bart-1)}}{\baru} \ipd{\baru'} \cX_{(s)}(\baru') ] - U_R(\vb{m}_{{\bart}}), + U_R(\vec{m}_{{\bart}}), \label{eq:classical_solution} \end{equation} which follows in spinor representation from~\eqref{eq:spinor_doubling_trick} and where $f_{(s),\, (\bart-1)} = f^I_{(\bart-1)}\, \tau_I$. @@ -1328,19 +1328,19 @@ In this case~\eqref{eq:symbolic_solutions_using_P} becomes \infty & \\ - n_{\vb{0}} + n_{0} & - n_{\vb{1}} + n_{1} & - n_{\vb{\infty}} + \ffa^{(L)} + n_{\infty} + \ffa^{(L)} & \omega \\ - -n_{\vb{0}} + 1 -\ffc^{(L)} + -n_{0} + 1 -\ffc^{(L)} & - -n_{\vb{1}} - \ffa^{(L)} - \ffb^{(L)} + \ffc^{(L)} + -n_{1} - \ffa^{(L)} - \ffb^{(L)} + \ffc^{(L)} & - -n_{\vb{\infty}} + \ffb^{(L)} + -n_{\infty} + \ffb^{(L)} & } }. @@ -1356,19 +1356,19 @@ The only possible solution compatible with~\eqref{eq:constraints_finite_X} is \infty & \\ - n_{\vb{0}} - 1 + n_{0} - 1 & - n_{\vb{1}} - 1 + n_{1} - 1 & - n_{\vb{\infty}} + 1 + n_{\infty} + 1 & \omega \\ - -n_{\vb{0}} + -n_{0} & - -n_{\vb{1}} + -n_{1} & - -n_{\vb{\infty}} + 2 + -n_{\infty} + 2 & } }, @@ -1376,10 +1376,10 @@ The only possible solution compatible with~\eqref{eq:constraints_finite_X} is \end{equation} that is $\ffa^{(L)} = -1$, $\ffb^{(L)} = 0$, $\ffc^{(L)} = 0$, $\ffA = -1$ and $\ffB = -1$. -In the general the solution is more complicated and it depends on the relation between the rotation vectors $\vb{n}_{\vb{0},\, \vb{1},\, \vb{\infty}}$, $\vb{m}_{\vb{0},\, \vb{1},\, \vb{\infty}}$. +In the general the solution is more complicated and it depends on the relation between the rotation vectors $\vec{n}_{0,\, 1,\, \infty}$, $\vec{m}_{0,\, 1,\, \infty}$. For each possible case the solution is however unique and it is given by \begin{enumerate} - \item $n_{\vb{0}} > m_{\vb{0}}$ and $n_{\vb{1}} > m_{\vb{1}}$: + \item $n_{0} > m_{0}$ and $n_{1} > m_{1}$: \begin{equation} \rP\qty{ \mqty{ @@ -1390,19 +1390,19 @@ For each possible case the solution is however unique and it is given by \infty & \\ - n_{\vb{0}} - 1 + n_{0} - 1 & - n_{\vb{1}} - 1 + n_{1} - 1 & - n_{\vb{\infty}} + 1 + n_{\infty} + 1 & \omega \\ - -n_{\vb{0}} + -n_{0} & - -n_{\vb{1}} + -n_{1} & - -n_{\vb{\infty}} + 2 + -n_{\infty} + 2 & } } @@ -1415,26 +1415,26 @@ For each possible case the solution is however unique and it is given by \infty & \\ - m_{\vb{0}} + m_{0} & - m_{\vb{1}} + m_{1} & - m_{\vb{\infty}} + m_{\infty} & \omega \\ - -m_{\vb{0}} + 1 + -m_{0} + 1 & - -m_{\vb{1}} + -m_{1} & - -m_{\vb{\infty}} + 1 + -m_{\infty} + 1 & } }, \label{eq:X_solution>>} \end{equation} - \item $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} < m_{\vb{1}}$ and $n_{\vb{\infty}} > m_{\vb{\infty}}$: + \item $n_{0} > m_{0}$, $n_{1} < m_{1}$ and $n_{\infty} > m_{\infty}$: \begin{equation} \rP\qty{ \mqty{ @@ -1445,19 +1445,19 @@ For each possible case the solution is however unique and it is given by \infty & \\ - n_{\vb{0}} - 1 + n_{0} - 1 & - n_{\vb{1}} + n_{1} & - n_{\vb{\infty}} + n_{\infty} & \omega \\ - -n_{\vb{0}} + -n_{0} & - -n_{\vb{1}} + -n_{1} & - -n_{\vb{\infty}} + 2 + -n_{\infty} + 2 & } } @@ -1470,26 +1470,26 @@ For each possible case the solution is however unique and it is given by \infty & \\ - m_{\vb{0}} + m_{0} & - m_{\vb{1}} - 1 + m_{1} - 1 & - m_{\vb{\infty}} + 1 + m_{\infty} + 1 & \omega \\ - -m_{\vb{0}} + -m_{0} & - -m_{\vb{1}} + -m_{1} & - -m_{\vb{\infty}} + 1 + -m_{\infty} + 1 & } }, \label{eq:X_solution><>} \end{equation} - \item $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} < m_{\vb{1}}$ and $n_{\vb{\infty}} < m_{\vb{\infty}}$: + \item $n_{0} > m_{0}$, $n_{1} < m_{1}$ and $n_{\infty} < m_{\infty}$: \begin{equation} \rP\qty{ \mqty{ @@ -1500,19 +1500,19 @@ For each possible case the solution is however unique and it is given by \infty & \\ - n_{\vb{0}} - 1 + n_{0} - 1 & - n_{\vb{1}} + n_{1} & - n_{\vb{\infty}} + 1 + n_{\infty} + 1 & \omega \\ - -n_{\vb{0}} + -n_{0} & - -n_{\vb{1}} + -n_{1} & - -n_{\vb{\infty}} + 1 + -n_{\infty} + 1 & } } @@ -1525,26 +1525,26 @@ For each possible case the solution is however unique and it is given by \infty & \\ - m_{\vb{0}} + m_{0} & - m_{\vb{1}} - 1 + m_{1} - 1 & - m_{\vb{\infty}} + m_{\infty} & \omega \\ - -m_{\vb{0}} + -m_{0} & - -m_{\vb{1}} + -m_{1} & - -m_{\vb{\infty}} + 2 + -m_{\infty} + 2 & } }, \label{eq:X_solution><<} \end{equation} - \item $n_{\vb{0}} < m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} > m_{\vb{\infty}}$: + \item $n_{0} < m_{0}$, $n_{1} > m_{1}$ and $n_{\infty} > m_{\infty}$: \begin{equation} \rP\qty{ \mqty{ @@ -1555,19 +1555,19 @@ For each possible case the solution is however unique and it is given by \infty & \\ - n_{\vb{0}} + n_{0} & - n_{\vb{1}} - 1 + n_{1} - 1 & - n_{\vb{\infty}} + n_{\infty} & \omega \\ - -n_{\vb{0}} + -n_{0} & - -n_{\vb{1}} + -n_{1} & - -n_{\vb{\infty}} + 2 + -n_{\infty} + 2 & } } @@ -1580,26 +1580,26 @@ For each possible case the solution is however unique and it is given by \infty & \\ - m_{\vb{0}} - 1 + m_{0} - 1 & - m_{\vb{1}} + m_{1} & - m_{\vb{\infty}} + 1 + m_{\infty} + 1 & \omega \\ - -m_{\vb{0}} + -m_{0} & - -m_{\vb{1}} + -m_{1} & - -m_{\vb{\infty}} + 1 + -m_{\infty} + 1 & } }, \label{eq:X_solution<>>} \end{equation} - \item $n_{\vb{0}} < m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} < m_{\vb{\infty}}$: + \item $n_{0} < m_{0}$, $n_{1} > m_{1}$ and $n_{\infty} < m_{\infty}$: \begin{equation} \rP\qty{ \mqty{ @@ -1610,19 +1610,19 @@ For each possible case the solution is however unique and it is given by \infty & \\ - n_{\vb{0}} + n_{0} & - n_{\vb{1}} - 1 + n_{1} - 1 & - n_{\vb{\infty}} + 1 + n_{\infty} + 1 & \omega \\ - -n_{\vb{0}} + -n_{0} & - -n_{\vb{1}} + -n_{1} & - -n_{\vb{\infty}} + 1 + -n_{\infty} + 1 & } } @@ -1635,26 +1635,26 @@ For each possible case the solution is however unique and it is given by \infty & \\ - m_{\vb{0}} - 1 + m_{0} - 1 & - m_{\vb{1}} + m_{1} & - m_{\vb{\infty}} + m_{\infty} & \omega \\ - -m_{\vb{0}} + -m_{0} & - -m_{\vb{1}} + -m_{1} & - -m_{\vb{\infty}} + 2 + -m_{\infty} + 2 & } }, \label{eq:X_solution<><} \end{equation} - \item $n_{\vb{0}} < m_{\vb{0}}$, $n_{\vb{1}} < m_{\vb{1}}$: + \item $n_{0} < m_{0}$, $n_{1} < m_{1}$: \begin{equation} \rP\qty{ \mqty{ @@ -1665,19 +1665,19 @@ For each possible case the solution is however unique and it is given by \infty & \\ - n_{\vb{0}} + n_{0} & - n_{\vb{1}} + n_{1} & - n_{\vb{\infty}} + n_{\infty} & \omega \\ - -n_{\vb{0}} + -n_{0} & - -n_{\vb{1}} + -n_{1} & - -n_{\vb{\infty}} + 1 + -n_{\infty} + 1 & } } @@ -1690,19 +1690,19 @@ For each possible case the solution is however unique and it is given by \infty & \\ - m_{\vb{0}} - 1 + m_{0} - 1 & - m_{\vb{1}} - 1 + m_{1} - 1 & - m_{\vb{\infty}} + 1 + m_{\infty} + 1 & \omega \\ - -m_{\vb{0}} + -m_{0} & - -m_{\vb{1}} + -m_{1} & - -m_{\vb{\infty}} + 2 + -m_{\infty} + 2 & } }. @@ -1717,12 +1717,12 @@ The parameters associated to this list of solutions are summarised in~\Cref{tab: \toprule & & & \ffA & \ffB & $\ffa^{(L)}$ & $\ffb^{(L)}$ & $\ffc^{(L)}$ & $\ffa^{(R)}$ & $\ffb^{(R)}$ & $\ffc^{(R)}$ \\ \midrule - $n_{\vb{0}} > m_{\vb{0}}$ & $n_{\vb{1}} > m_{\vb{1}}$ & $n_{\vb{\infty}} \lessgtr m_{\vb{\infty}}$ & -1 & -1 & -1 & 0 & 0 & 0 & +1 & +1 \\ - $n_{\vb{0}} > m_{\vb{0}}$ & $n_{\vb{1}} < m_{\vb{1}}$ & $n_{\vb{\infty}} > m_{\vb{\infty}}$ & -1 & -1 & -1 & +1 & 0 & 0 & 0 & +1 \\ - $n_{\vb{0}} > m_{\vb{0}}$ & $n_{\vb{1}} < m_{\vb{1}}$ & $n_{\vb{\infty}} < m_{\vb{\infty}}$ & -1 & -1 & 0 & 0 & 0 & -1 & +1 & +1 \\ - $n_{\vb{0}} < m_{\vb{0}}$ & $n_{\vb{1}} > m_{\vb{1}}$ & $n_{\vb{\infty}} > m_{\vb{\infty}}$ & -1 & -1 & -1 & +1 & +1 & 0 & 0 & 0 \\ - $n_{\vb{0}} < m_{\vb{0}}$ & $n_{\vb{1}} > m_{\vb{1}}$ & $n_{\vb{\infty}} < m_{\vb{\infty}}$ & -1 & -1 & 0 & 0 & +1 & -1 & +1 & 0 \\ - $n_{\vb{0}} < m_{\vb{0}}$ & $n_{\vb{1}} < m_{\vb{1}}$ & $n_{\vb{\infty}} \lessgtr m_{\vb{\infty}}$ & -1 & -1 & 0 & +1 & +1 & -1 & 0 & 0 \\ + $n_{0} > m_{0}$ & $n_{1} > m_{1}$ & $n_{\infty} \lessgtr m_{\infty}$ & -1 & -1 & -1 & 0 & 0 & 0 & +1 & +1 \\ + $n_{0} > m_{0}$ & $n_{1} < m_{1}$ & $n_{\infty} > m_{\infty}$ & -1 & -1 & -1 & +1 & 0 & 0 & 0 & +1 \\ + $n_{0} > m_{0}$ & $n_{1} < m_{1}$ & $n_{\infty} < m_{\infty}$ & -1 & -1 & 0 & 0 & 0 & -1 & +1 & +1 \\ + $n_{0} < m_{0}$ & $n_{1} > m_{1}$ & $n_{\infty} > m_{\infty}$ & -1 & -1 & -1 & +1 & +1 & 0 & 0 & 0 \\ + $n_{0} < m_{0}$ & $n_{1} > m_{1}$ & $n_{\infty} < m_{\infty}$ & -1 & -1 & 0 & 0 & +1 & -1 & +1 & 0 \\ + $n_{0} < m_{0}$ & $n_{1} < m_{1}$ & $n_{\infty} \lessgtr m_{\infty}$ & -1 & -1 & 0 & +1 & +1 & -1 & 0 & 0 \\ \bottomrule \end{tabular} \caption{Integer shifts in the parameters of the hypergeometric function.} @@ -1735,38 +1735,38 @@ The parameters associated to this list of solutions are summarised in~\Cref{tab: In the previous section we produced one solution for each ordering of the $n_{\omega_z}$ with respect to $m_{\omega_z}$. There are however other solutions connected to the $\Z_2$ equivalence class in the isomorphism between \SO{4} its double cover. -Given a solution $(\vb{n}_{\vb{0}},\, \vb{n}_{\vb{1}},\, \vb{n}_{\vb{\infty}}) \oplus (\vb{m}_{\vb{0}},\, \vb{m}_{\vb{1}},\, \vb{m}_{\vb{\infty}})$, we can in fact replace any couple of $\vb{n}$ and $\vb{m}$ by $\widehat{\vb{n}}$ and $\widehat{\vb{m}}$ and get an apparently new solution.\footnotemark{} +Given a solution $(\vec{n}_{0},\, \vec{n}_{1},\, \vec{n}_{\infty}) \oplus (\vec{m}_{0},\, \vec{m}_{1},\, \vec{m}_{\infty})$, we can in fact replace any couple of $\vec{n}$ and $\vec{m}$ by $\widehat{\vec{n}}$ and $\widehat{\vec{m}}$ and get an apparently new solution.\footnotemark{} \footnotetext{% We need to change two rotation vectors because the monodromies are constrained by~\eqref{eq:monodromy_relations}. } -For instance we could consider $(\widehat{\vb{n}}_{\vb{0}},\, \widehat{\vb{n}}_{\vb{1}},\, \vb{n}_{\vb{\infty}}) \oplus (\vb{m}_{\vb{0}},\, \widehat{\vb{m}}_{\vb{1}},\, \widehat{\vb{m}}_{\vb{\infty}})$. +For instance we could consider $(\widehat{\vec{n}}_{0},\, \widehat{\vec{n}}_{1},\, \vec{n}_{\infty}) \oplus (\vec{m}_{0},\, \widehat{\vec{m}}_{1},\, \widehat{\vec{m}}_{\infty})$. On the other hand the previous substitution would change the \SO{4} in both $\omega = 0$ and $\omega = \infty$: it does not represent a new solution. We are left therefore with three possibilities besides the original one: \begin{equation} \begin{split} - (\widehat{\vb{n}}_{\vb{0}},\, \widehat{\vb{n}}_{\vb{1}},\, \vb{n}_{\vb{\infty}}) + (\widehat{\vec{n}}_{0},\, \widehat{\vec{n}}_{1},\, \vec{n}_{\infty}) & \oplus - (\widehat{\vb{m}}_{\vb{0}},\, \widehat{\vb{m}}_{\vb{1}},\, \vb{m}_{\vb{\infty}}), + (\widehat{\vec{m}}_{0},\, \widehat{\vec{m}}_{1},\, \vec{m}_{\infty}), \\ - (\widehat{\vb{n}}_{\vb{0}},\, \vb{n}_{\vb{1}},\, \widehat{\vb{n}}_{\vb{\infty}}) + (\widehat{\vec{n}}_{0},\, \vec{n}_{1},\, \widehat{\vec{n}}_{\infty}) & \oplus - (\vb{m}_{\vb{0}},\, \widehat{\vb{m}}_{\vb{1}},\, \widehat{\vb{m}}_{\vb{\infty}}), + (\vec{m}_{0},\, \widehat{\vec{m}}_{1},\, \widehat{\vec{m}}_{\infty}), \\ - (\widehat{\vb{n}}_{\vb{0}},\, \vb{n}_{\vb{1}},\, \widehat{\vb{n}}_{\vb{\infty}}) + (\widehat{\vec{n}}_{0},\, \vec{n}_{1},\, \widehat{\vec{n}}_{\infty}) & \oplus - (\vb{m}_{\vb{0}},\, \widehat{\vb{m}}_{\vb{1}},\, \widehat{\vb{m}}_{\vb{\infty}}). + (\vec{m}_{0},\, \widehat{\vec{m}}_{1},\, \widehat{\vec{m}}_{\infty}). \end{split} \end{equation} -We can gauge fix the $\Z_2$ choice by letting $\vb{n}_{\vb{0}}^3,\, \vb{m}_{\vb{0}}^3 > 0$ as required by~\eqref{eq:maximal_torus_left} and~\eqref{eq:maximal_torus_right}. +We can gauge fix the $\Z_2$ choice by letting $\vec{n}_{0}^3,\, \vec{m}_{0}^3 > 0$ as required by~\eqref{eq:maximal_torus_left} and~\eqref{eq:maximal_torus_right}. We are thus left with two possible solutions \begin{align} - (\vb{n}_{\vb{0}},\, \vb{n}_{\vb{1}},\, \vb{n}_{\vb{\infty}}) + (\vec{n}_{0},\, \vec{n}_{1},\, \vec{n}_{\infty}) & \oplus - (\vb{m}_{\vb{0}},\, \vb{m}_{\vb{1}},\, \vb{m}_{\vb{\infty}}), + (\vec{m}_{0},\, \vec{m}_{1},\, \vec{m}_{\infty}), \\ - (\vb{n}_{\vb{0}},\, \widehat{\vb{n}}_{\vb{1}},\, \widehat{\vb{n}}_{\vb{\infty}}) + (\vec{n}_{0},\, \widehat{\vec{n}}_{1},\, \widehat{\vec{n}}_{\infty}) & \oplus - (\vb{m}_{\vb{0}},\, \widehat{\vb{m}}_{\vb{1}},\, \widehat{\vb{m}}_{\vb{\infty}}). + (\vec{m}_{0},\, \widehat{\vec{m}}_{1},\, \widehat{\vec{m}}_{\infty}). \end{align} These are the original and a modified solution obtained by acting with a parity operator $P_2$ on the rotation parameters at $\omega = 1,\, \infty$ on both left and right sector at the same time. We then need to ensure its independence in order to accept it as a possible solution. @@ -1775,30 +1775,30 @@ As shown in~\Cref{tab:coeffs_k}, there are only two different cases up to left-r The first is \begin{equation} \Big\lbrace - (n_{\vb{0}} > m_{\vb{0}},\, n_{\vb{1}} > m_{\vb{1}},\, n_{\vb{\infty}} > m_{\vb{\infty}} ),~ - (n_{\vb{0}} > m_{\vb{0}},\, \hat{n}_{\vb{1}} < \hat{m}_{\vb{1}},\, \hat{n}_{\vb{\infty}} < \hat{m}_{\vb{\infty}}) + (n_{0} > m_{0},\, n_{1} > m_{1},\, n_{\infty} > m_{\infty} ),~ + (n_{0} > m_{0},\, \hat{n}_{1} < \hat{m}_{1},\, \hat{n}_{\infty} < \hat{m}_{\infty}) \Big\rbrace, \end{equation} which is mapped to \begin{equation} \Big\lbrace - (n_{\vb{0}} < m_{\vb{0}},\, n_{\vb{1}} < m_{\vb{1}},\, n_{\vb{\infty}} < m_{\vb{\infty}} ),~ - (n_{\vb{0}} < m_{\vb{0}},\, \hat{n}_{\vb{1}} > \hat{m}_{\vb{1}},\, \hat{n}_{\vb{\infty}} > \hat{m}_{\vb{\infty}}) + (n_{0} < m_{0},\, n_{1} < m_{1},\, n_{\infty} < m_{\infty} ),~ + (n_{0} < m_{0},\, \hat{n}_{1} > \hat{m}_{1},\, \hat{n}_{\infty} > \hat{m}_{\infty}) \Big\rbrace \end{equation} by the left-right symmetry. The second is \begin{equation} \Big\lbrace - (n_{\vb{0}} > m_{\vb{0}},\, n_{\vb{1}} > m_{\vb{1}},\, n_{\vb{\infty}} < m_{\vb{\infty}} ),~ - (n_{\vb{0}} > m_{\vb{0}},\, \hat{n}_{\vb{1}} < \hat{m}_{\vb{1}},\, \hat{n}_{\vb{\infty}} > \hat{m}_{\vb{\infty}}) + (n_{0} > m_{0},\, n_{1} > m_{1},\, n_{\infty} < m_{\infty} ),~ + (n_{0} > m_{0},\, \hat{n}_{1} < \hat{m}_{1},\, \hat{n}_{\infty} > \hat{m}_{\infty}) \Big\rbrace, \end{equation} which is mapped to \begin{align} \Big\lbrace - (n_{\vb{0}} < m_{\vb{0}},\, n_{\vb{1}} < m_{\vb{1}},\, n_{\vb{\infty}} > m_{\vb{\infty}} ),~ - (n_{\vb{0}} < m_{\vb{0}},\, \hat{n}_{\vb{1}} > \hat{m}_{\vb{1}},\, \hat{n}_{\vb{\infty}} < \hat{m}_{\vb{\infty}}) + (n_{0} < m_{0},\, n_{1} < m_{1},\, n_{\infty} > m_{\infty} ),~ + (n_{0} < m_{0},\, \hat{n}_{1} > \hat{m}_{1},\, \hat{n}_{\infty} < \hat{m}_{\infty}) \Big\rbrace \end{align} by the same symmetry. @@ -1808,19 +1808,19 @@ We first perform the computations common to both cases and then we explicitly sp 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)} + a^{(L)} & = n_{0} + n_{1} + n_{\infty} + \ffa^{(L)} \\ - b^{(L)} & = n_{\vb{0}} + n_{\vb{1}} - n_{\vb{\infty}} + \ffb^{(L)} + b^{(L)} & = n_{0} + n_{1} - n_{\infty} + \ffb^{(L)} \\ - c^{(L)} & = 2\, n_{\vb{0}} + \ffc^{(L)} + c^{(L)} & = 2\, n_{0} + \ffc^{(L)} \end{cases}, \qquad \begin{cases} - a^{(R)} & = m_{\vb{0}} + m_{\vb{1}} + m_{\vb{\infty}} + \ffa^{(R)} + a^{(R)} & = m_{0} + m_{1} + m_{\infty} + \ffa^{(R)} \\ - b^{(R)} & = m_{\vb{0}} + m_{\vb{1}} - m_{\vb{\infty}} + \ffb^{(R)} + b^{(R)} & = m_{0} + m_{1} - m_{\infty} + \ffb^{(R)} \\ - c^{(R)} & = 2\, m_{\vb{0}} + 1 + \ffc^{(R)} + c^{(R)} & = 2\, m_{0} + 1 + \ffc^{(R)} \end{cases}. \end{equation} The values of the constants are in \Cref{tab:coeffs_k}. @@ -1830,8 +1830,8 @@ The first solution reads: \begin{split} \ipd{\omega} \cX_1 & = - (-\omega)^{n_{\vb{0}} + m_{\vb{0}} - 1}\, - (1-\omega)^{n_{\vb{1}} + m_{\vb{1}} - 1}\, + (-\omega)^{n_{0} + m_{0} - 1}\, + (1-\omega)^{n_{1} + m_{1} - 1}\, \\ & \times \mqty( @@ -1859,19 +1859,19 @@ The parameters of the second solution read \begin{cases} \hat{a}^{(L)} & = - n_{\vb{0}} + \hat{n}_{\vb{1}} + \hat{n}_{\vb{\infty}} + \hat{\ffa}^{(L)} + n_{0} + \hat{n}_{1} + \hat{n}_{\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)} + n_{0} + \hat{n}_{1} - \hat{n}_{\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)} + 2\, n_{0} + \hat{\ffc}^{(L)} = c^{(L)} - \ffc^{(L)} + \hat{\ffc}^{(L)} \end{cases} @@ -1880,19 +1880,19 @@ The parameters of the second solution read \begin{cases} \hat{a}^{(R)} & = - m_{\vb{0}} + \hat{n}_{\vb{1}} + \hat{n}_{\vb{\infty}} + \hat{\ffa}^{(R)} + m_{0} + \hat{n}_{1} + \hat{n}_{\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)} + m_{0} + \hat{n}_{1} - \hat{n}_{\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)} + 2\, m_{0} + \hat{\ffc}^{(R)} = c^{(R)} - \ffc^{(R)} + \hat{\ffc}^{(R)} \end{cases} @@ -1903,8 +1903,8 @@ 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 $\hat{n}_{\vb{\infty}} < \hat{m}_{\vb{\infty}}$. +Consider $n_{0} > m_{0}$, $n_{1} > m_{1}$ and $n_{\infty} > m_{\infty}$. +The associated second solution is $n_{0} > m_{0}$, $\hat{n}_{1} < \hat{m}_{1}$ and $\hat{n}_{\infty} < \hat{m}_{\infty}$. Its parameters are: \begin{equation} \begin{cases} @@ -1938,8 +1938,8 @@ we can write the second solution as \begin{split} \ipd{\omega} \cX_2 & = - (-\omega)^{n_{\vb{0}} + m_{\vb{0}} - 1}\, - (1-\omega)^{n_{\vb{1}} + m_{\vb{1}}}\, + (-\omega)^{n_{0} + m_{0} - 1}\, + (1-\omega)^{n_{1} + m_{1}}\, \\ & \times \mqty( @@ -1963,8 +1963,8 @@ In this solution the left basis is exactly the same as in the first solution~\eq \paragraph{Case 2} -Consider now the second option $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} < m_{\vb{\infty}}$. -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: +Consider now the second option $n_{0} > m_{0}$, $n_{1} > m_{1}$ and $n_{\infty} < m_{\infty}$. +For the second solution we have $n_{0} > m_{0}$, $\hat{n}_{1} < \hat{m}_{1}$ and $\hat{n}_{\infty} > \hat{m}_{\infty}$ and the parameters are explicitly: \begin{equation} \begin{cases} \hat{a}^{(L)} & = c^{(L)} - a^{(L)} - 1 @@ -1998,8 +1998,8 @@ Using Euler relation we write the second solution for the second case as \begin{split} \ipd{\omega} \cX_2 & = - (-\omega)^{n_{\vb{0}} + m_{\vb{0}} - 1}\, - (1-\omega)^{n_{\vb{1}} + m_{\vb{1}}}\, + (-\omega)^{n_{0} + m_{0} - 1}\, + (1-\omega)^{n_{1} + m_{1}}\, \\ & \times \mqty( @@ -2040,11 +2040,11 @@ Explicitly we impose the four real equations in spinorial formalism \finiteint{\omega}{0}{1} \ipd{\omega} \cX(\omega) + - U_L^{\dagger}(\vb{n}_{{\bart}}) + U_L^{\dagger}(\vec{n}_{{\bart}}) \qty[ \finiteint{\bomega}{0}{1} \ipd{\bomega} \cX(\bomega) ] - U_R(\vb{m}_{{\bart}}) + U_R(\vec{m}_{{\bart}}) = f_{{\bart+1}\, (s)} - f_{{\bart-1}\, (s)}, \end{equation} @@ -2064,52 +2064,52 @@ The Abelian solution emerges from this construction as a limit and produces the \centering \begin{tabular}{@{}rr|cc|cr|c@{}} \toprule - $\vb{n}_{\vb{0}}$ & - $\vb{n}_{\vb{\infty}}$ & + $\vec{n}_{0}$ & + $\vec{n}_{\infty}$ & \multicolumn{2}{c|}{relations} & - $n_{\vb{1}}$ & - $\vb{n}_{\vb{1}}$ & - $\sum\limits_{t} \vb{n}_{\vb{t}}$ + $n_{1}$ & + $\vec{n}_{1}$ & + $\sum\limits_{t} \vec{n}_{\vec{t}}$ \\ \midrule - $n_{\vb{0}}\, \vb{k}$ & - $n_{\vb{\infty}}\, \vb{k}$ & - $n_{\vb{0}} + n_{\vb{\infty}} < \frac{1}{2}$ & - $n_{\vb{0}} \lessgtr n_{\vb{\infty}}$ & - $n_{\vb{0}} + n_{\vb{\infty}}$ & - $-n_{\vb{1}}\, \vb{k}$ & - $\vb{0}$ + $n_{0}\, \vec{k}$ & + $n_{\infty}\, \vec{k}$ & + $n_{0} + n_{\infty} < \frac{1}{2}$ & + $n_{0} \lessgtr n_{\infty}$ & + $n_{0} + n_{\infty}$ & + $-n_{1}\, \vec{k}$ & + $0$ \\ - $n_{\vb{0}}\, \vb{k}$ & - $n_{\vb{\infty}}\, \vb{k}$ & - $n_{\vb{0}} + n_{\vb{\infty}} > \frac{1}{2}$ & - $n_{\vb{0}} \lessgtr n_{\vb{\infty}}$ & - $1 - (n_{\vb{0}} + n_{\vb{\infty}})$ & - $n_{\vb{1}}\, \vb{k}$ & - $\vb{k}$ + $n_{0}\, \vec{k}$ & + $n_{\infty}\, \vec{k}$ & + $n_{0} + n_{\infty} > \frac{1}{2}$ & + $n_{0} \lessgtr n_{\infty}$ & + $1 - (n_{0} + n_{\infty})$ & + $n_{1}\, \vec{k}$ & + $\vec{k}$ \\ - $n_{\vb{0}}\, \vb{k}$ & - $-n_{\vb{\infty}}\, \vb{k}$ & - $n_{\vb{0}} + n_{\vb{\infty}} \lessgtr \frac{1}{2}$ & - $n_{\vb{0}} > n_{\vb{\infty}}$ & - $n_{\vb{0}} - n_{\vb{\infty}}$ & - $-n_{\vb{1}}\, \vb{k}$ & - $\vb{0}$ + $n_{0}\, \vec{k}$ & + $-n_{\infty}\, \vec{k}$ & + $n_{0} + n_{\infty} \lessgtr \frac{1}{2}$ & + $n_{0} > n_{\infty}$ & + $n_{0} - n_{\infty}$ & + $-n_{1}\, \vec{k}$ & + $0$ \\ - $n_{\vb{0}}\, \vb{k}$ & - $-n_{\vb{\infty}}\, \vb{k}$ & - $n_{\vb{0}} + n_{\vb{\infty}} \lessgtr \frac{1}{2}$ & - $n_{\vb{0}} < n_{\vb{\infty}}$ & - $-n_{\vb{0}} + n_{\vb{\infty}}$ & - $n_{\vb{1}}\, \vb{k}$ & - $\vb{0}$ + $n_{0}\, \vec{k}$ & + $-n_{\infty}\, \vec{k}$ & + $n_{0} + n_{\infty} \lessgtr \frac{1}{2}$ & + $n_{0} < n_{\infty}$ & + $-n_{0} + n_{\infty}$ & + $n_{1}\, \vec{k}$ & + $0$ \\ \bottomrule \end{tabular} \caption{Abelian limit of \SU{2} monodromies} \label{tab:Abelian_composition} \end{table} -Here we compute the parameter $\vb{n}_{\vb{1}}$ given two Abelian rotation in $\omega = 0$ and $\omega = \infty$ using the standard expression for two \SU{2} element multiplication given in~\eqref{eq:product_in_SU2} in~\Cref{sec:isomorphism}. +Here we compute the parameter $\vec{n}_{1}$ given two Abelian rotation in $\omega = 0$ and $\omega = \infty$ 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{figure}[tbp] @@ -2118,7 +2118,7 @@ Results are shown in~\Cref{tab:Abelian_composition}. \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 $P_2$.} + This corresponds to the cases $n_{0} + n_{\infty}< \frac{1}{2}$ and $n_{0}< n_{\infty}$ which are exchanged under the parity $P_2$.} \label{fig:Abelian_angles_1} \end{figure} @@ -2128,25 +2128,25 @@ Results are shown in~\Cref{tab:Abelian_composition}. \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 $P_2$.} + This corresponds to the cases $n_{0} + n_{\infty}> \frac{1}{2}$ and $n_{0}> n_{\infty}$ which are exchanged under the parity $P_2$.} \label{fig:Abelian_angles_2} \end{figure} 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 $\{ n_{\vb{1}} = 1 - (n_{\vb{0}} + n_{\vb{\infty}}),\, \hat{n}_{\vb{1}} = n_{\vb{0}} - \hat{n}_{\vb{\infty}} \}$. +into two sets $\{ n_{1} = n_{0} + n_{\infty},\, \hat{n}_{1} = -n_{0} + \hat{n}_{\infty} \}$ and $\{ n_{1} = 1 - (n_{0} + n_{\infty}),\, \hat{n}_{1} = n_{0} - \hat{n}_{\infty} \}$. 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. +Since $n^3_{0} > 0$ we can then fix the orientation of $D_{{1}}$. +$D_{{2}}$ is then fixed relatively to $D_{{1}}$ by the sign of $n^3_{\infty}$. +The sign of $n^3_{1}$ 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 +The relation between the usual Abelian paramter $\epsilon_{\vec{t}}$ and $n_{\vec{t}}^3$ is \begin{equation} - \varepsilon_{\vb{t}} + \varepsilon_{\vec{t}} = - n_{\vb{t}}^3 + \theta(-n^3_{\vb{t}}) + n_{\vec{t}}^3 + \theta(-n^3_{\vec{t}}) \label{eq:Abelian_vs_n_simple_case}, \end{equation} when all $m = 0$. @@ -2165,7 +2165,7 @@ when all $m = 0$. \subsubsection{The Abelian Limit of the Left Solutions} -We can then compute the basis element for the entries of~\Cref{tab:coeffs_k} for any value of $n_{\vb{1}}$ given in~\Cref{tab:Abelian_composition}. +We can then compute the basis element for the entries of~\Cref{tab:coeffs_k} for any value of $n_{1}$ given in~\Cref{tab:Abelian_composition}. For simplicity we consider the left sector of the solution and drop the notation identifying it to avoid cluttering the equations. The right sector follows in a similar way. @@ -2182,114 +2182,114 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h \begin{tabular}{@{}ccc@{}} \toprule $\qty( \ffa^{(L)},\, \ffb^{(L)},\, \ffc^{(L)} )$ & - $n_{\vb{1}}$ & + $n_{1}$ & $\qty( \cB^{(L)}( z ) )^T$ \\ \midrule \multirow{4}{*}{$(-1,\, 0,\, 0)$} & - $n_{\vb{0}} + n_{\vb{\infty}}$ & - $\mqty( (1 - z)^{-2\, n_{\vb{\infty}} - 2\, n_{\vb{0}} + 1} & 0 )$ + $n_{0} + n_{\infty}$ & + $\mqty( (1 - z)^{-2\, n_{\infty} - 2\, n_{0} + 1} & 0 )$ \\ & - $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ & + $1 - \qty( n_{0} + n_{\infty} )$ & $\mqty( 1 & 0 )$ \\ & - $n_{\vb{0}} - n_{\vb{\infty}}$ & - $\mqty( 1 & (-z)^{1 - 2\, n_{\vb{0}}} )$ + $n_{0} - n_{\infty}$ & + $\mqty( 1 & (-z)^{1 - 2\, n_{0}} )$ \\ & - $- n_{\vb{0}} + n_{\vb{\infty}}$ & + $- n_{0} + n_{\infty}$ & $\mqty( 1 & 0 )$ \\ \midrule \multirow{4}{*}{$(-1,\, 1,\, 0)$} & - $n_{\vb{0}} + n_{\vb{\infty}}$ & - $\mqty( \hyp{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 1} - {2\, n_{\vb{0}} + 1} - {2\, n_{\vb{0}}} + $n_{0} + n_{\infty}$ & + $\mqty( \hyp{2\, n_{\infty} + 2\, n_{0} - 1} + {2\, n_{0} + 1} + {2\, n_{0}} {z} & 0 )$ \\ & - $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ & + $1 - \qty( n_{0} + n_{\infty} )$ & $\mqty( 1 & 0 )$ \\ & - $n_{\vb{0}} - n_{\vb{\infty}}$ & - $\mqty( 0 & (-z)^{1 - 2\, n_{\vb{0}}} )$ + $n_{0} - n_{\infty}$ & + $\mqty( 0 & (-z)^{1 - 2\, n_{0}} )$ \\ & - $- n_{\vb{0}} + n_{\vb{\infty}}$ & - $\mqty( 0 & (1-z)^{2\, n_{\vb{0}} - 2\, n_{\vb{\infty}}}\, (-z)^{1 - 2\, n_{\vb{0}}} )$ + $- n_{0} + n_{\infty}$ & + $\mqty( 0 & (1-z)^{2\, n_{0} - 2\, n_{\infty}}\, (-z)^{1 - 2\, n_{0}} )$ \\ \midrule \multirow{4}{*}{$(0,\, 0,\, 0)$} & - $n_{\vb{0}} + n_{\vb{\infty}}$ & - $\mqty( (1-z)^{-2\, n_{\vb{\infty}} - 2\, n_{\vb{0}}} & 0 )$ + $n_{0} + n_{\infty}$ & + $\mqty( (1-z)^{-2\, n_{\infty} - 2\, n_{0}} & 0 )$ \\ & - $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ & - $\mqty( 0 & (1-z)^{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 2}\, (-z)^{1 - 2\, n_{\vb{0}}} )$ + $1 - \qty( n_{0} + n_{\infty} )$ & + $\mqty( 0 & (1-z)^{2\, n_{\infty} + 2\, n_{0} - 2}\, (-z)^{1 - 2\, n_{0}} )$ \\ & - $n_{\vb{0}} - n_{\vb{\infty}}$ & - $\mqty( (1-z)^{2\, n_{\vb{\infty}} - 2\, n_{\vb{0}}} & 0 )$ + $n_{0} - n_{\infty}$ & + $\mqty( (1-z)^{2\, n_{\infty} - 2\, n_{0}} & 0 )$ \\ & - $- n_{\vb{0}} + n_{\vb{\infty}}$ & + $- n_{0} + n_{\infty}$ & $\mqty( 1 & 0 )$ \\ \midrule \multirow{4}{*}{$(-1,\, 1,\, 1)$} & - $n_{\vb{0}} + n_{\vb{\infty}}$ & - $\mqty( (1-z)^{-2\, n_{\vb{\infty}} - 2\, n_{\vb{0}}} + 1 & 0 )$ + $n_{0} + n_{\infty}$ & + $\mqty( (1-z)^{-2\, n_{\infty} - 2\, n_{0}} + 1 & 0 )$ \\ & - $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ & + $1 - \qty( n_{0} + n_{\infty} )$ & $\mqty( 1 & 0 )$ \\ & - $n_{\vb{0}} - n_{\vb{\infty}}$ & - $\mqty( 0 & (-z)^{-2\, n_{\vb{0}}})\, \hyp{-1}{1 - 2\, n_{\vb{\infty}}}{1 - 2\, n_{\vb{0}}}{z} )$ + $n_{0} - n_{\infty}$ & + $\mqty( 0 & (-z)^{-2\, n_{0}})\, \hyp{-1}{1 - 2\, n_{\infty}}{1 - 2\, n_{0}}{z} )$ \\ & - $- n_{\vb{0}} + n_{\vb{\infty}}$ & - $\mqty( 0 & (1-z)^{-2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} + 1}\, (-z)^{-2\, n_{\vb{0}}} )$ + $- n_{0} + n_{\infty}$ & + $\mqty( 0 & (1-z)^{-2\, n_{\infty} + 2\, n_{0} + 1}\, (-z)^{-2\, n_{0}} )$ \\ \midrule \multirow{4}{*}{$(0,\, 0,\, 1)$} & - $n_{\vb{0}} + n_{\vb{\infty}}$ & - $\mqty( 0 & (-z)^{-2\, n_{\vb{0}}} )$ + $n_{0} + n_{\infty}$ & + $\mqty( 0 & (-z)^{-2\, n_{0}} )$ \\ & - $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ & - $\mqty( 0 & (1-z)^{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 1}\, (-z)^{-2\, n_{\vb{0}}} )$ + $1 - \qty( n_{0} + n_{\infty} )$ & + $\mqty( 0 & (1-z)^{2\, n_{\infty} + 2\, n_{0} - 1}\, (-z)^{-2\, n_{0}} )$ \\ & - $n_{\vb{0}} - n_{\vb{\infty}}$ & - $\mqty( 0 & (-z)^{-2\, n_{\vb{0}}}) )$ + $n_{0} - n_{\infty}$ & + $\mqty( 0 & (-z)^{-2\, n_{0}}) )$ \\ & - $- n_{\vb{0}} + n_{\vb{\infty}}$ & + $- n_{0} + n_{\infty}$ & $\mqty( 1 & 0 )$ \\ \midrule \multirow{4}{*}{$(0,\, 1,\, 1)$} & - $n_{\vb{0}} + n_{\vb{\infty}}$ & - $\mqty( (1-z)^{-2\, n_{\vb{\infty}} -2\, n_{\vb{0}}} & 0 )$ + $n_{0} + n_{\infty}$ & + $\mqty( (1-z)^{-2\, n_{\infty} -2\, n_{0}} & 0 )$ \\ & - $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ & - $\mqty( 0 & (1-z)^{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 2}\, (-z)^{-2\, n_{\vb{0}}} )$ + $1 - \qty( n_{0} + n_{\infty} )$ & + $\mqty( 0 & (1-z)^{2\, n_{\infty} + 2\, n_{0} - 2}\, (-z)^{-2\, n_{0}} )$ \\ & - $n_{\vb{0}} - n_{\vb{\infty}}$ & - $\mqty( 0 & (-z)^{-2\, n_{\vb{0}}}) )$ + $n_{0} - n_{\infty}$ & + $\mqty( 0 & (-z)^{-2\, n_{0}}) )$ \\ & - $- n_{\vb{0}} + n_{\vb{\infty}}$ & - $\mqty( 0 & (1-z)^{2\, n_{\vb{0}} - 2\, n_{\vb{\infty}}}\, (-z)^{-2\, n_{\vb{0}}} )$ + $- n_{0} + n_{\infty}$ & + $\mqty( 0 & (1-z)^{2\, n_{0} - 2\, n_{\infty}}\, (-z)^{-2\, n_{0}} )$ \\ \bottomrule \end{tabular} @@ -2300,20 +2300,20 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h \subsubsection{The \texorpdfstring{$\SU{2}_L$}{Left SU(2)} Limit} -We recover the non Abelian \SU{2} solution by considering $m_{\vb{t}}\sim 0$. +We recover the non Abelian \SU{2} solution by considering $m_{\vec{t}}\sim 0$. This is the first specific case shown in~\Cref{sec:true_basis}. In this scenario the left solution $\cB^{(L)}$ is always the same and matches the previous computation, however the right sector seems to give different solutions when different Abelian limits are taken. -Studying all possible solutions we find that all of them give the same answer in the limit $m_{\vb{t}} \to 0$, i.e.\ both $\cB^{(R)} = \mqty(1 & 0)^T$ and $\cB^{(R)} = \mqty(0 & 1)^T$.\footnotemark{} +Studying all possible solutions we find that all of them give the same answer in the limit $m_{\vec{t}} \to 0$, i.e.\ both $\cB^{(R)} = \mqty(1 & 0)^T$ and $\cB^{(R)} = \mqty(0 & 1)^T$.\footnotemark{} \footnotetext{% - We write ``possible solutions'' because $m_{\vb{1}} = 1 - \qty( m_{\vb{0}} + m_{\vb{\infty}} )$ is not. + We write ``possible solutions'' because $m_{1} = 1 - \qty( m_{0} + m_{\infty} )$ is not. } -The difference is the solution obtained from $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} > m_{\vb{\infty}}$ or $n_{\vb{0}} > m_{\vb{0}}$, $\hat{n}_{\vb{1}} < \hat{m}_{\vb{1}}$ and $\hat{n}_{\vb{\infty}} < \hat{m}_{\vb{\infty}}$. +The difference is the solution obtained from $n_{0} > m_{0}$, $n_{1} > m_{1}$ and $n_{\infty} > m_{\infty}$ or $n_{0} > m_{0}$, $\hat{n}_{1} < \hat{m}_{1}$ and $\hat{n}_{\infty} < \hat{m}_{\infty}$. In any case the solution is factorised in the form $\cB^{(L)}(z) \otimes \mqty(C & C')^T$ which is expected since the right sector plays no role. \subsubsection{Relating the Abelian Angles with the Group Parameters} -Using the explicit form of the \SO{4} and $\SU{2} \times \SU{2}$ matrices, we can verify that when the left and right $\SU{2}$ parameters are $\vb{n} = n^3\, \vb{k}$ and $\vb{m} = m^3\, \vb{k}$ the rotation of the D-branes in the plane $14$ is a \SO{2} element +Using the explicit form of the \SO{4} and $\SU{2} \times \SU{2}$ matrices, we can verify that when the left and right $\SU{2}$ parameters are $\vec{n} = n^3\, \vec{k}$ and $\vec{m} = m^3\, \vec{k}$ the rotation of the D-branes in the plane $14$ is a \SO{2} element \begin{equation} \mqty( \cos(\theta) & \sin(\theta) \\ -\sin(\theta)& \cos(\theta) ), \qquad @@ -2323,13 +2323,13 @@ while in plane $23$ the angle is $\theta = n^3 + m^3$. Comparing with the case $m = 0$ given in~\eqref{eq:Abelian_vs_n_simple_case}, we then guess that the general relation between the group parameters and the usual Abelian angles is: \begin{equation} \begin{split} - \varepsilon_{\vb{t}} + \varepsilon_{\vec{t}} & = - n_{\vb{t}}^3 - m_{\vb{t}}^3 + \theta( -(n^3_{\vb{t}} - m_{\vb{t}}^3) ), + n_{\vec{t}}^3 - m_{\vec{t}}^3 + \theta( -(n^3_{\vec{t}} - m_{\vec{t}}^3) ), \\ - \varphi_{\vb{t}} + \varphi_{\vec{t}} & = - n_{\vb{t}}^3 + m_{\vb{t}}^3 + \theta( -(n^3_{\vb{t}} + m_{\vb{t}}^3) ). + n_{\vec{t}}^3 + m_{\vec{t}}^3 + \theta( -(n^3_{\vec{t}} + m_{\vec{t}}^3) ). \end{split} \label{eq:Abelian_vs_n_general_case} \end{equation} @@ -2337,7 +2337,7 @@ Comparing with the case $m = 0$ given in~\eqref{eq:Abelian_vs_n_simple_case}, we \subsubsection{Recovering the Abelian Result: an Example} -To show that the construction of the Abelian limit is indeed working, we consider the first case in~\Cref{sec:true_basis} with $n_{\vb{1}} = 1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ and $m_{\vb{1}} = -m_{\vb{0}} + m_{\vb{\infty}}$. +To show that the construction of the Abelian limit is indeed working, we consider the first case in~\Cref{sec:true_basis} with $n_{1} = 1 - \qty( n_{0} + n_{\infty} )$ and $m_{1} = -m_{0} + m_{\infty}$. This leads to two independent rational functions of $\omega_z$: \begin{equation} \begin{split} @@ -2361,13 +2361,13 @@ This leads to two independent rational functions of $\omega_z$: & = \mqty( 0 & C_1\, - (1-\omega_z)^{\varepsilon_{\vb{1}}-1}\, - (-\omega_z)^{\varepsilon_{\vb{0}}-1} + (1-\omega_z)^{\varepsilon_{1}-1}\, + (-\omega_z)^{\varepsilon_{0}-1} \\ 0 & C_2\, - (1-\omega_z)^{-\varphi_{\vb{1}}}\, - (-\omega_z)^{-\varphi_{\vb{0}}} + (1-\omega_z)^{-\varphi_{1}}\, + (-\omega_z)^{-\varphi_{0}} ), \end{split} \label{eq:Abelian_sol_example} @@ -2377,22 +2377,22 @@ This is the known result in the presence of Abelian rotations of the D-branes: w $\U{1}_1 \times \U{1}_2 \subset \SU{2}_L \times \SU{2}_R$. In particular we used~\eqref{eq:Abelian_vs_n_general_case} to write the relation between the usual Abelian angles and the group parameters as \begin{equation} - \varepsilon_{\vb{0}} = n_{\vb{0}} - m_{\vb{0}}, + \varepsilon_{0} = n_{0} - m_{0}, \qquad - \varepsilon_{\vb{1}} = n_{\vb{1}} - m_{\vb{1}}, + \varepsilon_{1} = n_{1} - m_{1}, \qquad - \varepsilon_{\vb{\infty}} = n_{\vb{\infty}} + m_{\vb{\infty}} + \varepsilon_{\infty} = n_{\infty} + m_{\infty} \end{equation} -such that $\sum\limits_{t} \varepsilon_{\vb{t}} = 1$, and +such that $\sum\limits_{t} \varepsilon_{\vec{t}} = 1$, and \begin{equation} - \varphi_{\vb{0}} = n_{\vb{0}} + m_{\vb{0}}, + \varphi_{0} = n_{0} + m_{0}, \qquad - \varphi_{\vb{1}} = n_{\vb{1}} + m_{\vb{1}}, + \varphi_{1} = n_{1} + m_{1}, \qquad - \varphi_{\vb{\infty}} = n_{\vb{\infty}} - m_{\vb{\infty}}, + \varphi_{\infty} = n_{\infty} - m_{\infty}, \label{eq:Abelian_rotation_second} \end{equation} -where $\sum\limits_{t} \varphi_{\vb{t}} = 2$, in order to approach the usual notation in the literature. +where $\sum\limits_{t} \varphi_{\vec{t}} = 2$, in order to approach the usual notation in the literature. As usual we have $\ipd{\omega_z} \cZ^1( \omega_z ) \neq \qty[ \ipd{\omega_z} \bcZ^1( \omega_z ) ]^*$. We can now build the Abelian solution to show the analytical structure of the limit. @@ -2413,18 +2413,18 @@ We have ) \end{equation} where we chose $R_{(\bart)} = \1_4$ so that $U_{(\bart)}$ in~\eqref{eq:Umatrices} is mapped to $(i \sigma_1, i \sigma_1) \in \SU{2} \times \SU{2}$. -Notice however that $\vb{n}_{\vb{t}} = n_{\vb{t}}^3\, \vb{k}$ implies that $v^3_{(t)} = 0$ in~\eqref{eq:special_UL_brane_t}. +Notice however that $\vec{n}_{\vec{t}} = n_{\vec{t}}^3\, \vec{k}$ implies that $v^3_{(t)} = 0$ in~\eqref{eq:special_UL_brane_t}. Hence $U_L$ and $U_R$ are always off diagonal and their action on~\eqref{eq:Abelian_sol_example} is to fill the first column. -From the previous relations we can also recover the usual holomorphicity $\barZ^1(\baru) = \qty[ Z^1(u) ]^*$ of the sector with $\sum\limits_t \varepsilon_{\vb{t}} = 1$ and $\barZ^2(\baru) = \qty[ Z^2(u) ]^*$ of the sector with $\sum\limits_t \varphi_{\vb{t}} = 2$. +From the previous relations we can also recover the usual holomorphicity $\barZ^1(\baru) = \qty[ Z^1(u) ]^*$ of the sector with $\sum\limits_t \varepsilon_{\vec{t}} = 1$ and $\barZ^2(\baru) = \qty[ Z^2(u) ]^*$ of the sector with $\sum\limits_t \varphi_{\vec{t}} = 2$. \subsubsection{Abelian Limits} -From the example in the previous section it is possible to consider both cases given in~\Cref{sec:true_basis} and all possible combinations of the expression of $n_{\vb{1}}$ and $m_{\vb{1}}$ for a total of $2 \times 4 \times 4 = 32$ possible combinations. +From the example in the previous section it is possible to consider both cases given in~\Cref{sec:true_basis} and all possible combinations of the expression of $n_{1}$ and $m_{1}$ for a total of $2 \times 4 \times 4 = 32$ possible combinations. In almost all cases (in fact all but six) the solution in spinorial formalism is a $2 \times 2$ matrix which has two non vanishing entries, hence two independent Abelian solutions. In the remaining cases the matrix has only one non vanishing entry but the constraints on $n$ and $m$ are not compatible, thus they should not be considered. -In the first case encountered in~\Cref{sec:true_basis} the inconsistent combinations are $\{ n_{\vb{1}} = n_{\vb{0}} + n_{\vb{\infty}},~ m_{\vb{1}} = 1 - (m_{\vb{0}} + m_{\vb{\infty}}) \}$ and $\{ n_{\vb{1}} = 1 - (n_{\vb{0}} + n_{\vb{\infty}}),~ m_{\vb{1}} = 1 - (m_{\vb{0}} + m_{\vb{\infty}}) \}$. -In the second case in~\Cref{sec:true_basis} the incompatible constraints appear when $n_{\vb{1}} = -n_{\vb{0}} + n_{\vb{\infty}}$. +In the first case encountered in~\Cref{sec:true_basis} the inconsistent combinations are $\{ n_{1} = n_{0} + n_{\infty},~ m_{1} = 1 - (m_{0} + m_{\infty}) \}$ and $\{ n_{1} = 1 - (n_{0} + n_{\infty}),~ m_{1} = 1 - (m_{0} + m_{\infty}) \}$. +In the second case in~\Cref{sec:true_basis} the incompatible constraints appear when $n_{1} = -n_{0} + n_{\infty}$. \subsection{The Physical Interpretation} diff --git a/sec/part1/introduction.tex b/sec/part1/introduction.tex index 18cce29..954533f 100644 --- a/sec/part1/introduction.tex +++ b/sec/part1/introduction.tex @@ -1308,12 +1308,12 @@ These stacks would separately lead to a $\U{3} \times \U{2}$ gauge theory. It would however be theory of pure force, without matter content. Moreover we should also worry about the extra \U{1} groups appearing: these need careful consideration but go beyond the necessary analysis for what follows. -Matter fields are notoriously fermions transforming in the bi-fundamental representation $(\vb{N}, \vb{M})$ of the \sm gauge group~\eqref{eq:intro:smgroup}. -For example left handed quarks in the \sm transform under the $(\vb{3}, \vb{2})$ representation of the group $\SU{3}_C \otimes \SU{2}_L$. +Matter fields are notoriously fermions transforming in the bi-fundamental representation $(\vec{N}, \vec{M})$ of the \sm gauge group~\eqref{eq:intro:smgroup}. +For example left handed quarks in the \sm transform under the $(\vec{3}, \vec{2})$ representation of the group $\SU{3}_C \otimes \SU{2}_L$. This is realised in string theory by a string stretched across two stacks of $3$ and $2$ D-branes as in the right of~\Cref{fig:dbranes:chanpaton}. The fermion would then be characterised by the charge under the gauge bosons living on the D-branes. The corresponding anti-particle would then simply be a string oriented in the opposite direction. -Things get complicated when introducing also left handed leptons transforming in the $(\vb{1}, \vb{2})$ representation: they cannot have endpoints on the same stack of D-branes as quarks since they do not have colour charge. +Things get complicated when introducing also left handed leptons transforming in the $(\vec{1}, \vec{2})$ representation: they cannot have endpoints on the same stack of D-branes as quarks since they do not have colour charge. We therefore need to introduce more D-branes to account for all the possible combinations. An additional issue comes from the requirement of chirality. @@ -1338,7 +1338,7 @@ The light spectrum is thus composed of the desired matter content alongside with \end{figure} It is therefore possible to recover a \sm-like construction using multiple D-branes at angles as in~\Cref{fig:dbranes:smbranes}, where the angles have been drawn perpendicular but can in principle be arbitrary~\cite{Ibanez:2001:GettingJustStandard,Grimm:2005:EffectiveActionType,Sheikh-Jabbari:1998:ClassificationDifferentBranes,Berkooz:1996:BranesIntersectingAngles}. -For instance quarks are localised at the intersection of the \emph{baryonic} stack of D-branes, yielding the colour symmetry generators, with the \emph{left} and \emph{right} stacks, leading to the $( \vb{3}, \vb{2} )$ and $( \vb{3}, \vb{1})$ representations. +For instance quarks are localised at the intersection of the \emph{baryonic} stack of D-branes, yielding the colour symmetry generators, with the \emph{left} and \emph{right} stacks, leading to the $( \vec{3}, \vec{2} )$ and $( \vec{3}, \vec{1})$ representations. The same applies to leptons created by strings attached to the \emph{leptonic} stack. Combinations of the additional \U{1} factors in the resulting gauge group finally lead to the definition of the hypercharge $Y$. diff --git a/sec/part2/divergences.tex b/sec/part2/divergences.tex index 2412f69..79540fa 100644 --- a/sec/part2/divergences.tex +++ b/sec/part2/divergences.tex @@ -13,7 +13,7 @@ In fact in Equation (6.16) of \cite{Liu:2002:StringsTimeDependentOrbifold} the f \begin{equation} A_4 \sim \int\limits_{q \sim \infty} \frac{\dd{q}}{\abs{q}} \ccA( q ) \end{equation} -where $\ccA_{\text{closed}}( q ) \sim q^{4 - \ap \norm{\vb{p}_{\perp}}^2}$ and $\ccA_{\text{closed}}( q ) \sim q^{1 - \ap \norm{\vb{p}_{\perp}}^2} \tr\qty( \liebraket{T_1}{T_2}_+ \liebraket{T_3}{T_4}_+ )$ ($T_i$ for $i = 1,\, 2,\, 3,\, 4$ are Chan-Paton matrices). +where $\ccA_{\text{closed}}( q ) \sim q^{4 - \ap \norm{\vec{p}_{\perp}}^2}$ and $\ccA_{\text{closed}}( q ) \sim q^{1 - \ap \norm{\vec{p}_{\perp}}^2} \tr\qty( \liebraket{T_1}{T_2}_+ \liebraket{T_3}{T_4}_+ )$ ($T_i$ for $i = 1,\, 2,\, 3,\, 4$ are Chan-Paton matrices). Moreover divergences in string amplitudes are not limited to four points: interestingly we show that the open string three point amplitude with two tachyons and the first massive state may be divergent when some \emph{physical} polarisations are chosen. The true problem is therefore not related to a gravitational issue but to the non existence of the effective field theory. In fact when we express the theory using the eigenmodes of the kinetic terms some coefficients do not exist, not even as a distribution. @@ -47,7 +47,7 @@ Since these terms arise from string theory also through the exchange of massive A deeper understanding of the subject requires the study of the polarisations of the massive state on the orbifold as seen from the covering Minkowski space before the computation of the overlap of the wave functions. We then go back to string theory and we verify that in the \nbo the open string three points amplitude with two tachyons and one first level massive string state does indeed diverge when some physical polarisation are chosen. -We then introduce the Generalized Null Boost Orbifold (\gnbo) as a generalization of the \nbo which still has a light-like singularity and is generated by one Killing vector. +We then introduce the generalised Null Boost Orbifold (\gnbo) as a generalization of the \nbo which still has a light-like singularity and is generated by one Killing vector. However in this model there are two directions associated with $\cA$, one compact and one non compact. We can then construct the scalar \qed and the effective field theory which extends it with the inclusion of higher order terms since all terms have a distributional interpretation~\cite{Estrada:2012:GeneralIntegral}. However if a second Killing vector is used to compactify the formerly non compact direction, the theory has again the same problems as in the \nbo. @@ -99,7 +99,7 @@ We will construct directly both the scalar and the spin-1 eigenfunctions which \subsubsection{Geometric Preliminaries} \label{sec:geometric_preliminaries_nbo} -In Minkowski spacetime $\ccM^{1,D-1}$ with coordinates $\qty(x^{\mu}) = \qty(x^+,\, x^-,\, x^2,\, \vb{x})$ +In Minkowski spacetime $\ccM^{1,D-1}$ with coordinates $\qty(x^{\mu}) = \qty(x^+,\, x^-,\, x^2,\, \vec{x})$ and metric \begin{equation} \dss[2]{s} @@ -108,7 +108,7 @@ and metric + \qty(\dd{x^2})^2 + \eta_{ij} \dd{x}^i \dd{x}^j, \end{equation} -we consider the following change of coordinates to $\qty(x^{\alpha}) = (u,\, v,\, z,\, \vb{x})$ +we consider the following change of coordinates to $\qty(x^{\alpha}) = (u,\, v,\, z,\, \vec{x})$ \begin{equation} \begin{cases} x^- & = u @@ -174,7 +174,7 @@ in such a way that where $\cK^{n}= e^{n\kappa}$, leads to the identifications \begin{equation} x= - \mqty( x^- \\ x^2 \\ x^+ \\ \vb{x} ) + \mqty( x^- \\ x^2 \\ x^+ \\ \vec{x} ) \equiv \cK^{n} x = @@ -182,14 +182,14 @@ where $\cK^{n}= e^{n\kappa}$, leads to the identifications x^- \\ x^2 + n \qty(2 \pi \Delta) x^- \\ x^+ + n \qty(2 \pi \Delta) x^2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 x^- \\ - \vb{x} + \vec{x} ) \end{equation} or to \begin{equation} - \qty( u,\, v,\, z,\, \vb{x} ) + \qty( u,\, v,\, z,\, \vec{x} ) \equiv - \qty( u,\, v,\, z + 2 \pi n,\, \vb{x} ) + \qty( u,\, v,\, z + 2 \pi n,\, \vec{x} ) \end{equation} in coordinates $\qty(x^{\alpha})$ where $\kappa = 2 \pi \ipd{z}$ is a global Killing vector. @@ -250,7 +250,7 @@ We study the eigenmodes of the Laplacian operator to diagonalize the scalar kine ) \\ & = - \int \dd[D-3]{\vb{x}}\, + \int \dd[D-3]{\vec{x}}\, \int \dd{u}\, \int \dd{v}\, \finiteint{z}{0}{2\pi} @@ -272,7 +272,7 @@ We study the eigenmodes of the Laplacian operator to diagonalize the scalar kine \end{equation} The solution to the equation of motion is enough when we want to perform the canonical quantization. Since we use Feynman diagrams we consider the path integral approach: we take off-shell modes and solve the eigenvalue problem $\square \phi_r = r \phi_r$. -Comparing with the flat case we see that $r$ is $2\, k_-\, k_+ - \norm{\vb{k}}^2$ when $k$ is the impulse in flat coordinates. +Comparing with the flat case we see that $r$ is $2\, k_-\, k_+ - \norm{\vec{k}}^2$ when $k$ is the impulse in flat coordinates. We therefore have \begin{equation} -2 \ipd{u} \ipd{v} \phi_r @@ -288,9 +288,9 @@ We therefore have \end{equation} Using Fourier transforms it follows that the eigenmodes are \begin{equation} - \phi_{\kmkr}\qty(u,\, v,\, z,\, \vb{x}) + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) = - e^{i k_+ v + i l z + i \vb{k} \cdot \vb{x}}\, + e^{i k_+ v + i l z + i \vec{k} \cdot \vec{x}}\, \tphi_{\kmkr}(u), \end{equation} with @@ -300,14 +300,14 @@ with \frac{1}{\sqrt{\qty( 2 \pi )^D~ \abs{2 \Delta k_+\, u}}} e^{ - i \frac{l^2}{2 \Delta^2 k_+} \frac{1}{u} - + i \frac{\norm{\vb{k}}^2 + r}{2 k_+} u + + i \frac{\norm{\vec{k}}^2 + r}{2 k_+} u }, \end{equation} and \begin{equation} - \phi^*_{\kmkr}\qty(u,\, v,\, z,\, \vb{x}) + \phi^*_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) = - \phi_{\mkmkr}\qty(u,\, v,\, z,\, \vb{x}). + \phi_{\mkmkr}\qty(u,\, v,\, z,\, \vec{x}). \end{equation} We chose the numeric factor in order to get a canonical normalisation: \begin{equation} @@ -316,14 +316,14 @@ We chose the numeric factor in order to get a canonical normalisation: \qty( \phi_{\kmkrN{1}},\, \phi_{\kmkrN{2}} ) \\ = & - \int \dd[D-1]{\vb{x}}\, + \int \dd[D-1]{\vec{x}}\, \int \dd{u}\, \int \dd{v}\, \finiteint{z}{0}{2\pi} \abs{\Delta u}\, \phi_{\kmkrN{1}}\, \phi_{\kmkrN{2}} \\ = & - \delta^{D-3}( \vb{k}_{\qty(1)} + \vb{k}_{\qty(2)})\, + \delta^{D-3}( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)})\, \delta( r_{(1)} - r_{(2)})\, \delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\, \delta_{l_{\qty(1)} + l_{\qty(2)},\, 0}. @@ -331,20 +331,20 @@ We chose the numeric factor in order to get a canonical normalisation: \end{equation} We can then perform the off-shell expansion \begin{equation} - \phi\qty(u,\, v,\, z,\, \vb{x}) + \phi\qty(u,\, v,\, z,\, \vec{x}) = - \int \dd[D-3]{\vb{k}} + \int \dd[D-3]{\vec{k}} \int \dd{k_+} \int \dd{r} \infinfsum{l} \cA_{\kmkr}\, - \phi_{\kmkr}\qty(u,\, v,\, z,\, \vb{x}), + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}), \end{equation} such that the scalar kinetic term becomes \begin{equation} \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \cA ] = - \int \dd[D-3]{\vb{k}}\, + \int \dd[D-3]{\vec{k}}\, \int \dd{k_+} \int \dd{r} \infinfsum{l} @@ -539,7 +539,7 @@ We get the solutions: \end{equation} then we can expand the off-shell fields as \begin{equation} - a_{\alpha}\qty(u,\, v,\, z,\, \vb{x} ) + a_{\alpha}\qty(u,\, v,\, z,\, \vec{x} ) = \int \ccD k \sum\limits_{% @@ -549,16 +549,16 @@ then we can expand the off-shell fields as } \infinfsum{l} \cE_{\kmkr\, \underline{\alpha}}\, - {a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vb{x} ), + {a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vec{x} ), \end{equation} -where ${a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vb{x}) = \tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)\, e^{i\, \qty( k_+ v + l z + \vb{k} \cdot \vb{x})}$ and $\int \ccD k = \int \dd[D-3]{\vb{k}} \int \dd{k_+} \int \dd{r}$. +where ${a}^{\underline{\alpha}}_{\kmkr\, \alpha}\qty(u,\, v,\, z,\, \vec{x}) = \tildea^{\underline{\alpha}}_{\kmkr\, \alpha}(u)\, e^{i\, \qty( k_+ v + l z + \vec{k} \cdot \vec{x})}$ and $\int \ccD k = \int \dd[D-3]{\vec{k}} \int \dd{k_+} \int \dd{r}$. We can also compute the normalisation as \begin{equation} \begin{split} \qty(a_{(1)},\, a_{(2)}) & = - \int \dd[D-3]{\vb{x}} + \int \dd[D-3]{\vec{x}} \int \dd{u} \int \dd{v} \finiteint{z}{0}{2\pi} @@ -572,7 +572,7 @@ We can also compute the normalisation as \cE_{\kmkrN{1}} \circ \cE_{\kmkrN{2}} \\ & \times - \delta^{D-3}( \vb{k}_{\qty(1)} + \vb{k}_{\qty(2)})\, + \delta^{D-3}( \vec{k}_{\qty(1)} + \vec{k}_{\qty(2)})\, \delta( r_{(1)} - r_{(2)})\, \delta( k_{\qty(1)\, +} + k_{\qty(2)\, +})\, \delta_{l_{\qty(1)} + l_{\qty(2)},\, 0}, @@ -601,7 +601,7 @@ Finally the Lorenz gauge reads - k_+\, \cE_{\kmkr\, \underline{u}} - - \frac{\vb{k}^2 + r}{2\, k_+} \cE_{\kmkr\, \underline{v}} + \frac{\vec{k}^2 + r}{2\, k_+} \cE_{\kmkr\, \underline{v}} = 0, \label{eq:explicit_orbifold_Lorenz} @@ -612,7 +612,7 @@ The photon kinetic term becomes \begin{equation} \rS_{\text{s}\qed}^{(\text{kinetic})}\qty[ \cE ] = - \int \dd[D-3]{\vb{k}} + \int \dd[D-3]{\vec{k}} \int \dd{k_+} \int \dd{r} \infinfsum{l}\, @@ -699,13 +699,13 @@ and we get:\footnotemark{} & = \finiteprod{i}{1}{3} \qty[% - \int \dd[D-3]{\vb{k}_{\qty(i)}}\, + \int \dd[D-3]{\vec{k}_{\qty(i)}}\, \dd{r_{\qty(i)}}\, \dd{k_{\qty(i)\, +}} \sum_{l_{\qty(i)}} ]\, \qty(2\pi)^{D-1} - \delta\qty(\finitesum{i}{1}{3} \vb{k}_{\qty(i)})\, + \delta\qty(\finitesum{i}{1}{3} \vec{k}_{\qty(i)})\, \delta\qty(\finitesum{i}{1}{3} k_{\qty(i)\, +})\, \\ & \times @@ -731,7 +731,7 @@ and we get:\footnotemark{} \\ & + \cE_{\kmkrN{1}\, \underline{v}}\, - \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vb{k}_{\qty(2)}) + \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vec{k}_{\qty(2)}) \\ & - \left. @@ -748,9 +748,9 @@ and we get:\footnotemark{} where \begin{equation} \begin{split} - \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vb{k}_{\qty(2)}) + \ccF\qty(k_{\qty(1)\, +},\, l_{\qty(1)},\, k_{\qty(2)\, +},\, l_{\qty(2)},\, r_{\qty(2)},\, \vec{k}_{\qty(2)}) & = - \frac{\norm{\vb{k}_{\qty(2)}}^2 + r_{\qty(2)}}{2\, k_{\qty(2)\, +}} \cI_{\qty{3}}^{\qty[0]} + \frac{\norm{\vec{k}_{\qty(2)}}^2 + r_{\qty(2)}}{2\, k_{\qty(2)\, +}} \cI_{\qty{3}}^{\qty[0]} + i \frac{k_{\qty(2)\, +}}{2\, k_{\qty(1)\, +}} @@ -813,7 +813,7 @@ which can be expressed using the modes as: & = \finiteprod{i}{1}{4} \qty[% - \int \dd[D-3]{\vb{k}_{\qty(i)}} + \int \dd[D-3]{\vec{k}_{\qty(i)}} \dd{k_{\qty(i)\, +}} \dd{r_{\qty(i)}} \sum_{l_{\qty(i)}} @@ -821,7 +821,7 @@ which can be expressed using the modes as: \qty(2\pi)^{D-1} \\ & \times - \delta\qty( \finitesum{i}{1}{4} \vb{k}_{\qty(i)} )\, + \delta\qty( \finitesum{i}{1}{4} \vec{k}_{\qty(i)} )\, \delta\qty( \finitesum{i}{1}{4} k_{\qty(i)\, +} )\, \delta_{l_{\qty(1)} + l_{\qty(2)} + l_{\qty(3)} + l_{\qty(4)},\, 0} \\ @@ -862,7 +862,7 @@ which can be expressed using the modes as: & - \left. \frac{g_4}{4}\, - \ccA\qty(\qty{k_+,\, l,\, \vb{k},\, r})\, + \ccA\qty(\qty{k_+,\, l,\, \vec{k},\, r})\, \cI_{\qty{4}}^{\qty[0]} \right\rbrace, \end{split} @@ -870,7 +870,7 @@ which can be expressed using the modes as: where \begin{equation} \begin{split} - \ccA\qty(\qty{k_+,\, l,\, \vb{k},\, r}) + \ccA\qty(\qty{k_+,\, l,\, \vec{k},\, r}) & = \qty(\cA_{\mkmkrN{1}})^*\, \qty(\cA_{\mkmkrN{2}})^*\, @@ -908,7 +908,7 @@ with \qquad B(u) = - -\qty(\vb{k}^2 + r) + -\qty(\vec{k}^2 + r) - i\, k_+\, \frac{1}{u} - @@ -999,7 +999,7 @@ We can perform the usual Fourier transform and the function $B(u)$ becomes B(u) & = - - (\vb{k}^2 + r) + (\vec{k}^2 + r) - i\, k_+\, \frac{u}{u^2 + \epsilon^2} - @@ -1039,93 +1039,1030 @@ However if $\ap \sim \epsilon^2$ as natural in string theory we do not solve the and the curvature terms are no longer singular. -%%% TODO %%% -\subsection{A Hope from Twisted State Background} +\subsubsection{A Hope from Twisted State Background} -It is clear from the previous discussion that the true problem is -associated with the dipole string and its charge neutral states -since the charged ones can be cured rather trivially by a Wilson line. +The issue with the divergences is associated with the dipole string and its charge neutral states since the charged ones can be cured rather trivially by a Wilson line. -On the other side we know that the usual time-like orbifolds are well -defined because of a presence of a $B_{\mu\nu}$ background and this -field is sourced by strings. -So we can think of switching on such a background in the open string. -For open strings $B$ is equivalent to $F$ so we can consider what -happens to an open string in an electromagnetic background. +On the other hand we know that the usual time-like orbifolds are well defined because of a presence of a $B_{\mu\nu}$ background and this field is sourced by strings. +We may switch on such a background in the open string. +For open strings $F$ is equivalent to such $B$ field so we can consider what happens to an open string in an electromagnetic background. -The choice of such a background is limited first of all -by the request that it must be an exact string solution, i.e. that it -satisfies the e.o.m derived from the DBI. -If a closed string winds the compact direction $z$ -is coupled to $B_{z u}$, $B_{z v}$ and $B_{z i}$ but if we choose +The choice of such a background is limited first of all by the request that it must be an exact string solution, i.e.\ it needs to obey the \eom derived from the Dirac--Born--Infeld action. +If a closed string winds the compact direction $z$ then it is coupled to $B_{z u}$, $B_{z v}$ and $B_{z i}$ but if we choose \begin{equation} -\frac{1}{2\pi\ap} B = f(u) d u \wedge d z + \frac{1}{2\pi\ap} B(u) + = + f(u) \dd{u} \wedge \dd{z}. \label{eq:F_bck} - . - \end{equation} +\end{equation} then - \begin{equation} - \det(g+ 2\pi \ap f ) = \det(g) - , - \end{equation} -therefore it is a solution of open string e.o.m. for any -$f(u,v,z,x^i)$. - Suppose that the action for a real neutral scalar $\phi$ is given by - (as the 2 tachyons -- 2 photons amplitude suggests) - \begin{align} - S_{\mbox{scalar kin}} - =& - \int_\Omega d^D x\, - \sqrt{- \det g} - \frac{1}{2} - \Bigl( - -g^{\alpha\beta} - \ipd{\alpha}\phi\, - \ipd{\beta} \phi - -M^2 \phi^2 - + c_1 - \qty(\ap)^2\, \ipd{\mu} \phi \ipd{\nu}\phi - f^{\mu\kappa} f^{\nu}_{~\kappa} - \Bigr) - \nonumber\\ - = - & - \int d^{D-3} \vb{x}\, - \int d u\, \int d v\, \int_0^{2\pi} d z\, - | \Delta u| - \frac{1}{2} - \Biggl( - 2\ipd{u} \phi\,\ipd{v} \phi\, - \nonumber\\ - & +\begin{equation} + \det( g + 2 \pi \ap f(u) ) = \det(g). +\end{equation} +It is therefore a solution of the open string \eom for any $f\qty(u,\, v,z,x^i)$. +As the two-tachyons---two-photons amplitude suggests, suppose that the action for a real neutral scalar $\phi$ is given by: +\begin{equation} + \begin{split} + S_{\text{scalar}}^{(\text{kinetic})}\qty[ \phi ] + & = + \int\limits_{\Omega} \dd[D]{x}\, + \sqrt{- \det g}\, + \frac{1}{2} + \qty( + - g^{\alpha\beta} + \ipd{\alpha} \phi\, \ipd{\beta} \phi + - M^2 \phi^2 + + c_1 + \qty(\ap)^2\, \ipd{\mu} \phi\, \ipd{\nu} \phi + \tensor{f}{^{\mu}^\kappa} \tensor{f}{^{\nu}_\kappa} + ) + \\ + & = + \int \dd[D-3]{\vec{x}}\, + \int \dd{u}\, + \int \dd{v}\, + \finiteint{z}{0}{2\pi} + \abs{\Delta u}\, + \frac{1}{2}\, + \Biggl( + 2\, \ipd{u} \phi\,\ipd{v} \phi\, + \\ + & - + \frac{1}{\qty(\Delta u )^2} \qty(\ipd{z} \phi)^2 - - \frac{1}{(\Delta u )^2} - (\ipd{z} \phi)^2 - - - (\vec \partial \phi)^2 - - - M^2 \phi^2 - +c_1 \qty(\ap)^2 - \frac{1}{(\Delta u)^2} (\ipd{v} \phi)^2 f^2(u) - \Biggr) - , -\end{align} + \eta^{ij} \ipd{i}\phi\, \ipd{j} \phi + - + M^2 \phi^2 + + + c_1 \qty(\ap)^2 \frac{1}{\qty(\Delta u)^2} \qty(\ipd{v} \phi)^2 f^2(u) + \Biggr) + , + \end{split} +\end{equation} Performing the same steps as before we get \begin{equation} - B(u) - = - (-\vb{k}^2-r) - + - (-i k_+) \frac{1}{u} - + - \frac{(-l^2 + c_1 \qty(\ap)^2 f(u)^2 k_+^2)}{\Delta^2\, u^2} - , + B(u) + = + - (\vec{k}^2 + r) + - + i\, k_+\, \frac{1}{u} + + + \frac{\qty(c_1 \qty(\ap)^2 f(u)^2 k_+^2 - l^2)}{\Delta^2\, u^2}, \end{equation} -so even for a constant $f(u)=f_0$ we get a solution which solves the issues. -Notice however that the ``trivial'' solution $f=f_0 d u \wedge d z$ is -not so trivial in Minkowski coordinates $f=\frac{f_0}{x^-} d x^- -\wedge d x^2$. -Though appealing, the study of the string in the presence of this non trivial background needs a deeper analysis which goes beyond the scope of this paper. +so even for a constant $f(u) = f_0$ we get a solution which solves the issues. +Notice however that the ``trivial'' solution $f = f_0 \dd{u} \wedge \dd{z}$ is not trivial in Minkowski coordinates where it reads $f = \frac{f_0}{x^-} \dd{x^-} \wedge \dd{x^2}$. +Though appealing, the study of the string in the presence of this non trivial background needs a deeper analysis and it surely is a direction to cover in the future. + + +\subsection{NBO Eigenfunction from the Covering Space} + +We recover the eigenfunctions from the covering Minkowski space in order to elucidate the connection between the polarizations in \nbo and in Minkowski. +Moreover we generalise the result to a symmetric two index tensor which is the polarization of the first massive state to compute the two-tachyons--one-massive-state amplitude in the next section and to show that it diverges. + + +\subsubsection{Spin 0 Wave Function from Minkowski space} + +We start with the usual plane wave in flat space and we express it in the new coordinates (we do not write the dependence on $\vec{x}$ since it is trivial): +\begin{equation} + \begin{split} + \psi_{k_+\, k_-\, k_2}\qty(x^+,\, x^-,\, x^2) + & = + e^{i\, \qty( k_+ x^+ + k_- x^- + k_2 x^2 )} + \\ + & = + e^{% + i\, \qty[% + k_+ v + + + \frac{2\, k_+ k_- - k_2^2}{2 k_+} u + + + \frac{1}{2} \Delta^2 k_+ u + \qty( z + \frac{k_2}{\Delta k_+} )^2 + ] + } + \\ + & = + \psi_{k_+\, k_-\, k_2}\qty(u,\, v,\, z). + \end{split} +\end{equation} +The corresponding wave function on the \nbo is obtained by the periodicity of $z$. +This can be done in two ways either in $\qty(x^{\mu})$ coordinates or in $\qty(x^{\alpha}) = \qty(u\, v\, z)$. +From the first we study how the map to the orbifold gives the function a dependence on the equivalence class of momenta. +Implementing the projection on periodic $z$ functions we get: +\begin{equation} + \begin{split} + \Psi_{\qty[k_+\, k_-\, k_2]}\qty(\qty[x^+,\, x^-,\, x^2]) + & = + \infinfsum{n} + \psi_{k_+\, k_-\, k_2}\qty( \cK^n\qty(x^+,\, x^-,\, x^2) ) + \\ + & = + \infinfsum{n} + \psi_{\cK^{-n}\qty( k_+\, k_-\, k_2 )}\qty( x^+,\, x^-,\, x^2 ), + \end{split} +\end{equation} +where we write $\qty[k_+\, k_-\, k_2]$ since the function depends on the equivalence class of $\qty(k_+\, k_-\, k_2)$ only. +The equivalence relation is given by +\begin{equation} + k = + \mqty( + k_+\\ k_-\\ k_2 + ) + \equiv + \cK^{-n} k + = + \mqty( + k_+ + \\ + k_- + n \qty(2 \pi \Delta) k_2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 k_+ + \\ + k_2 + n \qty(2 \pi \Delta) k_+ + ). + \end{equation} +It allows us to choose a representative with +\begin{equation} + \begin{cases} + 0 \le \frac{k_2}{\Delta \abs{k_+}} < 2 \pi, + & \qquad + k_+ \neq 0 + \\ + 0 \le \frac{k_-}{\Delta \abs{k_2}} < 2 \pi, + & \qquad + k_+ = 0, \quad k_2 \neq 0 + \end{cases}. +\end{equation} + +If we perform the computation in $\qty(u,\, v,\, z)$ coordinates we get: +\begin{equation} + \begin{split} + \Psi_{\qty[k_+\, k_-\, k_2]}\qty(u,\, v,\, z) + & = + \infinfsum{n} + \psi_{k_+\, k_-\, k_2}\qty(u,\, v,\, z + 2 \pi n) + \\ + & = + \infinfsum{n} + e^{% + i\, \qty[% + k_+ v + + + \frac{r}{2 k_+} u + + + \frac{1}{2} \qty(2 \pi \Delta)^2 k_+ u + \qty[ n + \frac{1}{2\pi} \qty( z + \frac{k_2}{\Delta k_+} ) ]^2 + ] + }, + \end{split} +\end{equation} +with $r = 2\, k_+ k_- - k_2^2$ and $\Im(k_+ u) > 0$, i.e.\ $k_+ u = \abs{k_+ u} e^{i \epsilon}$ and $\pi > \epsilon > 0$. +There is no separate dependence on $z$ and on $\frac{k_2}{\Delta k_+}$: we could fix the range $0 \le z + \frac{k_2}{\Delta k_+} < 2\pi$. +However this symmetry is broken when considering the photon eigenfunction. + +We can now use the Poisson resummation +\begin{equation} + \infinfsum{n} + e^{i\, a\, (n + b)^2} + = + \int \dd{s}\, + \delta_P(s) e^{i\, a\, (s + b)^2} + = + \qty(2\pi)^2 + \frac{e^{-i\, \qty( \frac{\pi}{4} + \frac{1}{2} arg(a) ) }}{2 \sqrt{\pi \abs{a}}} + \infinfsum{m} e^{-i\, \frac{\pi^2 m^2}{a} + i\, 2 \pi b m}, +\end{equation} +to finally get:\footnotemark{} +\footnotetext{% + In the expression we insert the variables $\vec{k}$ and $\vec{x}$ for completeness. + We also set $r = 2\, k_+ k_- -k_2^2 - \vec{k}^2$. +} +\begin{equation} + \begin{split} + \Psi_{[k_+\, k_-\, k_2\, \vec{k}]}\qty(u,\, v,\, z,\, \vec{x}) + & = + \sqrt{2\pi}~ + \frac{2 e^{-i \frac{\pi}{4}}}{\Delta} + \\ + & \times + \infinfsum{l} + \qty[ + \frac{1}{\sqrt{\abs{k_+ u}}} + e^{% + i\, \qty[% + k_+ v + + + l z + - + \frac{l^2}{2 \Delta^2 k_+}\, \frac{1}{u} + + + \frac{r + \vec{k}^2}{2 k_+} u + + + \vec{k} \cdot \vec{x} + ] + } + ] + e^{i\, l\, \frac{k_2}{\Delta k_+}} + \\ + & = + \cN\, + \infinfsum{l} + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) + e^{i\, l\, \frac{k_2}{\Delta k_+}}, + \end{split} +\label{eq:Psi_phi} +\end{equation} +when $k_+ \neq 0$ and where +\begin{equation} + \cN + = + \sqrt{\frac{\qty(2\pi)^D}{\pi \Delta}} + \frac{e^{-i \frac{\pi}{4}}}{\pi}. +\end{equation} +The fact that $\Psi$ depends only on the equivalence class $\qty[k_+\, k_-\, k_2\, k]$ allows us to restrict $0 \le \frac{k_2}{\Delta\, \abs{k_+}} < 2 \pi$ so that we can invert the previous expression and get: +\begin{equation} + \begin{split} + \phi_{\kmkr}\qty(u,\, v,\, z,\, \vec{x}) + & = + \frac{1}{\cN}\, + \frac{1}{2 \pi \Delta \abs{k_+}} + \finiteint{k_2}{0}{2 \pi \Delta \abs{k_+}} + e^{-i\, l\, \frac{k_2}{\Delta k_+}}\, + \Psi_{\qty[k_+\, k_-\, k_2\, k]}\qty(u,\, v,\, z,\, \vec{x}). + \end{split} +\end{equation} + + +\subsubsection{Spin 1 Wave Function from Minkowski space} + +We go through the steps in the previous case for an electromagnetic wave. +We concentrate on $x^+$, $x^-$ and $x^2$ coordinates and reinstate $\vec{x}$ at the end. +We start with the usual plane wave in flat space $\psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}$ and we express it in both Minkowski and orbifold coordinates. +We use the notation $\psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}$ to stress that it is the eigenfunction and not the field which is obtained as +\begin{equation} + A_{\mu}(x)\, \dd{x}^{\mu} + = + \int \dd[3]{k}\, + \sum_{\qty{\epsilon_+,\, \epsilon_-,\, \epsilon_2}} + \psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}, +\end{equation} +where the sum is performed over $\epsilon_+$, $\epsilon_-$, $\epsilon_2$ independent and compatible with $k$. +The explicit expression for the eigenfunction with constant $\epsilon_+$, $\epsilon_-$ and $\epsilon_2$ is:\footnotemark{} +\footnotetext{% + We introduce the normalization factor $\cN$ in order to have a less cluttered relation between $\epsilon$ and $\cE$. +} +\begin{equation} + \begin{split} + \cN\, + \psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}\qty(x^+,\, x^-,\, x^2) + & = + \qty(\epsilon_+ \dd{x^+} + \epsilon_- \dd{x^-} + \epsilon_2 \dd{x^2})\, + e^{i\, \qty( k_+ x^+ + k_- x^- + k_2 x^2 )} + \\ + & = + \qty( \epsilon_u\, \dd{u} + \epsilon_z\, \dd{z} + \epsilon_v\, \dd{v})\, + \\ + & \times + e^{% + i\, \qty[% + k_+ v + + + \frac{2\, k_+ k_- - k_2^2}{2 k_+} u + + + \frac{1}{2} \Delta^2 k_+ u \qty( z + \frac{k_2}{\Delta k_+})^2 + ] + } + \\ + & = + \cN + \psi^{[1]}_{k_+\, k_-\, k_2;\, \epsilon_+\, \epsilon_-\, \epsilon_2}\qty(u,\, v,\, z), + \end{split} +\end{equation} +with +\begin{equation} + \begin{split} + \epsilon_v & = \epsilon_+, + \\ + \epsilon_u(z) + & = + \epsilon_- + (\Delta z)\, \epsilon_2 + (\frac{1}{2} \Delta^2 z^2)\, \epsilon_+, + \\ + \epsilon_z(u,\, z) + & = + \qty(\Delta u)\, + \qty(\epsilon_2 + \Delta z\, \epsilon_+ ). + \end{split} +\end{equation} +Notice that we are not imposing any gauge condition. +Moreover if $(\epsilon_+,\, \epsilon_-,\, \epsilon_2)$ are constant then $(\epsilon_u,\, \epsilon_v,\, \epsilon_z)$ are generic functions. +It is worth stressing that they are not the polarizations in the orbifold which are in any case constant: the fact that they depend on the coordinates is simply the statement that not all eigenfunctions of the vector d'Alembertian are equal. + +Building the corresponding function on the orbifold amounts to summing the images created by the orbifold group: +\begin{equation} + \begin{split} + \cN + \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) + = + \infinfsum{n} + \vec{\epsilon} \cdot \qty( \cK^{-n} \dd{x})~ + \psi_{k}\qty( \cK^{-n} x) + = + \infinfsum{n} + \cK^n + \vec{\epsilon} \cdot \dd{x}~ + \psi_{\cK^n k}\qty(x). + \end{split} +\end{equation} +Under the action of the Killing vector $\epsilon$ transforms exactly as the $k$ since it is induced by $\epsilon \cdot \cK^n \dd{x} = \cK^{-n} \epsilon \cdot \dd{x}$, that is: +\begin{equation} + \epsilon + = + \mqty(% + \epsilon_+ \\ \epsilon_2 \\ \epsilon_- + ) + \equiv + \cK^{-n} \epsilon + = + \mqty(% + \epsilon_+ + \\ + \epsilon_2 + n\, \qty(2 \pi \Delta)\, \epsilon_+ + \\ + \epsilon_- + n\, \qty(2 \pi \Delta)\, \epsilon_2 + \frac{1}{2} n^2 \qty(2 \pi \Delta)^2 \epsilon_+ + ). +\end{equation} +However the pair $\qty(\vec{k},\, \vec{\epsilon})$ transforms with the same $n$ since both are ``dual'' to $x$, i.e.\ their transformation rules are dictated by $x$. +There is therefore only one equivalence class $\qty[\vec{k},\, \vec{\epsilon}]$ and not two separate classes $\qty[\vec{k}]$, $\qty[\vec{\epsilon}]$. +In other words, a representative of the combined equivalence class is the one with $0 \le k_2 < 2 \pi \Delta \abs{k_+}$ when $k_+ \neq 0$. + +%%% TODO %%% + +We now proceed to find the eigenfunctions on the orbifold +in orbifold coordinates. +We notice that $\dd{u}, \dd{v}$ and $\dd{z}$ are +invariant and therefore their coefficients in $a$ are as well. +So we write +\begin{align} +\cN + \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) + =& + \sum_n \epsilon \cdot ( \cK^{ n } \dd x)\, + \psi_{k}( \cK^{ n } x) +\nonumber\\ + =& + \dd{v}\, + \left[ \epsilon_+ \sum_n \psi_{ k}( \cK^{ n } x) \right] + + + \dd{z}\, (\Delta u) + \left[ + \epsilon_2 \sum_n \psi_{ k}( \cK^{ n } x) + + + \epsilon_+ \Delta \sum_n (z + 2\pi n) \psi_{ k}(\cK^{ n }x) + \right] + \nonumber\\ + &+ + \dd{u} + \left[ + \epsilon_- \sum_n \psi_{ k}(\cK^{ n }x) + + + \epsilon_2 \Delta \sum_n (z + 2\pi n) \psi_{ k}(\cK^{ n }x) + + + \frac{1}{2} + \epsilon_+ \Delta^2 + \sum_n (z + 2\pi n)^2 \psi_{ k}(\cK^{ n }x) + \right] +. +\end{align} +From direct computation we get\footnote{Notice that these expressions may be written using Hermite polynomials.} +\begin{align} + & + \sum_n (z + 2\pi n) \psi_{ k}(\cK^{ n }x) + = + \qty( + \frac{1}{i \Delta\, u} + \frac{\partial}{\partial k_2} + - + \frac{k_2}{\Delta k_+} + ) \Psi_{[k]}\qty(\qty[x]) + \nonumber\\ +& + \sum_n (z + 2\pi n)^2 \psi_{ k}(\cK^{ n }x) + = + \qty( + \frac{1}{i \Delta\, u} + \frac{\partial}{\partial k_2} + - + \frac{k_2}{\Delta k_+} + )^2 \Psi_{[k]}\qty(\qty[x]) + . +\label{eq:sum_z_psi} +\end{align} + Then it follows that +\begin{align} +\cN + \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) + = + & + \dd{v}\, + \left[ \epsilon_+\, \Psi_{[k]}\qty(\qty[x]) + \vphantom{\frac{\partial}{\partial k_2}} + \right] + \nonumber\\ + & + + + \dd{z}\, (\Delta u) + \left[ + \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} + \Psi_{[k]}\qty(\qty[x]) + + + \epsilon_+ \, \frac{-i}{u} \frac{\partial}{\partial k_2} \Psi_{[k]}\qty(\qty[x]) + \right] + \nonumber\\ + &+ + \dd{u} + \Biggl[ + \qty( + \epsilon_- - \epsilon_2 \frac{k_2}{k_+} + + \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2 + ) + \, \Psi_{[k]}\qty(\qty[x]) + + + \frac{i}{2 u} \frac{\epsilon_+}{k_+ } \, \Psi_{[k]}\qty(\qty[x]) + \nonumber\\ + &\phantom{d u [} + + + \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} + \frac{-i}{ u} \frac{\partial}{\partial k_2} \Psi_{[k]}\qty(\qty[x]) + + + \frac{1}{2} + \epsilon_+ + \frac{-1}{ u^2} \frac{\partial^2}{\partial k_2{}^2} \Psi_{[k]}\qty(\qty[x]) + \Biggr] + , + \label{eq:a_uvz_from_covering} +\end{align} +where many coefficients of $\Psi$ or its derivatives contain +$k_2$. +They cannot be expressed using the quantum +numbers of the orbifold $\kmkr$ but are invariant on the orbifold and +therefore are new orbifold quantities which we can interpret as +orbifold polarizations. +Using \eqref{eq:Psi_phi} we can finally write +\begin{align} +% \cN + \Psi^{[1]}_{[k,\, \epsilon]}\qty(\qty[x]) + = +% \cN + \sum_l + & +% \frac{1}{\sqrt{|k_+|}} +\phi_{\kmkr}(u,v,z,\vec{x}) + e^{ i l \frac{k_2}{ \Delta k_+} } + \Bigg\{ + \dd{v} \Bigl[ \epsilon_+ \Bigr] + \nonumber\\ + &+ + \dd{z}\, (\Delta u) + \Biggl[ + \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} + + + \epsilon_+ \frac{1}{\Delta u} \frac{l}{k_+} + \Biggr ] + \nonumber\\ + &+ + \dd{u} + \Biggl[ +% \epsilon_- + \qty( + \epsilon_- - \epsilon_2 \frac{k_2}{k_+} + + \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2 + ) + + + \frac{i}{2 u} \frac{\epsilon_+}{k_+ } + \nonumber\\ + & + \phantom{ \dd{u} [ } + + + \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} + \frac{1}{u} \frac{l}{\Delta k_+} + + + \epsilon_+ \frac{1}{2 u^2} + \qty(\frac{l}{ \Delta k_+} )^2 + \Biggr] + \Bigg\} + . + \label{eq:spin1_from_covering} +\end{align} + +If we compare with \eqref{eq:Orbifold_spin1_pol} we find +\begin{align} + \cE_{\kmkr\, \underline{v}} &= \epsilon_+ + \nonumber\\ + \cE_{\kmkr\, \underline{z}} &= \mathrm{sign}(u) + \frac{\epsilon_2 k_+ -\epsilon_+ k_2}{k_+} + \nonumber\\ + \cE_{\kmkr\, \underline{u}} &= + \epsilon_- - \epsilon_2 \frac{k_2}{k_+} + + \frac{1}{2} \epsilon_+ \qty( \frac{k_2}{k_+} )^2 + , + \label{eq:eps_calE} +\end{align} +which implies that the true polarizations $(\epsilon_+, \epsilon_-, \epsilon_2)$ +and +$\cE_{\kmkr\, \underline{*}}$ are constant as it turns out from direct computation. + +A different way of reading the previous result is that the +polarizations on the orbifold are the coefficients of the highest power +of $u$. + +We can also invert the previous relations to get +\begin{align} + \epsilon_+ + &= + \cE_{\kmkr\, \underline{v}} + \nonumber\\ + \epsilon_2 + &= + \cE_{\kmkr\, \underline{z}} \mathrm{sign}(u) + \frac{k_2}{k_+} \cE_{\kmkr\, \underline{v}} + \nonumber\\ + \epsilon_- &= + \cE_{\kmkr\, \underline{u}} + + \frac{k_2}{k_+} \cE_{\kmkr\, \underline{z}} \mathrm{sign}(u) + + \frac{1}{2} \qty( \frac{k_2}{k_+} )^2 \cE_{\kmkr\, \underline{v}} + , + \label{eq:calE_eps} +\end{align} +and use them in Lorenz gauge $k \cdot \epsilon=0$ in order to get the +expression of Lorenz gauge with orbifold polarizations. +If the definition of orbifold polarizations is right the result cannot +depend on $k_2$ since $k_2$ is not a quantum number of orbifold +eigenfunctions. +Taking in account $k_- = \frac{\vec{k}^2+ k_2^2 + r}{2 k_+}$ in $k \cdot +\epsilon=0$ +we get +exactly the expression for the Lorenz gauge for orbifold polarizations +\eqref{eq:Lorenz_gauge}. + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Tensor Wave Function from Minkowski space} +Once again, we can use the analysis of the previous section in the case of a +second order symmetric tensor wave function. +Again we suppress the dependence on $\vec{x}$ and $\vec{k}$ with a caveat: the Minkowskian polarizations $S_{+\, i}$, $S_{-\, i}$ and $S_{2\, i}$ do transform non trivially, therefore we give the full expressions in Appendix~\ref{app:NO_tensor_wave} even if these components contribute in a somewhat trivial way since they behave effectively as a vector of the orbifold. + +We start with the usual wave in flat space and we express either in +the Minkowskian coordinates +\begin{alignat}{4} +\cN + \psi^{[2]}_{k\, S}(x^+,x^-,x^2) + &= S_{\mu \nu}\, \psi_k(x)\, \dd x^{\mu}\, \dd x^{\nu} + \nonumber\\ + &= + % (\epsilon_+ d x^+ + \epsilon_- dx^- + \epsilon_2 d x^2) + % \otimes + % (\etp d x^+ + \etm dx^- + \etd d x^2) +% \qty( + ( + S_{+\, +}\, \dd x^+\, \dd x^+ + & + + + 2 S_{+\, 2}\, \dd x^+\, \dd x^2 + & + + + 2 S_{+\, -}\, \dd x^+\, \dd x^- + \nonumber\\ + & + & + + + 2 S_{2\, 2}\, \dd x^2\, \dd x^2 + & + + + 2 S_{2\, -}\, \dd x^2\, \dd x^- + \nonumber\\ + & + & + & + + + 2 S_{-\, -}\, \dd x^-\, \dd x^- + ) + e^{i\qty( k_+ x^+ + k_- x^- + k_2 x^2 )} + \nonumber\\ + , +\end{alignat} +or in orbifold coordinates +\begin{align} +\cN + \psi^{[2]}_{k\, S}(x) + =& S_{\alpha \beta}\, \psi_k(x)\, \dd x^\alpha\, \dd x^\beta + \nonumber\\ + =& + \Bigl\{ + (\dd{v})^2\, + [S_{+\, +}] + \nonumber\\ + & + + + \dd{v}\, \dd{z}\,\Delta u + [ 2 S_{+\, 2} + + S_{+\, +} \Delta z ] + \nonumber\\ + & + + + \dd{v}\, \dd{u}\, + [ 2 S_{+\, -} + + 2 S_{+\, 2} \Delta z + + S_{+\, +} \Delta^2 z^2 ] + \nonumber\\ + & + + + \dd{z}^2\,\Delta^2 u^2\, + [ S_{2\, 2} + + 2 S_{+\, 2} \Delta z + + S_{+\, +} \Delta^2 z^2 ] + % \nonumber\\ + % & + % + + % d z\, d v\, + % [ S_{2\, 2} \Delta^2 u^2 + S_{+\, 2} \Delta^3 u^2 z + % + \frac{1}{4} S_{+\, +} \Delta^4 u^2 z^2 ] + \nonumber\\ + & + + + \dd{z}\, \dd{v}\, \Delta u\, + [ 2 S_{-\, 2} + + 2 (S_{2\, 2} + S_{+\, -} ) \Delta z + + 3 S_{+\, 2} \Delta^2 z^2 + + S_{+\, +} \Delta^3 z^3 + ] + \nonumber\\ + & + + + \dd{u}^2\, + [ + S_{-\, -} + + 2 S_{-\, 2} \Delta z + + (S_{2\, 2} + S_{+\, -}) \Delta^2 z^2 + + S_{+\, 2} \Delta^3 z^3 + + \frac{1}{4} S_{+\, +} \Delta^4 z^4 + ] + \Bigr\} + \nonumber\\ + &\times + e^{i\left[ k_+ v + + + \frac{2 k_+ k_- - k_2^2}{2 k_+} u + + \frac{1}{2} \Delta^2 k_+ u \qty( z+ \frac{k_2}{\Delta k_+})^2 + \right]} +. +\end{align} +Now we define the tensor on the orbifold as a sum over all images as +\begin{align} +\cN + \Psi^{[2]}_{[k\, S]}\qty(\qty[x]) + &= + \sum_n ( \cK^{ n }d x) \cdot S \cdot ( \cK^{ n } \dd x)~ + \psi_{k}( \cK^{ n } x) + \nonumber\\ + &= + \sum_n \dd x \cdot ( \cK^{ - n }S ) \cdot \dd x~ + \psi_{ \cK^{ -n } k}( x) + . + \end{align} + In the last line we have defined the induced action of the Killing vector on + $(k, S)$ which can be explicitely written as + \begin{align} + \cK^{-n} + \qty( + \begin{array}{c} + S_{ + + }\\ + S_{ + 2 }\\ + S_{ + - }\\ + S_{ 2 2 }\\ + S_{ 2 - }\\ + S_{ - - } + \end{array} + ) + = + % + \qty( + \begin{array}{c} + S_{ + + }\\ + S_{ + 2 } + n \Delta S_{ + + }\\ + S_{ + - } + n \Delta S_{ + 2 } + \frac{1}{2} n^2 \Delta^2 S_{ + + }\\ + S_{ 2 2 } + 2 n \Delta S_{ + 2 } + n^2 \Delta^2 S_{ + + }\\ + S_{ 2 - } + n \Delta (S_{ 2 2 } +S_{ + - }) + + \frac{3}{2} n^2 \Delta^2 S_{ + 2 } + \frac{1}{2} n^3 \Delta^3 S_{ + + }\\ + S_{ - - } + 2 n \Delta S_{ - 2 } + + n^2 \Delta^2 (S_{ 2 2 } + S_{ + - } ) + n^3 \Delta^3 S_{ + 2 } + + \frac{1}{4} n^4 \Delta^4 S_{ + + } + \end{array} + ) +. + \end{align} + + In orbifold coordinates + to compute the tensor on the orbifold simply + amounts to sum over all the shifts + $z \rightarrow (z+2\pi n)$ and the use of the generalization of + \eqref{eq:sum_z_psi}, i.e. to substitute + $(\Delta\,z)^j \psi_k \rightarrow + \qty( + \frac{1}{i u} \frac{\partial}{\partial k_2} +- \frac{ k_2}{ \Delta k_+} + )^j + \Psi_{[k]}\qty(\qty[x])$. + When expressing all in the $\phi$ basis + this last step is equivalent to + $(\Delta\, z)^j \psi_k \rightarrow + \qty( \frac{l}{\Delta\, u\, k_+} )^j + + \dots + $. + We identify the basic polaritazions on the orbifold by + considering the highest power in $u$ and get + \begin{align} + \cS_{u\,u} + &= + \frac{1}{4}{{K^4\,S_{+\,+}}} + +K^2\,S_{+\,-} + -K^3\,S_{+\,2} + +S_{-\,-} + -2\,K\,S_{-\,2} + +S_{2\,2}\,K^2 +\nonumber\\ + \cS_{u\,v} + &= + \frac{1}{2} {{K^2\,S_{+\,+}}} + +S_{+\,-} + -K\,S_{+\,2} + \nonumber\\ + \cS_{u\,z} + &= + - \frac{1}{2} {{K^3\,S_{+\,+}}} + -K\,S_{+\,-} + +\frac{3}{2} {{K^2\,S_{+\,2}}} + +S_{-\,2} + -K\, S_{2\,2} +\nonumber\\ + \cS_{v\,v} + &= + S_{+\,+} +\nonumber\\ + \cS_{v\,z} + &= + S_{+\,2}-K\,S_{+\,+} +\nonumber\\ + \cS_{z\,z} + &= + K^2\,S_{+\,+}-2\,K\,S_{+\,2}+S_{2\,2} +. + \end{align} +with $K= \frac{k_2}{ k_+}$. +The previous equations can be inverted into +\begin{align} +S_{-\,-} +&= +% {{K^2\,\qty(4\,\cS_{z\,z}+4\,\cS_{u\,v})+4\,K^3\,\cS_{v\,z}+K^4\,\cS_{v\,v}+8\,K\,\cS_{u\,z}+4\,\cS_{u\,u}}\over{4}} + K^2\,\qty(\cS_{z\,z}+\cS_{u\,v}) + +K^3\,\cS_{v\,z} + +\frac{1}{4} K^4\,\cS_{v\,v} + +2\,K\,\cS_{u\,z} + +\cS_{u\,u} +\nonumber\\ +S_{+\,-} +&= +%{{2\,K\,\cS_{v\,z}+K^2\,\cS_{v\,v}+2\,\cS_{u\,v}}\over{2}} + K\,\cS_{v\,z} + +\frac{1}{2} K^2\,\cS_{v\,v} + +\cS_{u\,v} +\nonumber\\ +S_{-\,2} +&= +% {{K\,\qty(2\,\cS_{z\,z}+2\,\cS_{u\,v})+3\,K^2\,\cS_{v\,z}+K^3\,\cS_{v\,v}+2\,\cS_{u\,z}}\over{2}} + K\,\qty(\cS_{z\,z}+\cS_{u\,v}) + +\frac{3}{2}\,K^2\,\cS_{v\,z} + +\frac{1}{2} K^3\,\cS_{v\,v} + +\cS_{u\,z} +\nonumber\\ +S_{+\,+} +&= +\cS_{v\,v} +\nonumber\\ +S_{+\,2} +&= +\cS_{v\,z}+K\,\cS_{v\,v} +\nonumber\\ +S_{2\,2} +&= +\cS_{z\,z}+2\,K\,\cS_{v\,z}+K^2\,\cS_{v\,v} +. +\end{align} +Since we plan to use the previous quantities in the case of the first +massive string state we compute the relevant quantities: the trace +\begin{equation} + \tr(S)=\cS_{z\,z}-2\,\cS_{u\,v} +\end{equation} +and the transversality conditions +\begin{align} + % + % v + trans ~\cS_{v} + =& + (k\cdot S)_{+} + = + -\frac{\qty(r + \vec{k}^2)}{2\, k_+}\, + \cS_{v\,v} + -k_{+}\,\cS_{u\,v}, + \nonumber\\ + % + % z + trans ~\cS_{z} + =& + (k\cdot S)_{2} + -K (k\cdot S)_{+} += + -\frac{\qty(r + \vec{k}^2)}{2\, k_+}\, + \cS_{v\,z} + -k_{+}\,\cS_{u\,z}, +\nonumber\\ + trans ~\cS_{u} + =& + (k\cdot S)_{-} + -K (k\cdot S)_{2} + %+ \frac{1}{2} K^2 (k\cdot S)_{+} + + \frac{1}{2} K^2 (k\cdot S)_{+} + = + -\frac{\qty(r + \vec{k}^2)}{2\, k_+}\, + \cS_{u\,v} + -k_{+}\,\cS_{u\,u} + . +\end{align} +where we used $k_-= (r+\vec{k}^2+k_2^2)/(2 k_+)$. +These conditions correctly do no depend on $K$ since $k_2$ is +not an orbifold quantum number. + + +The final expression for the orbifold symmetric tensor is + \begin{align} + % \cN + \Psi^{[2]}_{[k,\, S]}\qty([x]) + & + = +% \cN + \sum_l +\phi_{\kmkr}(u,v,z,\vec{x}) + e^{ i l \frac{k_2}{ \Delta k_+} } +\nonumber\\ +% +% v v + & + \Big\{ + (\dd{v})^2\, + [\cS_{v v} ] + \nonumber\\ +% +% v z + & + + +2 + \Delta\, u\, + \dd{v}\, \dd{z}\, +% [ 2 S_{+\, 2} +% + S_{+\, +} \frac{l}{ \Delta\, u\,k_+} ] + \Bigl[ + \cS_{v\,z} + + + \qty( + \frac{L \cS_{v\,v}}{\Delta} + ) + \frac{1}{u} + \Bigr] + \nonumber\\ +% +% v u + & + + + 2 + \dd{v}\, \dd{u}\, +% [ 2 S_{+\, -} +% + 2 S_{+\, 2} \frac{l}{\Delta\, u\, k_+} +% + S_{+\, +} \qty(\frac{l}{\Delta\, u\, k_+})^2 ] +\Bigl[ +\cS_{u\,v} ++ +\qty( +\frac{L\,\cS_{v\,z}}{\Delta}+\frac{i\,\cS_{v\,v}}{2\,k_{+}} +) +\frac{1}{u} ++ +\qty( +\frac{L^2\,\cS_{v\,v}}{2\,\Delta^2} +) +\frac{1}{u^2} +\Bigr] + \nonumber\\ + % +% z z + & + + + (\Delta\, u)^2 + \dd{z}^2\, +% [ S_{2\, 2} +% + 2 S_{+\, 2} \frac{l}{\Delta\, u\, k_+} +% + S_{+\, +} \qty(\frac{l}{\Delta\, u\, k_+})^2 ]\, + \Bigl[ + \cS_{z\,z} ++ +\qty( +\frac{2\,L\,\cS_{v\,z}}{\Delta}+\frac{i\,\cS_{v\,v}}{k_{+}} +) + \frac{1}{u} + + +\qty( +\frac{L^2\,\cS_{v\,v}}{\Delta^2} +) +\frac{1}{u^2} + \Bigr] + \nonumber\\ + % + % z v + & + + + 2 + \Delta u\, + \dd{z}\, \dd{u}\, + % [ 2 S_{-\, 2} + % + 2 ( S_{2\, 2} + S_{+\, -} ) \frac{l}{\Delta\, u\, k_+} + % + 3 S_{+\, 2} \qty(\frac{l}{\Delta\, u\,k_+})^2 + % + S_{+\, +} \qty(\frac{l}{\Delta\, u\, k_+})^3 + % ]\, + \Bigl[ + \cS_{u\,z} ++ +\qty( + \frac{L\,\cS_{z\,z}}{\Delta} + +\frac{3\,i\,\cS_{v\,z}}{2\,k_{+}}+\frac{L\,\cS_{u\,v}}{\Delta} +) + \frac{1}{u} + + +\qty( + \frac{3\,L^2\,\cS_{v\,z}}{2\,\Delta^2} + +\frac{3\,i\,L\,\cS_{v\,v}}{2\,\Delta\,k_{+}} +) + \frac{1}{u^2} + \nonumber\\ + & + \phantom{+\Delta u\,d z\, d v\,} + + +\qty( +\frac{L^3\,\cS_{v\,v}}{2\,\Delta^3} +) +\frac{1}{u^3} + \Bigr] + \nonumber\\ + % + % u u + & + + + \dd{u}^2\, + % [ + % S_{-\, -} + % + 2 S_{-\, 2} \frac{l}{\Delta\, u\,k_+} + % + (S_{2\, 2} + S_{+\, -}) \qty(\frac{l}{\Delta\, u\,k_+})^2 + % + S_{+\, 2} \qty(\frac{l}{\Delta\, u\,k_+})^3 + % + \frac{1}{4} S_{+\, +} \qty(\frac{l}{\Delta\, u\,k_+})^4 + % ] + % + \Bigl[ +\cS_{u\,u} ++ +\qty( +\frac{i\,\cS_{z\,z}}{k_{+}}+\frac{2\,L\,\cS_{u\,z}}{\Delta}+\frac{i\,\cS_{u\,v}}{k_{+}} +) + \frac{1}{u} + + +\qty( + \frac{L^2\,\cS_{z\,z}}{\Delta^2}+\frac{3\,i\,L\,\cS_{v\,z}}{\Delta\,k_{+}} + -\frac{3\,\cS_{v\,v}}{4\,k_{+}^2}+\frac{L^2\,\cS_{u\,v}}{\Delta^2} +) + \frac{1}{u^2} + \nonumber\\ + & + \phantom{ + d u^2\, } + + +\qty( +\frac{L^3\,\cS_{v\,z}}{\Delta^3}+\frac{3\,i\,L^2\,\cS_{v\,v}}{2\,\Delta^2\,k_{+}} +) + \frac{1}{u^3} + + +\qty( +\frac{L^4 \cS_{v\,v}}{4 \Delta^4} +) +\frac{1}{u^4} + \Bigr] + \Bigr\} + , + \end{align} + with $L=\frac{l}{k_+}$. \subsection{Summary and Conclusions} @@ -1133,7 +2070,7 @@ In the previous analysis it seems that string theory cannot do better than field Moreover when spacetime becomes singular, the string massive modes are not anymore spectators. Everything seems to suggest that issues with spacetime singularities are hidden into contact terms and interactions with massive states. This would explain in an intuitive way why the eikonal approach to gravitational scattering works well: the eikonal is indeed concerned with three point massless interactions. -In fact it appears that the classical and quantum scattering on an electromagnetic wave~\cite{Jackiw:1992:ElectromagneticFieldsMassless} or gravitational wave \cite{tHooft:1987:GravitonDominanceUltrahighenergy} in \bo and \nbo are well behaved. +In fact it appears that the classical and quantum scattering on an electromagnetic wave~\cite{Jackiw:1992:ElectromagneticFieldsMassless} or gravitational wave~\cite{tHooft:1987:GravitonDominanceUltrahighenergy} in \bo and \nbo are well behaved. From this point of view the ACV approach~\cite{Soldate:1987:PartialwaveUnitarityClosedstring,Amati:1987:SuperstringCollisionsPlanckian} may be more sensible, especially when considering massive external states~\cite{Black:2012:HighEnergyString}. Finally it seems that all issues are related with the Laplacian associated with the space-like subspace with vanishing volume at the singularity. If there is a discrete zero eigenvalue the theory develops divergences.