Correction of an error on central charges

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

sec/part1/introduction.tex

@@ -572,19 +572,21 @@ which show that $c_{\text{ghost}} = - 26$.
 The central charge is therefore cancelled in the full theory (bosonic string and reparametrisation ghosts) when the spacetime dimensions are $D = 26$.
 In fact let $\cT_{\text{full}} = \cT + \cT_{\text{ghost}}$ and $\overline{\cT}_{\text{full}} = \overline{\cT} + \overline{\cT}_{\text{ghost}}$, then:
 \begin{equation}
-  \eval{\cT_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
-  =
-  \eval{\overline{\cT}_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
-  =
-  c + c_{\text{ghost}}
-  =
-  \frac{D}{2} - 13
-  =
-  0
-  \quad
-  \Leftrightarrow
-  \quad
-  D = 26.
+  \begin{split}
+    \eval{\cT_{\text{full}}( z ) \cT_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
+    =
+    \eval{\overline{\cT}_{\text{full}}( \barz ) \overline{\cT}_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
+    & =
+    \frac{c + c_{\text{ghost}}}{2}
+    \\
+    & =
+    \frac{D}{2} - 13
+    =
+    0
+    \\
+    & \Leftrightarrow
+    D = 26.
+  \end{split}
 \end{equation}
 $\cT_{\text{full}}$ and $\overline{\cT}_{\text{full}}$ are then primary fields with conformal weight $-2$.
 
@@ -695,19 +697,21 @@ These are conformal fields with conformal weights $\qty( \frac{3}{2},\, 0 )$ and
 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 $\cT_{\text{full}} = \cT + \cT_{\text{ghost}}$ and $\overline{\cT}_{\text{full}} = \overline{\cT} + \overline{\cT}_{\text{ghost}}$ the central charge vanishes only in 10-dimensional spacetime:
 \begin{equation}
-  \eval{\cT_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
-  =
-  \eval{\overline{\cT}_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
-  =
-  c + c_{\text{ghost}}
-  =
-  \frac{3}{2}\, D - 15
-  =
-  0
-  \quad
-  \Leftrightarrow
-  \quad
-  D = 10.
+  \begin{split}
+    \eval{\cT_{\text{full}}( z ) \cT_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
+    =
+    \eval{\overline{\cT}_{\text{full}}( \barz ) \overline{\cT}_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
+    & =
+    \frac{c + c_{\text{ghost}}}{2}
+    \\
+    & =
+    \frac{3}{4}\, D - \frac{15}{2}
+    =
+    0
+    \\
+    & \Leftrightarrow
+    D = 10.
+  \end{split}
   \label{eq:super:dimensions}
 \end{equation}