From 3b1e4a4f833fa2bd689d46f109d6072f55b1975e Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Wed, 18 Nov 2020 10:59:50 +0100 Subject: [PATCH] Correction of an error on central charges Signed-off-by: Riccardo Finotello --- sec/part1/introduction.tex | 56 ++++++++++++++++++++----------------- thesis.pdf | Bin 3105865 -> 3108938 bytes 2 files changed, 30 insertions(+), 26 deletions(-) diff --git a/sec/part1/introduction.tex b/sec/part1/introduction.tex index ec3fe8c..bf0fc1e 100644 --- a/sec/part1/introduction.tex +++ b/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} diff --git a/thesis.pdf b/thesis.pdf index 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