Up to NS fermions

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>

sec/app/parameters.tex

@@ -15,9 +15,9 @@ In the main text we set
 where $\cL(\vb{n}_{\vb{\infty}}) \in \SU{2}$.
 The previous equation implies
 \begin{equation}
-  \left( D\, \rM_{\vb{\infty}}\, D^{-1} \right)^\dagger
+  \qty( D\, \rM_{\vb{\infty}}\, D^{-1} )^\dagger
   =
-  \left( D\, \rM_{\vb{\infty}}\, D^{-1} \right)^{-1},
+  \qty( D\, \rM_{\vb{\infty}}\, D^{-1} )^{-1},
 \end{equation}
 which can be rewritten as
 \begin{equation}
@@ -147,7 +147,7 @@ which is satisfied by:
     k'_{ab} \in \Z,
   \end{split}
 \end{equation}
-where $p^{(L)},\, q^{(L)} \in \left\lbrace 0, 1 \right\rbrace$.
+where $p^{(L)},\, q^{(L)} \in \qty{ 0, 1 }$.
 Notice that changing the value of $p^{(L)}$ corresponds to swapping $a$ and $b$: since the hypergeometric function is symmetric in those parameters we can fix $p^{(L)}=0$.
 Redefining $k'$ we can always set $q^{(L)}=0$.
 We therefore have:
@@ -170,16 +170,16 @@ We cannot thus choose a vanishing $k_{\delta^{(L)}_{\vb{\infty}}}$ but we have t
 
 We find a third relation by considering the entry
 \begin{equation}
-  \Im\left(
+  \Im\qty(
     e^{+2\pi i \delta_{\vb{\infty}}^{(L)}}\,
     D^{(L)}\,
     \rM_{\vb{\infty}}^{(L)}\,
-    \left( D^{(L)} \right)^{-1}
-  \right)_{11}
+    \qty( D^{(L)} )^{-1}
+  )_{11}
   =
-  \Im\left(
+  \Im\qty(
     \cL(n_{\vb{\infty}})
-  \right)_{11}.
+  )_{11}.
 \end{equation}
 Using
 \begin{equation}
@@ -221,7 +221,7 @@ We then write
   \qquad
   k_{abc}\in \Z,
 \end{equation}
-with $f^{(L)} \in \left\lbrace 0, 1 \right\rbrace$.
+with $f^{(L)} \in \qty{ 0, 1 }$.
 The request
 \begin{equation}
   A
@@ -284,10 +284,10 @@ So far we can summarise the results in
 
 $K^{(L)}$ is finally determined from
 \begin{equation}
-  \left( D^{(L)}\, \rM_{\vb{\infty}}\, \left( D^{(L)} \right)^{-1} \right)_{21}
+  \qty( D^{(L)}\, \rM_{\vb{\infty}}\, \qty( D^{(L)} )^{-1} )_{21}
   =
   e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\,
-  \left( \cL(n_{\vb{\infty}}) \right)_{21},
+  \qty( \cL(n_{\vb{\infty}}) )_{21},
   \label{eq:fixing_K_21}
 \end{equation}
 and get:
@@ -310,7 +310,7 @@ We check the consistency condition \eqref{eq:K_consistency_condition} using~\eqr
 The result is
 \begin{equation}
   \begin{split}
-    \left( K^{(L)} \right)^{-1}
+    \qty( K^{(L)} )^{-1}
     & =
     \frac{(-1)^{m_a + m_b + m_c}}{2 \pi^2}\,
     \cG(1 - a^{(L)},\, 1 - b^{(L)},\, 2 - c^{(L)})\,
@@ -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}
-  \left( D^{(L)}\, \rM_{\vb{\infty}}\, \left( D^{(L)} \right)^{-1} \right)_{12}
+  \qty( D^{(L)}\, \rM_{\vb{\infty}}\, \qty( D^{(L)} )^{-1} )_{12}
   =
   e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\,
-  \left( \cL(n_{\vb{\infty}}) \right)_{12},
+  \qty( \cL(n_{\vb{\infty}}) )_{12},
 \end{equation}
 instead of \eqref{eq:fixing_K_21}.