From 28c749e5683ce3c75e2d37febe82f969c8873a95 Mon Sep 17 00:00:00 2001
From: Riccardo Finotello <riccardo.finotello@gmail.com>
Date: Fri, 25 Sep 2020 17:28:21 +0200
Subject: [PATCH] Up to NS fermions

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
---
 img/complex-plane.pdf      | Bin 0 -> 47883 bytes
 img/in-annihilators.pdf    | Bin 0 -> 61229 bytes
 sec/app/isomorphism.tex    |  20 +-
 sec/app/parameters.tex     |  28 +-
 sec/part1/dbranes.tex      | 220 +++++-----
 sec/part1/fermions.tex     | 829 +++++++++++++++++++++++++++++++++++--
 sec/part1/introduction.tex | 116 +++---
 thesis.cls                 |   6 +-
 thesis.tex                 |   5 +
 9 files changed, 984 insertions(+), 240 deletions(-)
 create mode 100644 img/complex-plane.pdf
 create mode 100644 img/in-annihilators.pdf

diff --git a/img/complex-plane.pdf b/img/complex-plane.pdf
new file mode 100644
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diff --git a/sec/app/isomorphism.tex b/sec/app/isomorphism.tex
index 95a471b..ee48cad 100644
--- a/sec/app/isomorphism.tex
+++ b/sec/app/isomorphism.tex
@@ -33,7 +33,7 @@ The parametrisation is such that:
   U(\widehat{\vb{n}})
   \label{eq:U_props}
 \end{eqnarray}
-where $\sigma^2$ is the second Pauli matrix, $\widetilde{\vb{n}} = \left( -n^1, n^2, -n^3 \right)$ and $\widehat{\vb{n}} = - \left(\frac{1}{2} -n \right)\, \frac{\vb{n}}{n}$.
+where $\sigma^2$ is the second Pauli matrix, $\widetilde{\vb{n}} = \qty( -n^1, n^2, -n^3 )$ and $\widehat{\vb{n}} = - \qty(\frac{1}{2} -n )\, \frac{\vb{n}}{n}$.
 
 The group product of two elements $U(\vb{n} \circ \vb{m} ) = U(\vb{n})\,  U(\vb{m})$ has an explicit realisation as:
 \begin{equation}
@@ -58,17 +58,17 @@ 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 = \left( i\, \1_2,\, \vb{\sigma} \right),
+  \tau_I = \qty( i\, \1_2,\, \vb{\sigma} ),
 \end{equation}
-where $\vb{\sigma} = \left( \sigma^1,\, \sigma^2,\, \sigma^3 \right)$ are the Pauli matrices.
+where $\vb{\sigma} = \qty( \sigma^1,\, \sigma^2,\, \sigma^3 )$ are the Pauli matrices.
 It is possible to show that:
 \begin{equation}
   \begin{split}
-    \left( \tau_I \right)^{\dagger}
+    \qty( \tau_I )^{\dagger}
     & =
     \eta_{IJ}\, {\tau}^I,
     \\
-    \left( \tau^I \right)^*
+    \qty( \tau^I )^*
     & =
     -\sigma_2\, \tau_I\, \sigma_2,
   \end{split}
@@ -85,7 +85,7 @@ The following relations are then a natural consequence:
   & = &
   2\, \eta_{IJ},
   \\
-  \tr(\tau_I \left( \tau_J \right)^{\dagger})
+  \tr(\tau_I \qty( \tau_J )^{\dagger})
   & = &
   2\, \delta_{IJ}.
 \end{eqnarray}
@@ -99,7 +99,7 @@ We can recover the components using the previous properties:
   X^I
   =
   \frac{1}{2}\, \delta^{IJ}\,
-  \tr(X_{(s)} \left( \tau_J \right)^{\dagger})
+  \tr(X_{(s)} \qty( \tau_J )^{\dagger})
   =
   \frac{1}{2}\, \eta^{IJ}\, \tr(X_{(s)} \tau_J),
 \end{equation}
@@ -126,7 +126,7 @@ A rotation in spinor representation is defined as:
 \end{equation}
 and it is equivalent to:
 \begin{equation}
-  \left( X' \right)^I
+  \qty( X' )^I
   =
   \tensor{R}{^I_J}\,
   X^J
@@ -137,7 +137,7 @@ through
   =
   \frac{1}{2}
   \tr(
-    \left( \tau_I \right)^{\dagger}\,
+    \qty( \tau_I )^{\dagger}\,
     U_{L}(\vb{n})\,
     \tau_J\,
     U_{R}^{\dagger}(\vb{m})
@@ -167,7 +167,7 @@ From the second equation in \eqref{eq:tau_props} and the first equation in \eqre
 \end{equation}
 Furthermore the direct computation of the determinant of $R$ using the parametrisation~\eqref{eq:su2parametrisation} shows that $\det R = 1$.
 Finally the explicit choice of the basis $\tau$ ensures $R$ to be a real matrix which ensures $R \in \SO{4}$.
-Since $\left\lbrace U_{L},\, U_{R} \right\rbrace$ and $\left\lbrace -U_{L},\, -U_{R} \right\rbrace$ generate the same \SO{4} matrix then the correct isomorphism takes the form:
+Since $\qty{ U_{L},\, U_{R} }$ and $\qty{ -U_{L},\, -U_{R} }$ generate the same \SO{4} matrix then the correct isomorphism takes the form:
 \begin{equation}
   \SO{4}
   \cong
diff --git a/sec/app/parameters.tex b/sec/app/parameters.tex
index a799e0e..23cafea 100644
--- a/sec/app/parameters.tex
+++ b/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}.
 
diff --git a/sec/part1/dbranes.tex b/sec/part1/dbranes.tex
index fc75a81..34f4662 100644
--- a/sec/part1/dbranes.tex
+++ b/sec/part1/dbranes.tex
@@ -32,10 +32,10 @@ Using the path integral approach we can in fact separate the classical contribut
   \right\rangle
   =
   \cN
-  \left(
-    \left\lbrace x_{(t)},\, \rM_{(t)} \right\rbrace_{1 \le t \le N_B}
-  \right)\,
-  e^{-S_E\left( \left\lbrace x_{(t)}, \rM_{(t)} \right\rbrace_{1 \le t \le N_B} \right)},
+  \qty(
+    \qty{ x_{(t)},\, \rM_{(t)} }_{1 \le t \le N_B}
+  )\,
+  e^{-S_E\qty( \qty{ x_{(t)}, \rM_{(t)} }_{1 \le t \le N_B} )},
 \end{equation}
 where $\rM_{(t)}$ (for $1 \le t \le N_B$) are the monodromies induced by the twist fields, $N_B$ is the number of D-branes and $x_{(t)}$ are the intersection points on the worldsheet.
 Even though quantum corrections are crucial to the complete determination of the normalisation of the correlator, the classical contribution of the Euclidean action represents the leading term of the Yukawa couplings.
@@ -119,15 +119,15 @@ The rotation $R_{(t)}$ is actually defined in the Grassmannian:
   \in
   \mathrm{Gr}(2, 4)
   =
-  \frac{\SO{4}}{\rS\left( \OO{2} \times \OO{2} \right)},
+  \frac{\SO{4}}{\rS\qty( \OO{2} \times \OO{2} )},
 \end{equation}
 that is we just need to consider the left coset where $R_{(t)}$ is a representative of an equivalence class
 \begin{equation}
-  \left[ R_{(t)} \right]
+  \qty[ R_{(t)} ]
   =
-  \left\lbrace R_{(t)} \sim \cO_{(t)} R_{(t)} \right\rbrace,
+  \qty{ R_{(t)} \sim \cO_{(t)} R_{(t)} },
 \end{equation}
-where $\cO_{(t)} = \rS\left( \OO{2} \times \OO{2} \right)$ is defined as
+where $\cO_{(t)} = \rS\qty( \OO{2} \times \OO{2} )$ is defined as
 \begin{equation}
   \cO_{(t)}
   =
@@ -135,7 +135,7 @@ where $\cO_{(t)} = \rS\left( \OO{2} \times \OO{2} \right)$ is defined as
 \end{equation}
 with $\cO^{\parallel}_{t} \in \OO{2}$, $\cO^{\perp}_{t} \in \OO{2}$ and $\det \cO_{(t)} = 1$.
 The superscript $\parallel$ represents any of the coordinates parallel to the D-brane, while $\perp$ any of the orthogonal.
-Notice that the additional $\Z_2$ factor in $\rS\left( \OO{2} \times \OO{2} \right)$ with respect to $\SO{2} \times \SO{2}$ can be used to set $g_{(t)}^{\perp} \ge 0$.
+Notice that the additional $\Z_2$ factor in $\rS\qty( \OO{2} \times \OO{2} )$ with respect to $\SO{2} \times \SO{2}$ can be used to set $g_{(t)}^{\perp} \ge 0$.
 
 
 \subsubsection{Boundary Conditions for Branes at Angles}
@@ -150,7 +150,7 @@ We define the usual upper plane coordinates:
   =
   e^{\tau_E + i \sigma}
   & \in &
-  \ccH \cup \left\lbrace z \in \C \mid \Im z = 0 \right\rbrace,
+  \ccH \cup \qty{ z \in \C \mid \Im z = 0 },
   \\
   \bu
   =
@@ -158,20 +158,20 @@ We define the usual upper plane coordinates:
   =
   e^{\tau_E - i \sigma}
   & \in &
-  \overline{\ccH} \cup \left\lbrace z \in \C \mid \Im z = 0 \right\rbrace,
+  \overline{\ccH} \cup \qty{ z \in \C \mid \Im z = 0 },
 \end{eqnarray}
-where $\ccH = \left\lbrace z \in \C \mid \Im z > 0 \right\rbrace$ is the upper complex plane and $\overline{\ccH} = \left\lbrace z \in \C \mid \Im z < 0 \right\rbrace$ is the lower complex plane.
+where $\ccH = \qty{ z \in \C \mid \Im z > 0 }$ is the upper complex plane and $\overline{\ccH} = \qty{ z \in \C \mid \Im z < 0 }$ is the lower complex plane.
 In conformal coordinates $u$ and $\bu$, D-branes at $\sigma = 0$ and $\sigma = \pi$ are mapped onto the real axis $\Im z = 0$.
 We use the symbol $D_{(t)}$ to label both the brane and the interval representing it on the real axis of the upper half plane:
 \begin{equation}
-  D_{(t)} = \left[ x_{(t)}, x_{(t-1)} \right],
+  D_{(t)} = \qty[ x_{(t)}, x_{(t-1)} ],
   \qquad
   t = 2,\, 3,\, \dots,\, N_B,
   \qquad
   x_{(t)} < x_{(t-1)}.
 \end{equation}
 The points $x_{(t)}$ and $x_{(t-1)}$ represent the worldsheet intersection points of the brane $D_{(t)}$ with the branes $D_{(t+1)}$ and $D_{(t-1)}$ respectively.
-The choice of the intervals must be carefully considered: since the D-branes are defined modulo $N_B$, the shorthand for the interval $D_{(1)} = \left[ x_{(1)}, x_{(N_B)} \right]$ should actually be:
+The choice of the intervals must be carefully considered: since the D-branes are defined modulo $N_B$, the shorthand for the interval $D_{(1)} = \qty[ x_{(1)}, x_{(N_B)} ]$ should actually be:
 \begin{equation}
   D_{(1)}
   =
@@ -193,11 +193,11 @@ In the global coordinates system associated to the subspace $\R^4 \subset \ccM^{
     \frac{1}{4 \pi \ap}
     \iint\limits_{\R \times \R^+}
     \dd{x}\dd{y}\,
-    \left(
+    \qty(
       \ipd{x} X^I\, \ipd{x} X^J
       +
       \ipd{y} X^I\, \ipd{y} X^J
-    \right)\,
+    )\,
     \eta_{IJ},
   \end{split}
   \label{eq:string_action}
@@ -208,7 +208,7 @@ The \eom in these coordinates are:
   \ipd{u} \ipd{\bu} X^I( u, \bu )
   =
   \frac{1}{4}
-  \left( \ipd{x}^2 + \ipd{y}^2 \right) X^I( x+iy, x-iy )
+  \qty( \ipd{x}^2 + \ipd{y}^2 ) X^I( x+iy, x-iy )
   =
   0.
   \label{eq:string_equation_of_motion}
@@ -235,29 +235,29 @@ In the well adapted frame~\eqref{eq:well-adapt-embed} we describe an open string
   m = 3,\, 4,
   \label{eq:dirichlet_bc}
 \end{eqnarray}
-where $x \in D_{(t)} = \left[ x_t, x_{t-1} \right]$ and the index $i$ labels the Neumann boundary conditions while $m$ labels the Dirichlet coordinates associated to the direction orthogonal to the D-branes.
+where $x \in D_{(t)} = \qty[ x_t, x_{t-1} ]$ and the index $i$ labels the Neumann boundary conditions while $m$ labels the Dirichlet coordinates associated to the direction orthogonal to the D-branes.
 
 As the presence of $g_{(t)}^m$ in \eqref{eq:brane_rotation} and \eqref{eq:dirichlet_bc} may complicate the analysis, we consider the derivative along the boundary direction of \eqref{eq:dirichlet_bc} to remove the dependence on the translation vector.
 This procedure produces simpler boundary conditions which are nevertheless not equivalent to the original~\eqref{eq:neumann_bc} and \eqref{eq:dirichlet_bc}: they will be recovered later by adding further constraints.
 The simpler boundary conditions we consider in the global coordinates are:
 \begin{eqnarray}
-  \tensor{\left( R_{(t)} \right)}{^i_J}
+  \tensor{\qty( R_{(t)} )}{^i_J}
   \eval{\ipd{\sigma} X^J( \tau, \sigma )}_{\sigma = 0}
   & = &
-  i\, \tensor{\left( R_{(t)} \right)}{^i_J}
-  \left(
+  i\, \tensor{\qty( R_{(t)} )}{^i_J}
+  \qty(
     \ipd{u} X^J( x + i\, 0^+ ) - \ipd{\bu} \bX^J( x - i\, 0^+ )
-  \right)
+  )
   =
   0,
   \\
-  \tensor{\left( R_{(t)} \right)}{^m_J}
+  \tensor{\qty( R_{(t)} )}{^m_J}
   \eval{\ipd{\tau} X^J( \tau, \sigma )}_{\sigma = 0}
   & = &
-  i\, \tensor{\left( R_{(t)} \right)}{^m_J}
-  \left(
+  i\, \tensor{\qty( R_{(t)} )}{^m_J}
+  \qty(
     \ipd{u} X^J( x + i\, 0^+ ) + \ipd{\bu} \bX^J( x - i\, 0^+ )
-  \right)
+  )
   =
   0,
 \end{eqnarray}
@@ -268,7 +268,7 @@ With the introduction of the target space embedding of the worldsheet interactio
   \begin{cases}
     \ipd{u} X^I( x + i\, 0^+ )
     & =
-    \tensor{\left( U_{(t)} \right)}{^I_J}
+    \tensor{\qty( U_{(t)} )}{^I_J}
     \ipd{\bu} \bX^J( x - i\, 0^+ ),
     \qquad
     x \in D_{(t)}
@@ -283,11 +283,11 @@ In the last expression we introduced the matrix
 \begin{equation}
   U_{(t)}
   =
-  \left( R_{(t)} \right)^{-1}\,
+  \qty( R_{(t)} )^{-1}\,
   \cS\,
   R_{(t)}
   \in
-  \frac{\SO{4}}{\rS\left( \OO{2} \times \OO{2} \right)},
+  \frac{\SO{4}}{\rS\qty( \OO{2} \times \OO{2} )},
   \label{eq:Umatrices}
 \end{equation}
 where
@@ -298,7 +298,7 @@ where
   \label{eq:reflection_S}
 \end{equation}
 embeds the difference between Neumann and Dirichlet conditions.
-Given its definition $U_{(t)}$ is such that $U_{(t)} = \left( U_{(t)} \right)^{-1} = \left( U_{(t)} \right)^T$.
+Given its definition $U_{(t)}$ is such that $U_{(t)} = \qty( U_{(t)} )^{-1} = \qty( U_{(t)} )^T$.
 
 The target space vector $f_{(t)}$ recovers the apparent loss of information suffered when losing $g_{(t)}$.
 Consider for instance the embedding equations~\eqref{eq:dirichlet_bc} for any two intersecting D-branes $D_{(t)}$ and $D_{(t+1)}$.
@@ -324,7 +324,7 @@ we can compute the intersection point as:
 \begin{equation}
   f_{(t)}
   =
-  \left( \cR_{(t,\, t+1)} \right)^{-1}\,
+  \qty( \cR_{(t,\, t+1)} )^{-1}\,
   \cG_{(t,\, t+1)}.
 \end{equation}
 Information on $g_{(t)}$ is thus recovered through the global boundary conditions in the second equation in \eqref{eq:discontinuity_bc}.
@@ -363,7 +363,7 @@ The boundary conditions in terms of the doubling field are:
   \ipd{z} \cX( x_t + \eta - i\, 0^+ ),
   \label{eq:bottom_monodromy}
 \end{eqnarray}
-for $0 < \eta < \min\left( \abs{x_{(t-1)} - x_{(t)}}, \abs{x_{(t)} - x_{(t+1)}} \right)$ in order to consider only the two adjacent D-branes $D_{(t)}$ and $D_{(t+1)}$.
+for $0 < \eta < \min\qty( \abs{x_{(t-1)} - x_{(t)}}, \abs{x_{(t)} - x_{(t+1)}} )$ in order to consider only the two adjacent D-branes $D_{(t)}$ and $D_{(t+1)}$.
 Matrices $\cU_{(t,\, t+1)}$ and $\widetilde{\cU}_{(t,\, t+1)}$ are the non trivial monodromies arising from the rotation of the D-branes.
 
 Since the relative rotations between consecutive D-branes are non Abelian, for each interaction point there are two monodromies \cU and $\widetilde{\cU}$ depending on the location of the base point of the closed loop: one for paths starting in the upper plane \ccH and one for paths starting in $\overline{\ccH}$.
@@ -451,7 +451,7 @@ We define the spinor representation of $X$ as:
 \begin{equation}
   X_{(s)}( u, \bu ) = X^I( u, \bu )\, \tau_I,
 \end{equation}
-where $\tau = \left( i\, \1_2,\, \vb{\sigma} \right)$ and $\vb{\sigma}$ is the vector of the Pauli matrices.
+where $\tau = \qty( i\, \1_2,\, \vb{\sigma} )$ and $\vb{\sigma}$ is the vector of the Pauli matrices.
 Consider then:
 \begin{equation}
   \ipd{z} \cX_{(s)}( z )
@@ -470,7 +470,7 @@ Consider then:
 \end{equation}
 
 As in the real representation the discontinuities on the D-branes can be cast into monodromy factors with respect to the D-brane $D_{(\bt)}$. Branch cut structure and considerations on the homotopy group are left unchanged as long as we consider both left and right sectors of $\SU{2}_L \times \SU{2}_R$ at the same time.
-Let $0 < \eta < \min\left( \abs{x_{(t)} - x_{(t-1)}}, \abs{x_{(t+1)} - x_{(t)}} \right)$.
+Let $0 < \eta < \min\qty( \abs{x_{(t)} - x_{(t-1)}}, \abs{x_{(t+1)} - x_{(t)}} )$.
 We find:
 \begin{eqnarray}
   \ipd{z} \cX_{(s)}( x_t + e^{2 \pi i}( \eta + i\, 0^+) )
@@ -749,7 +749,7 @@ 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)
-  \left( \cB_{\vb{0},\, r}^{(R)}(\omega_z) \right)^T,
+  \qty( \cB_{\vb{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.
@@ -790,14 +790,14 @@ We impose:
   &&\begin{cases}
     D^{(L)}\,
     \rM_{\vb{0}}^{(L)}\,
-    \left( D^{(L)} \right)^{-1}
+    \qty( D^{(L)} )^{-1}
     =
     e^{-2\pi i \delta_{\vb{0}}^{(L)}}\,
     \cL(\vb{n}_{\vb{0}})
     \\
     D^{(R)}\,
     \rM_{\vb{0}}^{(R)}\,
-    \left( D^{(R)} \right)^{-1}
+    \qty( D^{(R)} )^{-1}
     =
     e^{-2\pi i \delta_{\vb{0}}^{(R)}}\,
     \cR^*(\vb{m}_{\vb{0}})
@@ -815,14 +815,14 @@ We impose:
   &&\begin{cases}
     D^{(L)},
     \rM_{\vb{\infty}}^{(L)}\,
-    \left( D^{(L)} \right)^{-1}
+    \qty( D^{(L)} )^{-1}
     =
     e^{-2\pi i \delta_{\vb{\infty}}^{(L)}}\,
     \cL(\vb{n}_{\vb{\infty}})
     \\
     D^{(R)}\,
     \rM_{\vb{\infty}}^{(R)}\,
-    \left( D^{(R)} \right)^{-1}
+    \qty( D^{(R)} )^{-1}
     =
     e^{-2\pi i \delta_{\vb{\infty}}^{(R)}}\,
     \cR^*(\vb{m}_{\vb{\infty}})
@@ -964,7 +964,7 @@ We find:
   \frac{n^1_{\vb{\infty}}+ i\, n^2_{\vb{\infty}}}{n_{\vb{\infty}}},
 \label{eq:K_factor_value}
 \end{eqnarray}
-where $f^{(L)} \in \left\lbrace 0,\, 1 \right\rbrace$.
+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 $\omega_{\bt+1} = 1$.
 Its dependence on the other parameters is encoded in~\eqref{eq:monodromy_relations}, where $\rM^+_{\vb{1}} = \rM^{-1}_{\vb{0}}\, \rM^{-1}_{\vb{\infty}}$, and the composition rule~\eqref{eq:product_in_SU2}:
@@ -1004,28 +1004,28 @@ We can use properties of the hypergeometric functions to show that any choice do
 Specifically we can start with certain values but we can recover the others through:
 \begin{equation}
   \rP
-  \left\lbrace
+  \qty{
     \mqty{
       0   & 1     & \infty &   \\
       0   & 0     & a      & z \\
       1-c & c-a-b & b      &
     }
-  \right\rbrace
+  }
   =
   (1-z)^{c-a-b}\,
   \rP
-  \left\lbrace
+  \qty{
     \mqty{
       0   & 1     & \infty &   \\
       0   & 0     & c-b    & z \\
       1-c & a+b-c & c-a    &
     }
-  \right\rbrace,
+  },
 \end{equation}
 where \rP is the Papperitz-Riemann symbol for the hypergeometric functions.
 We can then assign any admissible value to $f^{(L)}$ and $f^{(R)}$ and then recover the other through the identification:
 \begin{eqnarray}
-  f^{(L)}{}' & = & \left( 1 + f^{(L)} \right)~\text{mod}~2,
+  f^{(L)}{}' & = & \qty( 1 + f^{(L)} )~\text{mod}~2,
   \\
   \ffa_l' & = & \ffc_l - \ffb_l,
   \\
@@ -1073,7 +1073,7 @@ Using the \rP symbol the solutions can be symbolically written as
     \\
     &
     \times
-    \rP \left\lbrace
+    \rP \qty{
       \mqty{
           0
           &
@@ -1097,11 +1097,11 @@ Using the \rP symbol the solutions can be symbolically written as
           -n_{\vb{\infty}} + \ffb^{(L)}
           &
       }
-    \right\rbrace
+    }
     \\
     &
     \times
-    \rP \left\lbrace
+    \rP \qty{
       \mqty{
           0
           &
@@ -1125,7 +1125,7 @@ Using the \rP symbol the solutions can be symbolically written as
           -m_{\vb{\infty}} + \ffb^{(R)}
           &
       }
-      \right\rbrace.
+      }.
   \end{split}
   \label{eq:symbolic_solutions_using_P}
 \end{equation}
@@ -1226,9 +1226,9 @@ In fact we require the finiteness of the Euclidean action~\eqref{eq:action_doubl
 
 In principle it could appear obvious to use~\eqref{eq:contiguous_functions} to restrict the possible arbitrary integers to:
 \begin{eqnarray}
-  \ffa^{(L)} \in \left\lbrace -1,\, 0 \right\rbrace,
+  \ffa^{(L)} \in \qty{ -1,\, 0 },
   & \qquad &
-  \ffa^{(R)} \in \left\lbrace -1,\, 0 \right\rbrace,
+  \ffa^{(R)} \in \qty{ -1,\, 0 },
   \\
   \ffb^{(L)} = 0,
   & \qquad &
@@ -1248,15 +1248,15 @@ We could then use~\eqref{eq:reduction_F_F+} to write the possible solution as
     (1-\omega_z)^{n_{\vb{1}} + m_{\vb{1}}}
     \\
     & \times
-    \sum\limits_{\ffa^{(L,\,R)} \in \left\lbrace -1, 0 \right\rbrace} 
+    \sum\limits_{\ffa^{(L,\,R)} \in \qty{ -1, 0 }} 
     h(\omega_z,\, \ffa^{(L,R)})
     \times
     \\
     & \times
     \cB_{\vb{0}}^{(L)}(a^{(L)} + \ffa^{(L)},\, b,\, c;\, \omega_z)
-    \left(
+    \qty(
       \cB_{\vb{0}}^{(R)}(a^{(R)} + \ffa^{(R)},\, b,\, c;\, \omega_z)
-    \right)^T.
+    )^T.
   \end{split}
   \label{eq:doubling_field_expansion}
 \end{equation}
@@ -1285,10 +1285,10 @@ It can be verified that the convergence of the action both at finite and infinit
   \ipd{u'} \cX_{(s)}(u')
   +
   U_L^{\dagger}(\vb{n}_{{\bt}})
-  \left[
+  \qty[
     \finiteint{\bu'}{x_{(\bt-1)}}{\bu}
     \ipd{\bu'} \cX_{(s)}(\bu')
-  \right]
+  ]
   U_R(\vb{m}_{{\bt}}),
   \label{eq:classical_solution}
 \end{equation}
@@ -1319,7 +1319,7 @@ In this case~\eqref{eq:symbolic_solutions_using_P} becomes
 \begin{equation}
   (-\omega)^\ffA\,
   (1-\omega)^\ffB\,
-  \rP\left\lbrace
+  \rP\qty{
   \mqty{
     0
     &
@@ -1343,11 +1343,11 @@ In this case~\eqref{eq:symbolic_solutions_using_P} becomes
     -n_{\vb{\infty}} + \ffb^{(L)}
     &
   }
-  \right\rbrace.
+  }.
 \end{equation}
 The only possible solution compatible with~\eqref{eq:constraints_finite_X} is
 \begin{equation}
-  \rP\left\lbrace
+  \rP\qty{
   \mqty{
     0
     &
@@ -1371,7 +1371,7 @@ The only possible solution compatible with~\eqref{eq:constraints_finite_X} is
     -n_{\vb{\infty}} + 2
     &
   }
-  \right\rbrace,
+  },
   \label{eq:X_solution_pure_L}
 \end{equation}
 that is $\ffa^{(L)} = -1$, $\ffb^{(L)} = 0$, $\ffc^{(L)} = 0$, $\ffA = -1$ and  $\ffB = -1$.
@@ -1381,7 +1381,7 @@ For each possible case the solution is however unique and it is given by
 \begin{enumerate}
   \item $n_{\vb{0}} > m_{\vb{0}}$ and $n_{\vb{1}} > m_{\vb{1}}$:
     \begin{equation}
-      \rP\left\lbrace
+      \rP\qty{
         \mqty{
           0
           &
@@ -1405,8 +1405,8 @@ For each possible case the solution is however unique and it is given by
           -n_{\vb{\infty}} + 2
           &
         }
-      \right\rbrace
-      \rP\left\lbrace
+      }
+      \rP\qty{
         \mqty{
           0
           &
@@ -1430,13 +1430,13 @@ For each possible case the solution is however unique and it is given by
           -m_{\vb{\infty}} + 1
           &
         }
-      \right\rbrace,
+      },
       \label{eq:X_solution>>}
    \end{equation}
 
    \item $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} < m_{\vb{1}}$ and $n_{\vb{\infty}} > m_{\vb{\infty}}$:
      \begin{equation}
-       \rP\left\lbrace
+       \rP\qty{
          \mqty{
            0
            &
@@ -1460,8 +1460,8 @@ For each possible case the solution is however unique and it is given by
            -n_{\vb{\infty}} + 2
            &
          }
-       \right\rbrace
-       \rP\left\lbrace
+       }
+       \rP\qty{
          \mqty{
            0
            &
@@ -1485,13 +1485,13 @@ For each possible case the solution is however unique and it is given by
            -m_{\vb{\infty}} + 1
            &
          }
-       \right\rbrace,
+       },
        \label{eq:X_solution><>}
      \end{equation}
 
    \item $n_{\vb{0}} > m_{\vb{0}}$, $n_{\vb{1}} < m_{\vb{1}}$ and $n_{\vb{\infty}} < m_{\vb{\infty}}$:
      \begin{equation}
-       \rP\left\lbrace
+       \rP\qty{
          \mqty{
            0
            &
@@ -1515,8 +1515,8 @@ For each possible case the solution is however unique and it is given by
            -n_{\vb{\infty}} + 1
            &
          }
-       \right\rbrace
-       \rP\left\lbrace
+       }
+       \rP\qty{
          \mqty{
            0
            &
@@ -1540,13 +1540,13 @@ For each possible case the solution is however unique and it is given by
            -m_{\vb{\infty}} + 2
            &
          }
-       \right\rbrace,
+       },
        \label{eq:X_solution><<}
      \end{equation}
 
    \item $n_{\vb{0}} < m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} > m_{\vb{\infty}}$:
      \begin{equation}
-       \rP\left\lbrace
+       \rP\qty{
          \mqty{
            0
            &
@@ -1570,8 +1570,8 @@ For each possible case the solution is however unique and it is given by
            -n_{\vb{\infty}} + 2
            &
          }
-       \right\rbrace
-       \rP\left\lbrace
+       }
+       \rP\qty{
          \mqty{
            0
            &
@@ -1595,13 +1595,13 @@ For each possible case the solution is however unique and it is given by
            -m_{\vb{\infty}} + 1
            &
          }
-       \right\rbrace,
+       },
        \label{eq:X_solution<>>}
      \end{equation}
 
    \item $n_{\vb{0}} < m_{\vb{0}}$, $n_{\vb{1}} > m_{\vb{1}}$ and $n_{\vb{\infty}} < m_{\vb{\infty}}$:
      \begin{equation}
-       \rP\left\lbrace
+       \rP\qty{
          \mqty{
            0
            &
@@ -1625,8 +1625,8 @@ For each possible case the solution is however unique and it is given by
            -n_{\vb{\infty}} + 1
            &
          }
-       \right\rbrace
-       \rP\left\lbrace
+       }
+       \rP\qty{
          \mqty{
            0
            &
@@ -1650,13 +1650,13 @@ For each possible case the solution is however unique and it is given by
            -m_{\vb{\infty}} + 2
            &
          }
-       \right\rbrace,
+       },
        \label{eq:X_solution<><}
      \end{equation}
 
    \item $n_{\vb{0}} < m_{\vb{0}}$, $n_{\vb{1}} < m_{\vb{1}}$:
      \begin{equation}
-       \rP\left\lbrace
+       \rP\qty{
          \mqty{
            0
            &
@@ -1680,8 +1680,8 @@ For each possible case the solution is however unique and it is given by
            -n_{\vb{\infty}} + 1
            &
          }
-       \right\rbrace
-       \rP\left\lbrace
+       }
+       \rP\qty{
          \mqty{
            0
            &
@@ -1705,7 +1705,7 @@ For each possible case the solution is however unique and it is given by
            -m_{\vb{\infty}} + 2
            &
          }
-       \right\rbrace.
+       }.
        \label{eq:X_solution<<}
      \end{equation}
 \end{enumerate}
@@ -1737,7 +1737,7 @@ In the previous section we produced one solution for each ordering of the $n_{\o
 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{}
 \footnotetext{%
-  We need to change two rotation vectors because the monodromies are constrained by \eqref{eq:monodromy_relations}.
+  We need to change two rotation vectors because the monodromies are constrained by~\eqref{eq:monodromy_relations}.
 }
 For instance we could consider $(\widehat{\vb{n}}_{\vb{0}},\, \widehat{\vb{n}}_{\vb{1}},\, \vb{n}_{\vb{\infty}}) \oplus (\vb{m}_{\vb{0}},\, \widehat{\vb{m}}_{\vb{1}},\, \widehat{\vb{m}}_{\vb{\infty}})$.
 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.
@@ -2041,9 +2041,9 @@ Explicitly we impose the four real equations in spinorial formalism
   \ipd{\omega} \cX(\omega)
   +
   U_L^{\dagger}(\vb{n}_{{\bt}})
-  \left[
+  \qty[
     \finiteint{\bomega}{0}{1} \ipd{\bomega} \cX(\bomega)
-  \right]
+  ]
   U_R(\vb{m}_{{\bt}})
   =
   f_{{\bt+1}\, (s)} - f_{{\bt-1}\, (s)},
@@ -2181,9 +2181,9 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
   \centering
   \begin{tabular}{@{}ccc@{}}
     \toprule
-    $\left( \ffa^{(L)},\, \ffb^{(L)},\, \ffc^{(L)} \right)$ &
+    $\qty( \ffa^{(L)},\, \ffb^{(L)},\, \ffc^{(L)} )$ &
     $n_{\vb{1}}$                                            &
-    $\left( \cB^{(L)}( z ) \right)^T$
+    $\qty( \cB^{(L)}( z ) )^T$
     \\
     \midrule
     \multirow{4}{*}{$(-1,\, 0,\, 0)$} &
@@ -2191,7 +2191,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
     $\mqty( (1 - z)^{-2\, n_{\vb{\infty}} - 2\, n_{\vb{0}} + 1} & 0 )$
     \\
                                                       &
-    $1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
+    $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
     $\mqty( 1 & 0 )$
     \\
                                    &
@@ -2212,7 +2212,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
             & 0 )$
     \\
                                                       &
-    $1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
+    $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
     $\mqty( 1 & 0 )$
     \\
                                    &
@@ -2229,7 +2229,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
     $\mqty( (1-z)^{-2\, n_{\vb{\infty}} - 2\, n_{\vb{0}}} & 0 )$
     \\
                                                       &
-    $1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
+    $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
     $\mqty( 0 & (1-z)^{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 2}\, (-z)^{1 - 2\, n_{\vb{0}}} )$
     \\
                                    &
@@ -2246,7 +2246,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
     $\mqty( (1-z)^{-2\, n_{\vb{\infty}} - 2\, n_{\vb{0}}} + 1 & 0 )$
     \\
                                                       &
-    $1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
+    $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
     $\mqty( 1 & 0 )$
     \\
                                    &
@@ -2263,7 +2263,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
     $\mqty( 0 & (-z)^{-2\, n_{\vb{0}}} )$
     \\
                                                       &
-    $1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
+    $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
     $\mqty( 0 & (1-z)^{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 1}\, (-z)^{-2\, n_{\vb{0}}} )$
     \\
                                    &
@@ -2280,7 +2280,7 @@ Results are summarised in~\Cref{tab:Left_Abelian_solutions} where we left some h
     $\mqty( (1-z)^{-2\, n_{\vb{\infty}} -2\, n_{\vb{0}}} & 0 )$
     \\
                                                       &
-    $1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ &
+    $1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ &
     $\mqty( 0 & (1-z)^{2\, n_{\vb{\infty}} + 2\, n_{\vb{0}} - 2}\, (-z)^{-2\, n_{\vb{0}}} )$
     \\
                                    &
@@ -2305,7 +2305,7 @@ 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{}
 \footnotetext{%
-  We write ``possible solutions'' because $m_{\vb{1}} = 1 - \left( m_{\vb{0}} + m_{\vb{\infty}} \right)$ is not.
+  We write ``possible solutions'' because $m_{\vb{1}} = 1 - \qty( m_{\vb{0}} + m_{\vb{\infty}} )$ 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 $\hat{n}_{\vb{\infty}} < \hat{m}_{\vb{\infty}}$.
 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.
@@ -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 $n_{\vb{1}} = 1 - \left( n_{\vb{0}} + n_{\vb{\infty}} \right)$ 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 $n_{\vb{1}} = 1 - \qty( n_{\vb{0}} + n_{\vb{\infty}} )$ and $m_{\vb{1}} = -m_{\vb{0}} + m_{\vb{\infty}}$.
 This leads to two independent rational functions of $\omega_z$:
 \begin{equation}
   \begin{split}
@@ -2393,7 +2393,7 @@ such that $\sum\limits_{t} \varepsilon_{\vb{t}} = 1$, and
   \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.
-As usual we have $\ipd{\omega_z} \cZ^1( \omega_z ) \neq \left[ \ipd{\omega_z} \overline{\cZ}^1( \omega_z ) \right]^*$.
+As usual we have $\ipd{\omega_z} \cZ^1( \omega_z ) \neq \qty[ \ipd{\omega_z} \overline{\cZ}^1( \omega_z ) ]^*$.
 
 We can now build the Abelian solution to show the analytical structure of the limit.
 We have
@@ -2415,7 +2415,7 @@ We have
 where we chose $R_{(\bt)} = \1_4$ so that $U_{(\bt)}$ 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}.
 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 $\overline{Z}^1(\bu) = \left[ Z^1(u) \right]^*$ of the sector with $\sum\limits_t \varepsilon_{\vb{t}} = 1$ and $\overline{Z}^2(\bu) = \left[ Z^2(u) \right]^*$ of the sector with $\sum\limits_t \varphi_{\vb{t}} = 2$.
+From the previous relations we can also recover the usual holomorphicity $\overline{Z}^1(\bu) = \qty[ Z^1(u) ]^*$ of the sector with $\sum\limits_t \varepsilon_{\vb{t}} = 1$ and $\overline{Z}^2(\bu) = \qty[ Z^2(u) ]^*$ of the sector with $\sum\limits_t \varphi_{\vb{t}} = 2$.
 
 
 \subsubsection{Abelian Limits}
@@ -2445,14 +2445,14 @@ We get:
   \sum\limits_{m \in \{3, 4\}}
   g_{(t)\, m}\,
   \finiteint{x}{x_{(t)}}{x_{(t-1)}}
-  \tensor{\left( R_{(t)} \right)}{_{mI}}
-  \eval{\left( X_L'(x) - X_R'(x) \right)^I}_{y=0^+},
+  \tensor{\qty( R_{(t)} )}{_{mI}}
+  \eval{\qty( X_L'(x) - X_R'(x) )^I}_{y=0^+},
   \label{eq:area_tmp}
 \end{equation}
 where indices $I = 1,\, 2,\, 3,\, 4$ are summed over and $m = 3,\, 4$ are the transverse directions in the well adapted frame with respect to the D-brane.
 As the number of D-branes is defined modulo $N_B = 3$, $D_{(1)}$ is split on two separate intervals:
 \begin{equation}
-  \left[ x_{(1)},\, x_{(3)} \right]
+  \qty[ x_{(1)},\, x_{(3)} ]
   =
   \left[ x_{(1)},\, +\infty \right)
   \cup
@@ -2488,7 +2488,7 @@ Now~\eqref{eq:area_tmp} becomes:
     \sum\limits_{m \in \{3, 4\}}
     \eval{%
       g_{(t)\, m}\,
-      \tensor{\Im \left( R_{(t)} \right)}{_{mI}}
+      \tensor{\Im \qty( R_{(t)} )}{_{mI}}
       X_L^I(x+i0^+)
     }^{x = x_{(t-1)}}_{x = x_{(t)}}
     \\
@@ -2501,9 +2501,9 @@ Now~\eqref{eq:area_tmp} becomes:
   \end{split}
   \label{eq:action_with_imaginary_part}
 \end{equation}
-where $g^{(\perp)}_{(t)\, I} = \sum_{m \in \{3, 4\}} \tensor{\left( R_{(t)}^{-1} \right)}{_{mI}}\, g_{(t)\, m}$ is the transverse shift of $D_{(t)}$ in the global coordinates of the target space:
+where $g^{(\perp)}_{(t)\, I} = \sum_{m \in \{3, 4\}} \tensor{\qty( R_{(t)}^{-1} )}{_{mI}}\, g_{(t)\, m}$ is the transverse shift of $D_{(t)}$ in the global coordinates of the target space:
 \begin{equation}
-  g^{(\perp)}_{(t)\, I}\, (f_{(t-1)} - f_{(t)})^I = 0.
+  g^{(\perp)}_{(t)\, I}\, \qty(f_{(t-1)} - f_{(t)})^I = 0.
   \label{eq:g_perp_Delta_f}
 \end{equation}
 
@@ -2514,20 +2514,20 @@ In this case there are global complex coordinates for which the string solution
 \begin{equation}
   Z^i(u, \bu) = Z^i_L(u),
   \qquad
-  \overline{Z}^i(u, \bu) = \bar{Z}^i(\bu) = \left( Z^i_L(u) \right)^*,
+  \overline{Z}^i(u, \bu) = \bar{Z}^i(\bu) = \qty( Z^i_L(u) )^*,
 \end{equation}
 where $i = 1$ in the Abelian case and $i=1,\, 2$ in the \SU{2} case.
 We also have $f^i_{(t)} = Z^i_L(x_{(t)} + i\, 0^+)$.
 Equations~\eqref{eq:g_perp_Delta_f} and~\eqref{eq:action_with_imaginary_part} then become
 \begin{eqnarray}
-  \Re( g_{(t)\, i}^{(\perp)}\, \left( f_{(t-1)} - f_{(t)} \right)^i )
+  \Re( g_{(t)\, i}^{(\perp)}\, \qty( f_{(t-1)} - f_{(t)} )^i )
   & = &
   0,
   \\
   4 \pi \ap \eval{S_{\R^4}}_{\text{on-shell}}
   & = &
   -2 \finitesum{t}{1}{3}\,
-  \Im( g_{(t)\, i}^{(\perp)}\, \left( f_{(t-1)} - f_{(t)} \right)^i ),
+  \Im( g_{(t)\, i}^{(\perp)}\, \qty( f_{(t-1)} - f_{(t)} )^i ),
 \end{eqnarray}
 where the last equation shows that the action can be expressed using just the global data.
 
@@ -2540,10 +2540,10 @@ Since the action is positive then we can write
   =
   \frac{1}{2 \pi \ap}
   \finitesum{t}{1}{3}\,
-  \left(
+  \qty(
     \frac{1}{2} \abs{g^{(\perp)}_{(t)}} \,
     \abs{f_{(t-1)} - f_{(t)}}
-  \right),
+  ),
 \end{equation}
 where a factor $\frac{1}{2}$ comes from raising the complex index in $g^{(\perp)}_{(t)\, 1}$.
 The right hand side of the previous expression is the sum of the areas of the triangles having the interval between two intersection points on a given D-brane $D_{(t)}$ as base and the distance between the D-brane and the origin as height.
diff --git a/sec/part1/fermions.tex b/sec/part1/fermions.tex
index cbd3ace..1ea999c 100644
--- a/sec/part1/fermions.tex
+++ b/sec/part1/fermions.tex
@@ -26,11 +26,11 @@ Their two-dimensional Minkowski action defined on the strip $\Sigma$ is:
   \frac{T}{2}
   \infinfint{\tau}
   \finiteint{\sigma}{0}{\pi}
-  \left(
+  \qty(
     \frac{1}{2}\, \bpsi_i( \tau,\, \sigma )\,
-    \left( -i \gamma^{\alpha} \lrpartial{\alpha} \right)\,
+    \qty( -i \gamma^{\alpha} \lrpartial{\alpha} )\,
     \psi^i( \tau,\, \sigma ) 
-  \right),
+  ),
   \label{eq:cft-action_full}
 \end{equation}
 where the gamma matrices are
@@ -59,7 +59,7 @@ and the components of the massless fermions are
   =
   \mqty( -\psi_-^* & \psi_+^* ).
 \end{equation}
-We then define the lightcone coordinates $\xi_{\pm} = \tau \pm \sigma$ such that $\ipd{\pm} = \frac{1}{2}\, \left( \ipd{\tau} \pm \ipd{\sigma} \right)$.
+We then define the lightcone coordinates $\xi_{\pm} = \tau \pm \sigma$ such that $\ipd{\pm} = \frac{1}{2}\, \qty( \ipd{\tau} \pm \ipd{\sigma} )$.
 In components the action reads:
 \begin{equation}
   S
@@ -67,9 +67,9 @@ In components the action reads:
   i \frac{T}{4}
   \infinfint{\xi_+}
   \infinfint{\xi_-}
-  \left(
+  \qty(
     \psi^*_{-,\, i} \lrpartial{+} \psi^i_- + \psi^*_{+,\, i} \lrpartial{-} \psi^i_+
-  \right),
+  ),
   \label{eq:cft-action}
 \end{equation}
 so the \eom are:
@@ -102,12 +102,12 @@ Their solutions are the ``holomorphic'' functions $\psi_{+}^i(\xi_+)$ and $\psi_
 The boundary conditions are instead:
 \begin{equation}
   \eval{
-    \left(
+    \qty(
       \var{\psi}_{+,\, i}^* \psi_{+}^{ i} +
       \var{\psi}_{-,\, i}^* \psi_{-}^{ i} -
       \psi_{+,\, i}^* \var{\psi}_{+}^{ i} -
       \psi_{-,\, i}^* \var{\psi}_{-}^{ i}
-    \right)
+    )
   }_{\sigma = 0}^{\sigma = \pi} = 0.
   \label{eq:boundary-conditions}
 \end{equation}
@@ -116,10 +116,10 @@ We solve the constraint imposing the non trivial relations:
   \begin{cases}
     \psi_-^i( \tau, 0 )
     =
-    \tensor{\left( R_{(t)} \right)}{^i_j}
+    \tensor{\qty( R_{(t)} )}{^i_j}
     \psi^j_+( \tau, 0 ),
     & \qquad
-    \tau \in \left( \htau_{(t)}, \htau_{(t-1)} \right),
+    \tau \in \qty( \htau_{(t)}, \htau_{(t-1)} ),
     \\
     \psi_-^i( \tau, \pi )
     =
@@ -164,19 +164,19 @@ The boundary conditions become:
 \begin{equation}
   \Psi^i(\tau, 2 \pi )
   =
-  - \tensor{\left( R_{(t)} \right)}{^i_j}
+  - \tensor{\qty( R_{(t)} )}{^i_j}
   \Psi^j(\tau,\, 0 ),
   \qquad
-  \tau \in \left( \htau_{(t)},\, \htau_{(t-1)} \right).
+  \tau \in \qty( \htau_{(t)},\, \htau_{(t-1)} ).
 \end{equation}
 Using the equation of motion we get $\Psi^i(\tau,\, \phi) = \Psi^i(\tau + \phi)$ and the boundary conditions become the (pseudo-)periodicity conditions
 \begin{equation}
   \Psi^i(\tau + 2 \pi )
   =
-  - \tensor{\left( R_{(t)} \right)}{^i_j}
+  - \tensor{\qty( R_{(t)} )}{^i_j}
   \Psi^j(\tau ),
   \qquad
-  \tau \in \left( \htau_{(t)}, \htau_{(t-1)} \right).
+  \tau \in \qty( \htau_{(t)}, \htau_{(t-1)} ).
 \end{equation}
 The main issue is now to expand $\Psi$ in a basis of modes and proceed to its quantization.
 Even in the simplest case $N_f = 1$ the task of finding the Minkowskian modes turns out to be fairly complicated.
@@ -232,7 +232,7 @@ Integration over the surface $\Sigma' = [ \tau_i, \tau_f ] \times [ 0, \pi ]$ yi
 The current $j_{\tau}( \tau,\, \sigma )$ is thus conserved in time if
 \begin{equation}
   \finiteint{\tau}{\tau_i}{\tau_f}
-  \left( \eval{j_{\sigma}}_{\sigma = \pi} - \eval{j_{\sigma}}_{\sigma = 0} \right)
+  \qty( \eval{j_{\sigma}}_{\sigma = \pi} - \eval{j_{\sigma}}_{\sigma = 0} )
   =
   0.
   \label{eq:time-conservation}
@@ -253,7 +253,7 @@ We write it as
 \begin{equation}
   j_{\alpha}^a ( \tau,\, \sigma )
   =
-  \tensor{\left( \rT^a \right)}{^i_j}\,
+  \tensor{\qty( \rT^a )}{^i_j}\,
   \bpsi_i( \tau,\, \sigma )\, \gamma_{\alpha}\, \psi^j( \tau,\, \sigma ),
 \end{equation}
 where $\rT^a$ is a generator of $\U{N_f}$ such that $a = 1,\, 2,\, \dots,\, N_f^2$.\footnotemark{}
@@ -265,21 +265,21 @@ In components we have:
 \begin{eqnarray}
   j^a_{\tau}( \tau,\, \sigma )
   & = &
-  \tensor{\left( \rT^a \right)}{^i_j}\,
-  \left( \psi^*_{+,\, i} \psi^j_+ + \psi^*_{-,\, i} \psi^j_- \right)
+  \tensor{\qty( \rT^a )}{^i_j}\,
+  \qty( \psi^*_{+,\, i} \psi^j_+ + \psi^*_{-,\, i} \psi^j_- )
   \\
   j^a_{\sigma}( \tau,\, \sigma )
   & = &
-  \tensor{\left( \rT^a \right)}{^i_j}\,
-  \left( \psi^*_{+,\, i} \psi^j_+ - \psi^*_{-,\, i} \psi^j_- \right).
+  \tensor{\qty( \rT^a )}{^i_j}\,
+  \qty( \psi^*_{+,\, i} \psi^j_+ - \psi^*_{-,\, i} \psi^j_- ).
 \end{eqnarray}
 
 In order to define a conserved charge, we require:
 \begin{equation}
   \finiteint{\tau}{\tau_i}{\tau_f}
-  \left(
+  \qty(
     \eval{j_{\sigma}^a}_{\sigma = \pi} - \eval{j_{\sigma}^a}_{\sigma = 0}
-  \right)
+  )
   =
   0,
 \end{equation}
@@ -292,13 +292,13 @@ Moreover we have:
 \begin{equation}
   \eval{j_{\sigma}^a( \tau,\, \sigma )}_{\sigma = 0}
   =
-  \left[
+  \qty[
     \psi^*_+\,
-    \left( \rT^a - R_{(t)}^{\dagger} \rT^a R_{(t)} \right)\,
+    \qty( \rT^a - R_{(t)}^{\dagger} \rT^a R_{(t)} )\,
     \psi_+
-  \right]_{\sigma = 0},
+  ]_{\sigma = 0},
   \qquad
-  \tau \in \left( \htau_{(t)}, \htau_{(t-1)} \right).
+  \tau \in \qty( \htau_{(t)}, \htau_{(t-1)} ).
 \end{equation}
 In general
 \begin{equation}
@@ -323,7 +323,7 @@ Let $\alpha$ and $\beta$ be two arbitrary (bosonic) solutions to the \eom~\eqref
   =
   \cN
   \finiteint{\sigma}{0}{\pi}
-  \left( \alpha_{+,\, i}^* \beta_+^i + \alpha_{-,\, i}^* \beta_-^i \right),
+  \qty( \alpha_{+,\, i}^* \beta_+^i + \alpha_{-,\, i}^* \beta_-^i ),
   \label{eq:conserved-product}
 \end{equation}
 where $\cN \in \R$ is a normalisation constant and the integrand must not present non integrable singularities.
@@ -366,7 +366,7 @@ build the hypothetical charges:
   \cT_{\tau\tau}( \tau,\, \sigma )
   =
   \finiteint{\sigma}{0}{\pi}
-  \left( \cT_{++}( \tau + \sigma ) + \cT_{--}( \tau - \sigma ) \right),
+  \qty( \cT_{++}( \tau + \sigma ) + \cT_{--}( \tau - \sigma ) ),
   \label{eq:hamiltonian}
   \\
   \rP( \tau )
@@ -375,7 +375,7 @@ build the hypothetical charges:
   \cT_{\tau\sigma}( \tau,\, \sigma )
   =
   \finiteint{\sigma}{0}{\pi}
-  \left( \cT_{++}( \tau + \sigma ) - \cT_{--}( \tau - \sigma ) \right),
+  \qty( \cT_{++}( \tau + \sigma ) - \cT_{--}( \tau - \sigma ) ),
   \label{eq:momentum}
 \end{eqnarray}
 which are conserved if~\eqref{eq:time-conservation} holds.
@@ -389,16 +389,16 @@ reads:\footnotemark{}
 \begin{equation}
   \begin{split}
     & \finiteint{\tau}{\tau_i}{\tau_f}
-    \eval{\left( \cT_{++}( \tau + \sigma ) + \cT_{--}( \tau - \sigma ) \right)}_{\sigma = 0}^{\sigma = \pi}
+    \eval{\qty( \cT_{++}( \tau + \sigma ) + \cT_{--}( \tau - \sigma ) )}_{\sigma = 0}^{\sigma = \pi}
     \\
     & =
     - i \frac{T}{4}
     \int \Delta\tau
-    \left(
+    \qty(
       2 \eval{\psi_{+,\, i}^*\, \lrpartial{\tau} \psi_+^i}^{\sigma = \pi}_{\sigma = 0}
       -
-      \eval{\psi_{+,\, i}^* \tensor{\left( R_{(t)}^{\dagger} \lrpartial{\tau} R_{(t)} \right)}{^i_j} \psi_+^j}_{\sigma = 0}
-    \right)
+      \eval{\psi_{+,\, i}^* \tensor{\qty( R_{(t)}^{\dagger} \lrpartial{\tau} R_{(t)} )}{^i_j} \psi_+^j}_{\sigma = 0}
+    )
     \neq
     0.
   \end{split}
@@ -407,31 +407,31 @@ The corresponding condition for the Hamiltonian $\rH$ is:
 \begin{equation}
   \begin{split}
     & \finiteint{\tau}{\tau_i}{\tau_f}
-    \eval{\left( \cT_{++}( \tau + \sigma ) - \cT_{--}( \tau - \sigma ) \right)}_{\sigma = 0}^{\sigma = \pi}
+    \eval{\qty( \cT_{++}( \tau + \sigma ) - \cT_{--}( \tau - \sigma ) )}_{\sigma = 0}^{\sigma = \pi}
     \\
     & =
     - i \frac{T}{4}
     \int \Delta\tau
-    \left( \eval{\psi_{+,\, i}^* \tensor{\left( R_{(t)}^{\dagger} \lrpartial{\tau} R_{(t)} \right)}{^i_j} \psi_+^j}_{\sigma = 0} \right)
+    \qty( \eval{\psi_{+,\, i}^* \tensor{\qty( R_{(t)}^{\dagger} \lrpartial{\tau} R_{(t)} )}{^i_j} \psi_+^j}_{\sigma = 0} )
     = 0
     \quad
     \Leftrightarrow
     \quad
-    \left( \tau_i, \tau_f \right) \in \left( \htau_{(t)}, \htau_{(t-1)} \right).
+    \qty( \tau_i, \tau_f ) \in \qty( \htau_{(t)}, \htau_{(t-1)} ).
   \end{split}
 \end{equation}
 In both cases we used the shorthand graphical notation
 \begin{equation}
     \int \Delta \tau
     =
-    \left(
+    \qty(
       \int\limits_{\tau_i}^{\htau_{t_0}}
       +
       \finitesum{t}{t_0}{t_N - 1}
       \int\limits_{\htau_{(t)}}^{\htau_{(t+1)}}
       +
       \int\limits_{\htau_N}^{\tau_f}
-    \right)
+    )
     \dd{\tau}
 \end{equation}
 for simplicity.
@@ -441,7 +441,7 @@ These relations therefore prove that the generator of the $\sigma$-translations~
 
 \subsection{Basis of Solutions and Dual Modes}
 
-Let $\left\lbrace \psi_{n,\, \pm}^i \right\rbrace_{n \in \Z}$ be a complete basis of modes such that:
+Let $\qty{ \psi_{n,\, \pm}^i }_{n \in \Z}$ be a complete basis of modes such that:
 \begin{equation}
   \begin{cases}
     \psi_{n,\, +}^i( \tau, 0 ) = \qty( R_{(t)} )^i_j \psi_{n,\, -}^j(
@@ -457,13 +457,13 @@ The fields $\psi^i$ (and the fields $\Psi^i$) are then a superposition of such m
 \begin{equation}
   \psi^i_{\pm}( \xi_{\pm} )
   =
-  \sum\limits_{n \in \Z} b_n \psi^i_{n,\, \pm}( \xi_{\pm} )
+  \sum\limits_{n \in \Z} b_n\, \psi^i_{n,\, \pm}( \xi_{\pm} )
   \qquad
   \Rightarrow
   \qquad
   \Psi^i( \xi )
   =
-  \sum\limits_{n \in \Z} b_n \Psi^i_n( \xi ).
+  \sum\limits_{n \in \Z} b_n\, \Psi^i_n( \xi ).
 \label{eq:usual-mode-expansion}
 \end{equation}
 
@@ -488,7 +488,7 @@ We then define the conserved product for the ``double fields''~\eqref{eq:conserv
   \label{eq:conserved-product-dual-basis}
 \end{equation}
 In the previous expression we changed the notation of the product.
-We are in fact dealing with the space of solutions whose basis is $\left\lbrace \Psi_n \right\rbrace$ and a dual space with basis $\left\lbrace \dual{\Psi}_n \right\rbrace$ which is not required to span entirely the original space but only to be a subset of it in order to be able to compute the anti-commutation relations among the annihilation and construction operators in a well defined way as in~\eqref{eq:Mink_can_anticomm_rel_ann_des}.
+We are in fact dealing with the space of solutions whose basis is $\qty{ \Psi_n }$ and a dual space with basis $\qty{ \dual{\Psi}_n }$ which is not required to span entirely the original space but only to be a subset of it in order to be able to compute the anti-commutation relations among the annihilation and construction operators in a well defined way as in~\eqref{eq:Mink_can_anticomm_rel_ann_des}.
 Given the previous product we can extract the operators as
 \begin{eqnarray}
   \lconsprod{\dual{\Psi}_n}{\Psi} & = & b_n,
@@ -497,15 +497,15 @@ Given the previous product we can extract the operators as
 \end{eqnarray}
 As a consequence of the canonical anti-commutation relations
 \begin{equation}
-  \left[
+  \qty[
     \Psi^i\qty( \tau,\, \sigma ), \Psi^*_j\qty( \tau,\, \sigma' )
-  \right]_+
+  ]_+
   =
   \frac{2}{T}\, \tensor{\delta}{^i_j}\, \delta( \sigma - \sigma' ),
 \end{equation}
 we have then:
 \begin{equation}
-  \eval{\left[ b_n, b_m^{\dagger} \right]_+}_{\tau = \tau_0}
+  \eval{\qty[ b_n, b_m^{\dagger} ]_+}_{\tau = \tau_0}
   =
   \frac{2}{T} \cN \eval{\lconsprod{\dual{\Psi}_n}{\dual{\Psi}_m}}_{\tau = \tau_0}.
   \label{eq:Mink_can_anticomm_rel_ann_des}
@@ -516,4 +516,743 @@ Clearly this does not exclude the possibility to have singularities in $\Psi_m$
 Using the definition of the conserved product we therefore moved the focus from finding a consistent definition of the Fock space to the construction of the dual basis of modes.
 This task is however easier to address in a Euclidean formulation.
 
+
+\subsection{Point-like Defect CFT: the Euclidean Formulation}
+\label{sec:eucl_formulation}
+
+In the Euclidean reformulation the solution to the \eom might be easier to study than its Lorentzian worldsheet form.
+This is specifically the case when $R_{(t)} \in \U{1}^{N_f} \subset \U{N_f}$.
+The presence of a time dependent Hamiltonian is not standard and we can neither blindly apply the usual Wick rotation nor the usual CFT techniques.
+
+
+\subsubsection{Fields on the Strip}
+
+Performing the Wick rotation as $\tau_E = i \tau$ such that $e^{i S} = e^{-S_E}$, the Minkowskian action~\eqref{eq:cft-action} becomes:
+\begin{equation}
+  S_E
+  =
+  \frac{T}{2}
+  \iint \dd{\xi} \dd{\bxi}\,
+  \frac{1}{2}\,
+  \qty(
+    \hpsi_{E,\, +,\, i}^*\, \lrpartial{\bxi} \hpsi_{E,\, +}^i
+    +
+    \hpsi_{E,\, -,\, i}^*\, \lrpartial{\xi} \hpsi_{E,\, -}^i
+  ),
+  \label{eq:S_Eu_strip}
+\end{equation}
+where the Euclidean fermion on the strip is connected to the Minkowskian formulation through
+\begin{equation}
+  \hpsi_{E,\, \pm}^i( \xi,\, \bxi )
+  =
+  \psi_{\pm}^i( -i\xi,\, -i\bxi ).
+\end{equation}
+In the previous expressions we defined the coordinates $\xi = \tau_E + i \sigma$, $\bar \xi = \tau_E - i \sigma$ such that $\bxi = \xi^*$.
+Moreover we get $\ipd{\xi} = \pdv{\xi} = \frac{1}{2} \qty( \pdv{\tau_E} - i\pdv{\sigma} )$, $\ipd{\bxi} = \pdv{\bxi} = \frac{1}{2} \qty( \pdv{\tau_E} + i \pdv{\sigma} )$.
+As a consequence the Euclidean ``complex conjugation'' $\star$ (defined off-shell) acts as
+\begin{equation}
+  \qty[
+    \hpsi_{E,\, \pm}^i(\xi,\, \bxi)
+  ]^\star
+  =
+  \hpsi_{E,\, \pm i}^*(-\bxi,\, -\xi).
+  \label{eq:off-shell-Hermitian-conjugate}
+\end{equation}
+
+The \eom are as usual
+\begin{eqnarray}
+  \ipd{\xi} \hpsi_{E,\, -}^i( \xi,\, \bxi )
+  = &
+  \ipd{\bxi} \hpsi_{E,\, +}^i( \xi,\, \bxi )
+  = &
+  0,
+  \\
+  \ipd{\xi} \hpsi_{E,\, -,\, i}^*( \xi,\, \bxi )
+  = &
+  \ipd{\bxi} \hpsi_{E,\, +,\, i}^*( \xi,\, \bxi )
+  = &
+  0,
+\end{eqnarray}
+whose solutions are the holomorphic functions $\hpsi_{E,\, +}( \xi )$ and $\hpsi_{E,\, -}( \bxi )$, together with $\hpsi_{E,\, +}^*( \xi )$ and $\hpsi_{E,\, -}^*( \bxi )$.
+In these coordinates the boundary conditions~\eqref{eq:boundary-conditions-solutions} translate to:
+\begin{equation}
+  \begin{cases}
+    \hpsi_{E,\, -}^i( \tau_E - i\, 0^+ )
+    & =
+    \tensor{\qty( R_{(t)} )}{^i_j}\,
+    \hpsi_{E,\, +}^j(\tau_E + i\, 0^+ )
+    \\
+    \hpsi_{E,\, -,\, i}^{*}( \tau_E - i\, 0^+ )
+    & =
+    \tensor{\qty( R_{(t)}^* )}{_i^j}\,
+    \hpsi_{E,\, +,\, j}^*(\tau_E + i\, 0^+ )
+  \end{cases}
+  \label{eq:bc_eu_strip}
+\end{equation}
+for $\tau_E \in \qty( \htau_{E,\, (t)}, \htau_{E,\, (t-1)} )$ and
+\begin{equation}
+  \begin{cases}
+    \hpsi_{E,\, -}^i( \tau_E - i\, \pi )
+    & =
+    -\hpsi_{E,\, +}^i( \tau_E + i\, \pi )
+    \\
+    \hpsi_{E,\, -,\, i}^*( \tau_E - i\, \pi )
+    & =
+    -\hpsi_{E,\, +,\, i}^*( \tau_E + i\, \pi )
+  \end{cases},
+\end{equation}
+where $t = 1,\, 2,\, \dots,\, N$ and $\htau_{E,\, (t)}$ are the Wick-rotated locations of the $N$ zero-dimensional defects, analytically continued to a real value.
+
+The conserved product on the strip becomes:
+\begin{equation}
+  \consprod{\halpha^*_E}{\hbeta_E}
+  =
+  \cN
+  \finiteint{\sigma}{0}{\pi}
+  \qty(
+    \halpha^*_{E,\, +,\, i} \hbeta_{E,\, +}^i
+    +
+    \halpha^*_{E,\, -,\, i}  \hbeta_{E,\, -}^i
+  ),
+  \label{eq:euclidean-conserved-product-strip}
+\end{equation}
+where $\halpha^*_E$ and $\hbeta_E$ are the Euclidean counterparts of the generic solutions in the original definition of the product in~\eqref{eq:conserved-product}.
+In the Euclidean context we have to explicitly write $\halpha^*_E$ because it is no longer the ``complex conjugate''  of $\halpha_E$ in the traditional sense.
+The product is conserved only when it couples two solutions which have different boundary conditions as in~\eqref{eq:bc_eu_strip}.
+
+The definition of the stress-energy tensor in~\eqref{eq:stress-energy-tensor-lightcone} requires a change in the numerical pre-factor to use the usual CFT normalization.
+Introducing a spacetime variable central charge as well the components of the stress-energy tensor become:\footnotemark{}
+\footnotetext{%
+  The canonical coefficient in front of the CFT stress-energy tensor is such that the Euclidean Hamiltonian $\rL_{0}$ is normalized such that
+  \begin{equation*}
+    \cT_{\zeta\zeta}( \zeta ) = \sum_n \rL_{n} e^{-n \zeta}
+  \end{equation*}
+  (we are anticipating the double strip notation defined in the next subsection for simplicity).
+  We thus get:
+  \begin{equation*}
+     \rH_E
+     =
+     \rL_{0}
+     =
+     \int\limits_{0}^{2\pi} \frac{\dd{\phi}}{2 \pi}
+     \cT_{\zeta \zeta}( \tau_E + i\, \phi )
+  \end{equation*}
+  therefore $\cT_{\zeta\zeta}( \zeta ) = 2 \pi\, \cT^{(can)}_{\zeta\zeta}( \zeta )$.
+}
+\begin{equation}
+  \begin{split}
+    \cT_{\xi \xi}( \xi )
+    & =
+    - \frac{\pi T}{2}\,
+    \hpsi_{E,\, +,\, i}^*( \xi )\, \lrpartial{\xi} \hpsi^i_{E,\, +}( \xi )
+    +
+    \widehat{\cC}( \xi ),
+    \\
+    \cT_{\bxi \bxi}( \bxi )
+    & =
+    - \frac{\pi T}{2}\,
+    \hpsi_{E,\, -,\, i}^*( \bxi ) \lrpartial{\bxi}
+    \hpsi^i_{E,\, -}( \bxi )
+    +
+    \widehat{\overline{\cC}}( \bxi ),
+  \end{split}
+\end{equation}
+where $\widehat{\cC}$ and $\widehat{\overline{\cC}}$ are the leftover terms after the regularization of the singularities due to the normal ordering.
+The canonical anti-commutation relations are then
+\begin{equation}
+  \eval{
+    \qty[
+      \hpsi_{E,\, \pm}^i( \xi_1,\, \bxi_1),
+      \hpsi_{E,\, \pm,\, j}^*( \xi_2,\, \bxi_2 )
+    ]_+
+  }_{\Re\xi_1 = \Re\xi_2}
+  =
+  \frac{2}{T}\,
+  \tensor{\delta}{^i_j}\,
+  \delta\qty( \Im\xi_1 - \Im\xi_2 ).
+\end{equation}
+
+Given the Euclidean modes $\hpsi^i_{E,\, \pm,\, n}$ and $\hpsi^*_{E,\, \pm,\, n,\, i}$ where $n \in \Z$, we can then define the dual modes $\dual{\hpsi}^i_{E,\, n}$ and $\dual{\hpsi}^*_{E,\, n,\, i}$ such that the conserved product~\eqref{eq:euclidean-conserved-product-strip} between them gives:
+\begin{equation}
+  \lconsprod{\dual{\hpsi}^*_{E,\, n}}{\hpsi_{E,\, m}}
+  =
+  \lconsprod{\dual{\hpsi}_{E,\, n}}{\hpsi^*_{E,\, m}}
+  =
+  \delta_{n,m}.
+\end{equation}
+We can then expand the fields as
+\begin{equation}
+  \begin{cases}
+    \hpsi^i_{E,\, +}(\xi)
+    & =
+    \sum\limits_{n \in \Z} b_n\, \hpsi^i_{E,\, +,\, n}(\xi)
+    \\
+    \hpsi^i_{E,\, -}(\bxi)
+    & =
+    \sum\limits_{n \in \Z} b_n\, \hpsi^i_{E,\, -,\, n}(\bxi)
+  \end{cases}
+\end{equation}
+and
+\begin{equation}
+  \begin{cases}
+    \hpsi^*_{E,\, +,\, i}(\xi)
+    & =
+    \sum\limits_{n \in \Z} b^*_n\, \hpsi^*_{E,\, +,\, n,\, i}(\xi)
+    \\
+    \hpsi^*_{E,\, -,\, i}(\bxi)
+    & =
+    \sum\limits_{n \in \Z} b^*_n\, \hpsi^*_{E,\, -,\, n,\, i}(\bxi)
+  \end{cases}
+\end{equation}
+in order to extract the operators through the conserved product
+\begin{equation}
+  b_n
+  =
+  \lconsprod{\dual{\hpsi}^*_{E,\, n}}{\hpsi_{E}},
+  \qquad
+  b^*_n
+  =
+  \lconsprod{\dual{\hpsi}_{E,\, n}}{\hpsi^*_{E}},
+\end{equation}
+and get the anti-commutation relations at fixed Euclidean time as
+\begin{equation}
+  \eval{ \qty[ b_n,\, b^*_m ]_+ }_{\tau_E = \tau_{E,\, (0)}}
+  =
+  \frac{2 \cN}{T}
+  \lconsprod{\dual{\hpsi}^*_{E,\, n}}{\dual{\hpsi}_{E,\, m}}.
+\end{equation}
+
+
+\subsubsection{Double Strip Formalism and Doubling Trick}
+
+It is natural to use the doubling trick on the strip to simplify the previous expressions by gluing the holomorphic and anti-holomorphic fields along the $\sigma = \pi$ boundary.
+Define the coordinate $\zeta = \tau_E + i\, \phi$ with $0 \le \phi \le 2\pi$.
+We then have
+\begin{equation}
+  \Hpsi( \zeta )
+  =
+  \begin{cases}
+    \hpsi_{E,\, +}(\zeta)
+    &
+    \qfor
+    \phi = \sigma \in \qty[ 0, \pi ]
+    \\
+    -\hpsi_{E,\, -}(\zeta - 2 \pi i)
+    &
+    \qfor
+    \phi = 2\pi - \sigma \in \qty[ \pi, 2 \pi ]
+  \end{cases}
+\end{equation}
+on-shell (and similarly for $\Hpsi^*( \zeta )$ with the substitution $\hpsi_{E,\, \pm} \to \hpsi_{E,\, \pm}^*$).
+The ``complex conjugation'' $\star$ acts on the off-shell double fields as
+\begin{equation}
+  \qty[ \Hpsi^i(\zeta,\, \bzeta) ]^\star = \Hpsi_i^*(-\bzeta,\, -\zeta),
+\end{equation}
+while the boundary conditions are translated into
+\begin{equation}
+  \begin{cases}
+    \Hpsi^i( \tau_E + 2 \pi i^- )
+    =
+    -\tensor{\qty( R_{(t)} )}{^i_j}\,
+    \Hpsi^j( \tau_E + i\, 0^+ )
+    \\
+    \Hpsi^{* i}( \tau_E + 2 \pi i^- )
+    =
+    -\tensor{\qty( R_{(t)}^* )}{^i_j}\,
+    \Hpsi^{* j}( \tau_E + i\, 0^+ )
+  \end{cases}
+\end{equation}
+for $\tau_E \in \qty( \htau_{E,\, (t)}, \htau_{E,\, (t-1)} )$.
+The conserved product can then be defined as
+\begin{equation}
+  \consprod{\widehat{A}^*}{\widehat{B}}
+  =
+  \cN
+  \finiteint{\phi}{0}{2\pi}\,
+  \widehat{A}^*_i(\tau_E + i\, \phi )\,
+  \widehat{B}^i( \tau_E + i\, \phi ),
+\end{equation}
+where $\widehat{A}^*$ and $\widehat{B}$ are the double fields connected to $\halpha^*_E$ and $\hbeta_E$ in the previous definition on the strip.
+The holomorphic stress-energy tensor is then
+\begin{equation}
+  \cT_{\zeta \zeta}( \zeta )
+  =
+  - \frac{\pi T}{2}\,
+  \Hpsi_{i}^*( \zeta )\, \lrpartial{\zeta} \Hpsi^i( \zeta ) 
+  +
+  \widehat{\cC}(\zeta)
+\end{equation}
+and the canonical anti-commutation relations are now
+\begin{equation}
+  \eval{
+    \qty[
+      \Hpsi^i( \zeta_1 )
+      ,
+      \Hpsi_{j}^*( \zeta_2 )
+    ]_+
+  }_{\Re\zeta_1 = \Re\zeta_2}
+  =
+  \frac{2}{T}\, \tensor{\delta}{^i_j} \delta\qty( \Im\zeta_1 - \Im\zeta_2 ).
+\end{equation}
+
+
+The double field formulation shows that we need only one coefficient $b_n$ (or $b_n^*$) for both $\psi_{E,\, +}$ and $\psi_{E,\, -}$ (or for both $\psi^*_{E,\, +}$ and $\psi^*_{E,\, -}$).
+In fact, given the Euclidean modes $\Hpsi^i_{n}$ and $\Hpsi^*_{n,\, i}$ where $n \in \Z$, we define the dual modes $\dual{\Hpsi}^i_{n}$ and $\dual{\Hpsi}^*_{n,\, i}$ such that:
+\begin{equation}
+  \lconsprod{\dual{\Hpsi}^*_{n}}{\Hpsi_{m}}
+  =
+  \lconsprod{\dual{\Hpsi}_{n}}{\Hpsi^*_{m}}
+  =
+  \delta_{n,m}.
+\end{equation}
+We expand the double fields as
+\begin{equation}
+  \Hpsi^i(\zeta)
+  =
+  \sum\limits_{n \in \Z} b_n \Hpsi^i_{n}(\zeta),
+  \qquad
+  \Hpsi^*_{i}(\zeta)
+  =
+  \sum\limits_{n \in \Z} b^*_n \Hpsi^*_{n}(\zeta)
+\end{equation}
+Operators are then extracted as
+\begin{equation}
+  b_n
+  =
+  \lconsprod{\dual{\Hpsi}^*_{n}}{\Hpsi},
+  \qquad
+  b^*_n
+  =
+  \lconsprod{\dual{\Hpsi}_{n}}{\Hpsi^*}.
+  \label{eq:upper-half-extraction}
+\end{equation}
+Finally we get the anti-commutation relations as
+\begin{equation}
+  \eval{ \qty[ b_n, b^*_m ]_+ }_{\tau_E = \tau_{E,\, 0}}
+  =
+  \frac{2 \cN}{T} \lconsprod{\dual{\Hpsi}^*_{n}}{\dual{\Hpsi}_m}.
+\end{equation}
+
+
+\subsection{Fields on the Upper Half Plane}
+
+\begin{figure}[tbp]
+  \centering
+  \includegraphics[width=0.5\linewidth]{img/complex-plane}
+  \caption{%
+    Fields are glued on the $x < 0$ semi-axis with non trivial discontinuities for $x_t < x < x_{t-1}$ for $t = 1,\, 2,\, \dots,\, N$ and where $x_t = \exp( \htau_{E,\, (t)} )$.
+  }
+  \label{fig:complex-plane}
+\end{figure}
+
+We consider a set of coordinates on the upper half \ccH of the complex plane:
+\begin{equation}
+  u = e^{\xi} \in \ccH,
+\end{equation}
+where $\xi = \tau_E + i \sigma$ and $\sigma \in \qty[ 0, \pi ]$ define the usual strip, and $\ccH = \qty{ w \in \C \mid \Im w \ge 0 }$.
+These coordinates can then be extended to the entire complex plane by considering
+\begin{equation}
+  z = e^{\zeta} \in \C,
+\end{equation}
+where $\zeta = \tau_E + i \phi$ and $\phi \in \qty[0, 2\pi ]$ define the double strip.
+Under this change of coordinates the Euclidean action~\eqref{eq:S_Eu_strip} becomes
+\begin{equation}
+  \begin{split}
+    S_E
+    & =
+    \frac{T}{2}
+    \iint \dd{u}\dd{\bu}\,
+    \frac{1}{2}\,
+    \qty(
+      \frac{1}{u}\, \hpsi_{E,\, +,\, i}^* \lrpartial{\bu} \hpsi_{E,\, +}^i
+      +
+      \frac{1}{\bu}\, \hpsi_{E,\, -,\, i}^* \lrpartial{u} \hpsi_{E,\, -}^i
+    )
+    \\
+    & =
+    \frac{T}{2}
+    \iint \dd{u}\dd{\bu}\,
+    \frac{1}{2}\,
+    \qty(
+      \psi_{E,\, +,\, i}^* \lrpartial{\bu} \psi_{E,\, +}^i
+      +
+      \psi_{E,\, -,\, i}^* \lrpartial{u} \psi_{E,\, -}^i
+    ),
+  \end{split}
+\end{equation}
+where we introduce the off-shell field redefinitions:
+\begin{equation}
+  \psi_{E,\, +}^i(u,\, \bu)
+  =
+  \frac{1}{\sqrt{u}}\, \hpsi_{E,\, +}^i( \xi,\, \bxi ),
+  \qquad
+  \psi_{E,\, -}^i(u,\, \bu)
+  =
+  \frac{1}{\sqrt{\bu}}\, \hpsi_{E,\, -}^i( \xi,\, \bxi ).
+  \label{eq:euclidean-off-shell-redefinitions}
+\end{equation}
+Fields with the hat sign on top thus represent strip and double strip definitions, while fields without the hat sign are defined on $\ccH$ or $\C$.\footnotemark{}
+\footnotetext{%
+  We could have anticipated these redefinitions from a CFT argument where
+    \begin{equation*}
+      \psi( u ) = \eval{\qty( \dv{u}{\xi} )^{-\frac{1}{2}}
+        {\hpsi}(\xi)}_{\xi = \ln( u )},
+    \end{equation*}
+  but we cannot and do not rely on CFT properties since we have not shown that the theory is a CFT yet.
+}
+Notice that this is the result one would expect from the engineering dimension: in this case it works since the theory is essentially free.
+Using the redefinitions~\eqref{eq:euclidean-off-shell-redefinitions}, the off-shell ``complex conjugation'' $\star$ then becomes
+\begin{equation}
+  \qty[ \psi_{E,\, +,\, i}( u,\, \bu ) ]^\star
+  =
+  \frac{1}{\bu}\, \psi_{E,\, +,\, i}^*\qty( \frac{1}{\bu},\, \frac{1}{u} ),
+  \qquad
+  \qty[ \psi_{E,\, -,\, i}( u,\, \bu ) ]^\star
+  =
+  \frac{1}{u}\, \psi_{E,\, -,\, i}^*\qty( \frac{1}{\bu},\, \frac{1}{u} ).
+\end{equation}
+
+We choose to insert the cut of the square root on the real negative axis.
+The boundary conditions are then translated into
+\begin{equation}
+  \begin{cases}
+    \psi_{E,\, -}^i( x - i\, 0^+ )
+    =
+    \tensor{\qty( R_{(t)} )}{^i_j} \psi_{E,\, +}^j( x + i\, 0^+ )
+    \\
+    \psi^{*}_{E,\, -,\, i}( x - i\, 0^+ ) =
+    \tensor{\qty( R_{(t)}^* )}{_i^j} \psi^{*}_{E,\, +,\, j}( x + i\, 0^+ )
+  \end{cases}
+\end{equation}
+for $x \in \qty( x_{(t)}, x_{(t-1)} )$, where $x_{(t)} = \exp( \htau_{E,\, (t)} ) > 0$, and
+\begin{equation}
+  \psi_{E,\, -}^i( x - i\, 0^+ )
+  =
+  \psi_{E,\, +}^i( x + i\, 0^+ ),
+  \qquad
+  \psi^{*}_{E,\, -,\, i}( x - i\, 0^+ )
+  =
+  \psi^{*}_{E,\, +,\, i}( x + i\, 0^+ )
+\end{equation}
+for $x<0$.
+
+The product~\eqref{eq:euclidean-conserved-product-strip} is then
+\begin{equation}
+  \consprod{\alpha^*}{\beta}
+  =
+  -i \cN\,
+  \qty[
+    \int\limits_{\widehat{\Sigma}} \dd{u}
+    \alpha^*_{+,\, i}(u) \beta_+^i(u)
+    -
+    \int\limits_{\widehat{\overline{\Sigma}}} \dd{\bu}
+    \alpha^*_{-,\, i}(\bu) \beta_-^i(\bu)
+  ],
+  \label{eq:prod_H}
+\end{equation}
+where integrations are computed over two semi-circles $\widehat{\Sigma} = \qty{ u \in \C \mid \abs{u} = \exp( \htau_E ),\, 0 \le \Im u \le \pi }$ and $\widehat{\overline{\Sigma}} = \qty{ u \in \C \mid \abs{u} = \exp( \htau_E ),\, -\pi \le \Im u \le 0}$.
+The stress-energy tensor becomes:\footnotemark{}
+\footnotetext{%
+  While rewriting the operator part of the stress-energy tensor from the strip
+  formulation into the coordinates in $\ccH$ we actually get
+  \begin{equation*}
+    \cT_{\xi \xi}( \xi(u) ) = u^2\, \cT_{u u}( u ).
+  \end{equation*}
+  The reason of the presence of $u^2$ factor can be explained in two ways.
+  Using GR we know that $\cT_{\xi \xi}( \xi ) \dss[2]{\xi} = \cT_{u u}( u ) \dss[2]{u}$.
+  Another way is to notice that a translation in $\xi$ is a dilatation of $u$: the infinitesimal generator of $\xi$ translation must be the infinitesimal
+  generator of $u$ dilatation, that is:
+  \begin{equation*}
+    P_{\xi}
+    \sim
+    \int \dd{\sigma}\, \cT_{\xi \xi}
+    \sim
+    D_u
+    \sim 
+    \int \dd{u}\, u\, \cT_{u u}.
+  \end{equation*}
+}
+\begin{equation}
+  \begin{split}
+    \cT_{u u}( u )
+    & =
+    - \frac{\pi T}{2}\,
+    \psi_{E,\, +,\, i}^*( u )\, \lrpartial{u} \psi^i_{E,\, +}( u )
+    +
+    \widehat{\cC}(u ),
+    \\
+    \cT_{\bu \bu}( \bu )
+    & =
+    - \frac{\pi T}{2}\,
+    \psi_{E,\, -,\, i}^*( \bu )\, \lrpartial{\bu} \psi^i_{E,\, -}( \bu )
+    +
+    \widehat{\overline{\cC}}( \bu ).
+  \end{split}
+\end{equation}
+Finally the anti-commutation relations are
+\begin{equation}
+  \begin{cases}
+    \eval{
+      \qty[
+        \psi_{E,\, +}^i( u_1,\, \bu_1 )
+        ,
+        \psi_{E,\, +,\, j}^*( u_2,\, \bu_2 )
+      ]_+
+    }_{\abs{u_1} = \abs{u_2}}
+    & =
+    \frac{2}{T u_1}\, \tensor{\delta}{^i_j}\, \delta( \arg(u_1) - \arg(u_2) )
+    \\
+    \eval{
+      \qty[
+        \psi_{E,\, -}^i( u_1,\, \bu_1 )
+        ,
+        \psi_{E,\, -,\, j}^*( u_2,\, \bu_2 )
+      ]_+
+    }_{\abs{u_1} = \abs{u_2}}
+    & =
+    \frac{2}{T \bu_1}\, \tensor{\delta}{^i_j}\, \delta( \arg(u_1) - \arg(u_2) ),
+  \end{cases}
+\end{equation}
+which despite the strange look of the expression are perfectly compatible with the definition~\eqref{eq:upper-half-extraction} leading to:
+\begin{equation}
+  \qty[ b_n,\, b^*_m ]_+
+  =
+  \frac{2 \cN}{T}\,
+  \lconsprod{\dual{\hpsi}^*_{E,\, n}}{\dual{\hpsi}_{E,\, m}}
+  =
+  \frac{2 \cN}{T}\,
+  \lconsprod{\dual{\psi}^*_{E,\, n}}{\dual{\psi}_{E,\, m}}
+\end{equation}
+when the product $\lconsprod{\cdot}{\cdot}$ is defined according to~\eqref{eq:prod_H}.
+We expand the fields in modes as:
+\begin{equation}
+  \begin{cases}
+    \psi^i_{E,\, +}(u)
+    =
+    \sum\limits_{n \in \Z} b_n\, \psi^i_{E,\, +,\, n}(u)
+    \\
+    \psi^i_{E,\, -}(\bu)
+    =
+    \sum\limits_{n \in \Z} b_n\, \psi^i_{E,\, -,\, n}(\bu)
+  \end{cases}
+\end{equation}
+and
+\begin{equation}
+  \begin{cases}
+    \psi^*_{E,\, +,\, i}(u)
+    =
+    \sum\limits_{n \in \Z} b^*_n\, \psi^*_{E,\, +,\, n,\, i}(u)
+    \\
+    \psi^*_{E,\, -,\, i}(\bu)
+    =
+    \sum\limits_{n \in \Z} b^*_n\, \psi^*_{E,\, -,\, n,\, i}(\bu)
+  \end{cases}
+\end{equation}
+and $\dual{\psi}_{E,\, n}$ and $\dual{\psi}^*_{E,\, n}$ are the corresponding dual modes on the upper half plane.   
+
+
+\subsection{Fields on the Complex Plane and the Doubling Trick}
+  
+We use again the doubling trick to define the fields on the subset $\C \setminus \qty[ x_{(N)},\, x_{(1)} ]$:
+\begin{equation}
+  \Psi( z )
+  =
+  \begin{cases}
+    \psi_{E,\, +}(u)
+    & \qfor
+    z = u \in \ccH \setminus \qty[ x_{(N)}, x_{(1)} ]
+    \\
+    \psi_{E,\, -}(\bu)
+    & \qfor
+    z = \bu \in \overline{\ccH} \setminus \qty[ x_{(N)}, x_{(1)} ]
+  \end{cases}
+\end{equation}
+where $z = \exp( \tau_E + i\, \phi ) = x + i y$ and $\overline{\ccH} = \qty{ w \in \C \mid \Im w \le 0 }$.
+The same procedure applies also to $\Psi^*$ with the exchange $\psi_{E,\, \pm} \leftrightarrow \psi_{E,\, \pm}^*$.
+
+In this case the ``complex conjugation'' $\star$ acts off-shell as
+\begin{equation}
+  \qty[ \Psi^i( z, \bz ) ]^\star
+  =
+  \frac{1}{\bz}\, \Psi_i^*\qty(\frac{1}{\bz}, \frac{1}{z}).
+  \label{eq:complex-plane-conjugate}
+\end{equation}
+The boundary conditions then become:
+\begin{equation}
+  \begin{cases}
+    \Psi^i( x - i\, 0^+ )
+    & =
+    \tensor{\qty( R_{(t)} )}{^i_j} \Psi^j( x + i\, 0^+ ),
+    \\
+    \Psi^{*\, i}( x - i 0^+ )
+    & =
+    \tensor{\qty( R_{(t)}^* )}{^i_j} \Psi^{*\, j}( x + i\, 0^+ ),
+  \end{cases}
+  \label{eq:boundary-condition-euclidean}
+\end{equation}
+for $x \in \qty( x_{(t)}, x_{(t-1)} )$, where $x_{(t)} = \exp( \htau_{E,\, (t)} ) > 0$ for $t \in \qty{ 1,\, 2,\, \dots,\, N }$.
+When $x < 0$ we get
+\begin{equation}
+  \begin{cases}
+    \Psi( x - i\, 0^+ ) & = \Psi( x + i\, 0^+ ),
+    \\
+    \Psi^*( x - i\, 0^+ ) & = \Psi^*( x + i\, 0^+ )
+  \end{cases}.
+  \label{eq:gluing-conditions-euclidean}
+\end{equation}
+
+Given the relations $\dd{z} = i\, z\, \dd{\phi}$, we can write the conserved product \eqref{eq:prod_H} as:
+\begin{equation}
+  \consprod{A^*}{B}
+  =
+  2\pi \cN
+  \oint\limits_{\abs{z} = \exp( \tau_E )}
+  \frac{\dd{z}}{2 \pi i}\, A^*_i( z )\, B^i( z ),
+  \label{eq:conserved-product-complex-plane}
+\end{equation}
+where we explicitly stressed that the integral has to be performed at a fixed Euclidean time $\tau_E$: in the new coordinate on the plane the conserved product becomes a contour integral at a fixed radius from the origin.
+
+In the same way we can recast the stress-energy tensor components~\eqref{eq:stress-energy-tensor-lightcone} in the new coordinates:
+\begin{equation}
+  \cT( z )
+  =
+  - \frac{\pi T}{2}\,
+  \Psi^*_i( z )\, \lrpartial{z} \Psi^i( z )
+  +
+  \cC( z ),
+\end{equation}
+where $\cT = \cT_{zz}$ for simplicity.
+
+Finally the canonical anti-commutation relations between the fields are:
+\begin{equation}
+  \eval{
+    \qty[
+      \Psi^i( z_1 ),\,
+      \Psi_{j}^*( z_2 )
+    ]_+
+  }_{\abs{z_1} = \abs{z_2}}
+  =
+  \frac{2}{T z_1}\, \tensor{\delta}{^i_j}\, \delta( \arg(z_1) - \arg(z_2) ).
+\end{equation}
+The fields expansion in modes thus reads
+\begin{equation}
+  \Psi^i(z)
+  =
+  \sum\limits_{n \in \Z} b_n\, \Psi^i_{n}(z),
+  \qquad
+  \Psi^*_{i}(z)
+  =
+  \sum\limits_{n \in \Z} b^*_n\, \Psi^*_{n. i}(z).
+  \label{eq:complex-plane-mode-expansion}
+\end{equation}
+The anti-commutation relations among the operators are
+\begin{equation}
+  \qty[ b_n,\, b^*_m ]_+
+  =
+  \frac{2 \cN}{T}\, \lconsprod{\dual{\Psi}^*_{n}}{\dual{\Psi}_{m}},
+\end{equation}
+when we introduce the dual modes
+$\dual{\Psi}_{n}(z)$ and $\dual{\Psi}^*_{n}(z)$ whose normalization is
+\begin{equation}
+  \lconsprod{\dual{\Psi}^*_{n}}{{\Psi}_{m}}
+  =
+  \lconsprod{\dual{\Psi}_{n}}{{\Psi}^*_{m}}
+  =
+  \delta_{m,n}.
+\end{equation}
+
+
+\subsection{Algebra of Creation and Annihilation Operators}
+\label{sec:modes_and_algebra}
+  
+In this section we find the explicit expression of the modes which satisfy the \eom and the boundary conditions.
+We then compute the dual fields and finally the algebra of the creators and annihilators.
+
+
+\subsubsection{NS Complex Fermions}
+\label{sec:ns-complex-fermions}
+
+We start from NS complex fermions to show that the formalism can recover known results.
+Consider the usual definition:
+\begin{equation}
+  \begin{cases}
+    \psi_-^i( \tau, 0 )
+    & =
+    \psi_+^i( \tau, 0 ),
+    \\
+    \psi_-^i( \tau, \pi )
+    & =
+    -\psi_+^i( \tau, \pi )
+  \end{cases}
+\end{equation}
+for $\tau \in \R$, which can be recovered from~$\eqref{eq:boundary-conditions-solutions}$ setting $R_{(t)} \equiv \1$.
+In the Euclidean formulation we use~\eqref{eq:boundary-condition-euclidean} and~\eqref{eq:gluing-conditions-euclidean} to get:
+\begin{equation}
+  \begin{cases}
+    \Psi( x - i\, 0^+ ) & =  \Psi( x + i\, 0^+ )
+    \\
+    \Psi^*( x - i\, 0^+ ) & =  \Psi^*( x + i\, 0^+ )
+  \end{cases}
+\end{equation}
+for $x \in \R$.
+
+We define:
+\begin{eqnarray}
+  \Psi^i_{( n,\, i_0 )}( z )
+  & = &
+  \cN_{\Psi}\, \delta^i_{i_0}\, z^{-n},
+  \\
+  \dual{\Psi}_{( m,\, j_0 ),\, j}( z )
+  & = &
+  \qty( 2 \pi \cN\, \cN_{\Psi} )^{-1}\, \delta_{j, j_0}\, z^{m-1}
+\end{eqnarray}
+to recover the definition of the dual modes~\eqref{eq:conserved-product-dual-basis} using the Euclidean conserved product \eqref{eq:conserved-product-complex-plane}.
+We then proceed similarly for $\Psi^*$ in such a way that
+\begin{equation}
+  \lconsprod{\dual{\Psi}_{( n,\, i_0 )}^*}{\Psi_{( m,\, j_0 )}}
+  =
+  \lconsprod{\dual{\Psi}_{( m,\, j_0 )}}{\Psi^*_{( n,\, i_0 )}}
+  =
+  \delta_{n, m}\, \delta_{i_0, j_0}.
+\end{equation}
+As a consequence we find
+\begin{equation}
+  \lconsprod{\dual{\Psi}_{( n,\, i_0 )}^*}{\dual{\Psi}_{( m,\, i_1 )}}
+  =
+  \frac{1}{2 \pi \cN\, \cN^2_{\Psi}}\, \delta_{i_0, i_1}\, \delta_{n + m, 1}.
+\end{equation}
+
+Consider the  NS expansion in modes of the double fields:
+\begin{eqnarray}
+  \Psi^i( z )
+  & = &
+  \sum\limits_{n \in \Z}\,
+  \sum\limits_{i_0}\,
+  b_{(n,\, i_0)}\, \Psi^i_{( n,\, i_0 )}( z ),
+  \\
+  \Psi^{*}_i( z )
+  & = &
+  \sum\limits_{n \in \Z}\,
+  \sum\limits_{i_0}\,
+  b^*_{(n,\, i_0)}\, \Psi^{*}_{( n,\, i_0 ),\, i}( z ),
+\end{eqnarray}
+then
+\begin{eqnarray}
+  b_{( n,\, i_0 )}
+  & = &
+  \lconsprod{\dual{\Psi}_{( n,\, i_0 )}^*}{\Psi},
+  \\
+  b^*_{( n,\, i_0 )}
+  & = &
+  \lconsprod{\dual{\Psi}_{( n,\, i_0)}}{\Psi^*},
+\end{eqnarray}
+and
+\begin{equation}
+  \qty[ b_{( n, i_0 )},\, b^*_{( m, j_0 )} ]_+
+  =
+  \frac{1}{\pi T \cN_{\Psi}^2}\, \delta_{i_0, j_0}\, \delta_{n + m, 1}.
+  \label{eq:ns-algebra}
+\end{equation}
+
+
 % vim: ft=tex
diff --git a/sec/part1/introduction.tex b/sec/part1/introduction.tex
index 2fc3d52..18d4458 100644
--- a/sec/part1/introduction.tex
+++ b/sec/part1/introduction.tex
@@ -19,7 +19,7 @@ In particular we recall some results on the symmetries of string theory and how
 \subsection{Properties of String Theory and Conformal Symmetry}
 
 Strings are extended one-dimensional objects.
-They are curves in spacetime parametrized by a coordinate $\sigma \in \left[0, \ell \right]$.
+They are curves in spacetime parametrized by a coordinate $\sigma \in \qty[0, \ell ]$.
 When propagating they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}(\tau, \sigma)$ where $\mu = 0, 1, \dots, D - 1$ indexes the coordinates.
 Such surface can have different topologies according to the nature of the object propagating in spacetime: strings can be \emph{closed} if $X^{\mu}(\tau, 0) = X^{\mu}(\tau, \ell)$ or \emph{open} if the endpoints in $\sigma = 0$ and $\sigma = \ell$ do not coincide.
 
@@ -46,11 +46,11 @@ The \eom for the string $X^{\mu}(\tau, \sigma)$ is therefore
 \begin{equation}
   \frac{1}{\sqrt{- \det \gamma}}\,
   \ipd{\alpha}
-  \left(
+  \qty(
     \sqrt{- \det \gamma}\,
     \gamma^{\alpha\beta}\,
     \ipd{\beta} X^{\mu}
-  \right)
+  )
   =
   0,
   \qquad
@@ -66,14 +66,14 @@ In fact
   =
   - \frac{1}{4 \pi \ap}
   \sqrt{- \det \gamma}\,
-  \left(
+  \qty(
     \ipd{\alpha} X \cdot \ipd{\beta} X
     -
     \frac{1}{2}
     \gamma_{\alpha\beta}\,
     \gamma^{\lambda\rho}\,
     \ipd{\lambda} X \cdot \ipd{\rho} X
-  \right)
+  )
   =
   0
   \label{eq:conf:worldsheetmetric}
@@ -152,13 +152,13 @@ In fact the classical constraint on the tensor is simply
   \fdv{S_P[\gamma, X]}{\gamma^{\alpha\beta}}
   =
   -\frac{1}{\ap}
-  \left(
+  \qty(
     \ipd{\alpha} X \cdot \ipd{\beta} X
     -
     \frac{1}{2} \eta_{\alpha\beta}\,
     \eta^{\lambda\rho}\,
     \ipd{\lambda} X \cdot \ipd{\rho} X
-  \right)
+  )
   =
   0.
   \label{eq:conf:stringT}
@@ -188,33 +188,33 @@ while its conservation $\partial^{\alpha} T_{\alpha\beta} = 0$ becomes:\footnote
   Since we fix $\gamma_{\alpha\beta}(\tau, \sigma) \propto \eta_{\alpha\beta}$ we do not need to account for the components of the connection and we can replace the covariant derivative $\nabla^{\alpha}$ with a standard derivative $\partial^{\alpha}$.
 }
 \begin{equation}
-  \bpd T_{\xi\xi}( \xi, \bxi ) = \pd \bT_{\bxi\bxi}( \xi, \bxi ) = 0.
+  \bpd T_{\xi\xi}( \xi,\, \bxi ) = \pd \bT_{\bxi\bxi}( \xi,\, \bxi ) = 0.
 \end{equation}
 The last equation finally implies
 \begin{equation}
-  T_{\xi\xi}( \xi, \bxi ) = T_{\xi\xi}( \xi ) = T( \xi ),
+  T_{\xi\xi}( \xi,\, \bxi ) = T_{\xi\xi}( \xi ) = T( \xi ),
   \qquad
-  \bT_{\bxi\bxi}( \xi, \bxi ) = \bT_{\bxi\bxi}( \bxi ) = \bT( \bxi ),
+  \bT_{\bxi\bxi}( \xi,\, \bxi ) = \bT_{\bxi\bxi}( \bxi ) = \bT( \bxi ),
 \end{equation}
-which are respectively the holomorphic and the anti-holomorphic components of the 2-dimensional stress energy tensor.
+which are respectively the holomorphic and the anti-holomorphic components of the bidimensional stress energy tensor.
 
-The previous properties define what is known as a 2-dimensional \emph{conformal field theory} (\cft).
+The previous properties define what is known as a bidimensional \emph{conformal field theory} (\cft).
 Ordinary tensor fields
 \begin{equation}
-  \phi_{\omega, \bomega}( \xi, \bxi )
+  \phi_{\omega, \bomega}( \xi,\, \bxi )
   =
-  \phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}( \xi, \bxi )
-  \left( \dd{\xi} \right)^{\omega}
-  \left( \dd{\bxi} \right)^{\bomega}
+  \phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}( \xi,\, \bxi )
+  \qty( \dd{\xi} )^{\omega}
+  \qty( \dd{\bxi} )^{\bomega}
 \end{equation}
-can be classified according to their weight $\left( \omega, \bomega \right)$ referring to the holomorphic and anti-holomorphic parts respectively.
+are classified according to their weight $\qty( \omega,\, \bomega )$ referring to the holomorphic and anti-holomorphic parts respectively.
 In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$ maps the conformal fields to
 \begin{equation}
   \phi_{\omega, \bomega}( \chi, \bchi )
   =
-  \left( \dv{\chi}{\xi} \right)^{\omega}\,
-  \left( \dv{\bchi}{\bxi} \right)^{\bomega}\,
-  \phi_{\omega, \bomega}( \xi, \bxi ).
+  \qty( \dv{\chi}{\xi} )^{\omega}\,
+  \qty( \dv{\bchi}{\bxi} )^{\bomega}\,
+  \phi_{\omega, \bomega}( \xi,\, \bxi ).
 \end{equation}
 
 \begin{figure}[tbp]
@@ -238,9 +238,9 @@ In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$
 
 An additional conformal transformation
 \begin{equation}
-  z = e^{\xi} = e^{\tau_e + i \sigma} \in \left\lbrace z \in \C | \Im z \ge 0 \right\rbrace,
+  z = e^{\xi} = e^{\tau_e + i \sigma} \in \qty{ z \in \C | \Im z \ge 0 },
   \qquad
-  \bz = e^{\bxi} = e^{\tau_e - i \sigma} \in \left\lbrace z \in \C | \Im z \le 0 \right\rbrace
+  \bz = e^{\bxi} = e^{\tau_e - i \sigma} \in \qty{ z \in \C | \Im z \le 0 }
 \end{equation}
 maps the worldsheet of the string to the complex plane.
 On this Riemann surface the usual time ordering becomes a \emph{radial ordering} as constant time surfaces are circles around the origin (see the contours $\ccC_{(0)}$ and $\ccC_{(1)}$ in \Cref{fig:conf:complex_plane}). 
@@ -265,14 +265,14 @@ The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bome
     \liebraket{Q_{\epsilon, \bepsilon}}{\phi_{\omega, \bomega}( w, \bw )}
     \\
     & =
-    \cint{0} \ddz \epsilon(z) \left[ T(z), \phi_{\omega, \bomega}( w, \bw ) \right]
+    \cint{0} \ddz \epsilon(z) \qty[ T(z), \phi_{\omega, \bomega}( w, \bw ) ]
     +
-    \cint{0} \ddbz \bepsilon(\bz) \left[ \bT(\bz), \phi_{\omega, \bomega}( w, \bw ) \right]
+    \cint{0} \ddbz \bepsilon(\bz) \qty[ \bT(\bz), \phi_{\omega, \bomega}( w, \bw ) ]
     \\
     & =
-    \cint{w} \ddz \epsilon(z)\, \rR\!\left( T(z)\, \phi_{\omega, \bomega}( w, \bw ) \right)
+    \cint{w} \ddz \epsilon(z)\, \rR\!\qty( T(z)\, \phi_{\omega, \bomega}( w, \bw ) )
     +
-    \cint{\bw} \ddbz \bepsilon(\bz)\, \rR\!\left( \bT(\bz)\, \phi_{\omega, \bomega}( w, \bw ) \right),
+    \cint{\bw} \ddbz \bepsilon(\bz)\, \rR\!\qty( \bT(\bz)\, \phi_{\omega, \bomega}( w, \bw ) ),
   \end{split}
 \end{equation}
 where in the last passage we used the fact that the difference of ordered integrals becomes the contour integral of the radially ordered product computed surrounding $w$.
@@ -393,7 +393,7 @@ This ultimately leads to the quantum algebra
 known as Virasoro algebra, unique central extension of the classical de Witt algebra, with central charge $c$.
 Operators $L_n$ and $\bL_n$ are called Virasoro operators.\footnotemark{}
 \footnotetext{%
-  Notice that the subset of Virasoro operators $\left\lbrace L_{-1}, L_0, L_1 \right\rbrace$ forms a closed subalgebra generating the group $\SL{2}{\R}$.
+  Notice that the subset of Virasoro operators $\qty{ L_{-1}, L_0, L_1 }$ forms a closed subalgebra generating the group $\SL{2}{\R}$.
 }
 Notice that $L_0 + \bL_0$ is the generator of the dilations on the complex plane.
 In terms of radial quantization this translates to time translations and $L_0 + \bL_0$ can be considered to be the Hamiltonian of the theory.
@@ -592,13 +592,13 @@ In complex coordinates on the plane it is:
   =
   - \frac{1}{4 \pi}
   \iint \dd{z} \dd{\bz}
-  \left(
+  \qty(
     \frac{2}{\ap}\, \ipd{\bz} X^{\mu}\, \ipd{z} X^{\nu}
     +
     \psi^{\mu}\, \ipd{\bz} \psi^{\nu}
     +
     \bpsi^{\mu}\, \ipd{z} \bpsi^{\nu}
-  \right)
+  )
   \eta_{\mu\nu}.
   \label{eq:super:action}
 \end{equation}
@@ -641,7 +641,7 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet
     - \bepsilon( \bz )\, \ipd{\bz} \bX^{\mu}( \bz )
   \end{split}
 \end{equation}
-generated by the currents $J( z ) = \epsilon( z )\, T_F( z )$ and $\overline{J}( \bz ) = \bepsilon( \bz )\, \bT_F( \bz )$, where $\epsilon( z )$ and $\bepsilon( \bz ) = \left( \epsilon( z ) \right)^*$ are anti-commuting fermions and
+generated by the currents $J( z ) = \epsilon( z )\, T_F( z )$ and $\overline{J}( \bz ) = \bepsilon( \bz )\, \bT_F( \bz )$, where $\epsilon( z )$ and $\bepsilon( \bz ) = \qty( \epsilon( z ) )^*$ are anti-commuting fermions and
 \begin{equation}
   \begin{split}
     T_F( z )
@@ -681,7 +681,7 @@ The central charge associated to the Virasoro algebra is in this case given by b
 The central charge is therefore $c = \frac{3}{2} D$ for the \cft defined in~\eqref{eq:super:action}.
 
 As in the case of the bosonic string, in order to cancel the central charge we introduce the reparametrisation anti-commuting ghosts $b( z )$ and $c( z )$ and their anti-holomorphic components as well as the commuting \emph{superghosts} $\beta( z )$ and $\gamma( z )$ and their anti-holomorphic counterparts.
-These are conformal fields with conformal weights $\left( \frac{3}{2}, 0 \right)$ and $\left( -\frac{1}{2}, 0 \right)$.
+These are conformal fields with conformal weights $\qty( \frac{3}{2}, 0 )$ and $\qty( -\frac{1}{2}, 0 )$.
 Their central charge becomes $c_{\text{ghost}} = c_{bc} + c_{\beta\gamma} = -26 + 11 = -15$ (see \cref{note:conf:ghosts} for the general computation).
 When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\bT_{\text{full}} = \bT + \bT_{\text{ghost}}$ the central charge vanishes only in 10-dimensional spacetime:
 \begin{equation}
@@ -738,14 +738,14 @@ The tensor $J$ is called \emph{complex structure} if there exist a tensor $N$ su
 \begin{equation}
   \tensor{N}{^a_{bc}}\, v_p^b\, w_p^c
   =
-  \left(
+  \qty(
     \liebraket{v_p}{w_p}
     +
     J
-    \left( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} \right)
+    \qty( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} )
     -
     \liebraket{J\, v_p}{J\, w_p}
-  \right)^a
+  )^a
   =
   0
 \end{equation}
@@ -807,7 +807,7 @@ $(M, J, g)$ is a \emph{Kähler} manifold if:
 \begin{equation}
   \dd{\omega}
   =
-  \left( \pd + \bpd \right)
+  \qty( \pd + \bpd )
   \omega(z, \bz)
   =
   0,
@@ -833,7 +833,7 @@ In local coordinates a Hermitian metric is such that
 thus the Kähler form becomes $\omega = i g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}$.
 The relation~\eqref{eq:cy:kaehler} then translates into:
 \begin{equation}
-  \dd{\omega} = i\, \left( \pd + \bpd \right)\, g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}
+  \dd{\omega} = i\, \qty( \pd + \bpd )\, g_{a\overline{b}}\, \dd{z}^a \wedge \dd{\bz}^{\overline{b}}
   =
   0
   \quad
@@ -995,20 +995,20 @@ The usual mode expansion in conformal coordinates $X^{\mu}( z, \bz ) = X( z ) +
     x_0^{\mu}
     +
     i\, \sqrt{\frac{\ap}{2}}\,
-    \left(
+    \qty(
       - \alpha_0^{\mu}\, \ln{z}
       + \sum\limits_{n \in \Z \setminus \{0\}} \frac{\alpha_n^{\mu}}{n} z^{-n}
-    \right),
+    ),
     \\
     \bX^{\mu}( \bz )
     & =
     \overline{x}_0^{\mu}
     +
     i\, \sqrt{\frac{\ap}{2}}\,
-    \left(
+    \qty(
       - \balpha_0^{\mu}\, \ln{\bz}
       + \sum\limits_{n \in \Z \setminus \{0\}} \frac{\balpha_n^{\mu}}{n} \bz^{-n}
-    \right),
+    ),
   \end{split}
   \label{eq:tduality:modes}
 \end{equation}
@@ -1045,9 +1045,9 @@ respectively encoding the quantisation of the momentum for a compact coordinate
 We finally have
 \begin{equation}
   \begin{split}
-    \alpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \left( n\, \frac{\ap}{R} + m\, R \right),
+    \alpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \qty( n\, \frac{\ap}{R} + m\, R ),
     \\
-    \balpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \left( n\, \frac{\ap}{R} - m\, R \right),
+    \balpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \qty( n\, \frac{\ap}{R} - m\, R ),
   \end{split}
 \end{equation}
 
@@ -1058,24 +1058,24 @@ From~\eqref{eq:conf:Texpansion} and \eqref{eq:conf:bosonicstringT} we find
     L_0
     &=
     \frac{\ap}{2}\,
-    \left(
-      \left( \alpha_0^{D-1} \right)^2
+    \qty(
+      \qty( \alpha_0^{D-1} )^2
       +
-      \sum\limits_{i = 0}^{D-2}\, \left( \alpha_0^i \right)^2
+      \sum\limits_{i = 0}^{D-2}\, \qty( \alpha_0^i )^2
       +
-      \sum\limits_{n = 1}^{+\infty}\, \left( 2 \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a \right)
-    \right),
+      \sum\limits_{n = 1}^{+\infty}\, \qty( 2 \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a )
+    ),
     \\
     \bL_0
     &=
     \frac{\ap}{2}\,
-    \left(
-      \left( \balpha_0^{D-1} \right)^2
+    \qty(
+      \qty( \balpha_0^{D-1} )^2
       +
-      \sum\limits_{i = 0}^{D-2}\, \left( \balpha_0^i \right)^2
+      \sum\limits_{i = 0}^{D-2}\, \qty( \balpha_0^i )^2
       +
-      \sum\limits_{n = 1}^{+\infty}\, \left( 2 \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a \right)
-    \right),
+      \sum\limits_{n = 1}^{+\infty}\, \qty( 2 \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a )
+    ),
   \end{split}
 \end{equation}
 where $a$ is constant given by normal ordering, representing the zero point energy of the theory.
@@ -1084,14 +1084,14 @@ Imposing physical conditions~\eqref{eq:conf:physical} and the \emph{level matchi
   \begin{split}
     M^2
     & =
-    \frac{1}{\ap^2}\, \left( n\, \frac{\ap}{R} + m\, R \right)^2
+    \frac{1}{\ap^2}\, \qty( n\, \frac{\ap}{R} + m\, R )^2
     +
-    \frac{4}{\ap}\, \left( \rN + a \right)
+    \frac{4}{\ap}\, \qty( \rN + a )
     \\
     & =
-    \frac{1}{\ap^2}\, \left( n\, \frac{\ap}{R} - m\, R \right)^2
+    \frac{1}{\ap^2}\, \qty( n\, \frac{\ap}{R} - m\, R )^2
     +
-    \frac{4}{\ap}\, \left( \overline{\rN} + a \right),
+    \frac{4}{\ap}\, \qty( \overline{\rN} + a ),
   \end{split}
   \label{eq:dbranes:closedspectrum}
 \end{equation}
@@ -1129,7 +1129,7 @@ The usual mode expansion~\eqref{eq:tduality:modes} here leads to
   +
   i\, \sqrt{\frac{\ap}{2}}\,
   \sum\limits_{n \in \Z \setminus \{0\}}
-  \frac{\alpha_n^{\mu}}{n} \left( z^{-n} + \bz^{-n} \right)
+  \frac{\alpha_n^{\mu}}{n} \qty( z^{-n} + \bz^{-n} )
 \end{equation}
 and $\ell = \pi$.
 
@@ -1217,7 +1217,7 @@ The mass shell condition for open strings then becomes:\footnotemark{}
   The constant $a$ in~\eqref{eq:dbranes:closedspectrum} takes here the value $-1$ from the imposition of the canonical commutation relations and a $\zeta$-regularisation.
 }
 \begin{equation}
-  M^2 = \frac{1}{\ap} \left( N - 1 \right).
+  M^2 = \frac{1}{\ap} \qty( N - 1 ).
 \end{equation}
 
 Consider for a moment bosonic string theory and define the usual vacuum as
diff --git a/thesis.cls b/thesis.cls
index 0ffaa98..cc4dbfd 100644
--- a/thesis.cls
+++ b/thesis.cls
@@ -195,7 +195,7 @@
   \thispagestyle{plain}
 
   \noindent {\Large \sc Abstract} \\
-  \rule{0.99\textwidth}{\sepwidth}
+  \rule{0.99\linewidth}{\sepwidth}
 }
 {%
   \vfill
@@ -208,11 +208,11 @@
   \thispagestyle{plain}
 
   \noindent {\Large \sc Acknowledgements} \\
-  \rule{0.99\textwidth}{\sepwidth}
+  \rule{0.99\linewidth}{\sepwidth}
 }
 {%
   \vspace{\parskip}
-  \raggedleft\theauthor
+  \theauthor
   \vfill
 }
 
diff --git a/thesis.tex b/thesis.tex
index 5aab4cb..2ad8fc7 100644
--- a/thesis.tex
+++ b/thesis.tex
@@ -74,7 +74,12 @@
 \newcommand{\bomega}{\ensuremath{\overline{\omega}}}
 \newcommand{\bepsilon}{\ensuremath{\overline{\epsilon}}}
 \newcommand{\balpha}{\ensuremath{\overline{\alpha}}}
+\newcommand{\bzeta}{\ensuremath{\overline{\zeta}}}
+\newcommand{\halpha}{\ensuremath{\widehat{\alpha}}}
+\newcommand{\hbeta}{\ensuremath{\widehat{\beta}}}
 \newcommand{\htau}{\ensuremath{\widehat{\tau}}}
+\newcommand{\hpsi}{\ensuremath{\widehat{\psi}}}
+\newcommand{\Hpsi}{\ensuremath{\widehat{\Psi}}}
 
 
 %---- BEGIN DOCUMENT