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Diffstat (limited to 'topology.tex')
| -rw-r--r-- | topology.tex | 9 |
1 files changed, 5 insertions, 4 deletions
diff --git a/topology.tex b/topology.tex index 64a1215..6f8f9fa 100644 --- a/topology.tex +++ b/topology.tex @@ -1448,20 +1448,21 @@ and look for the place where such solutions become complex. Inserting these simp \begin{equation} \begin{aligned} \overline{\log\chi(\Omega)} - =\lim_{n\to0}\frac\partial{\partial n}\int dC\,dR\,d\hat\omega_0\,d\omega_1\,d\hat\omega_1\, + =\lim_{n\to0}\frac\partial{\partial n}\int dC\,dr_d\,d\hat\omega_0\,d\hat\omega_1\, \exp N\Bigg\{ \frac i2\hat\omega_0\operatorname{Tr}(C-I) -in\hat\omega_1E \qquad\\ - -i\frac12n\omega_1\hat\omega_1r_df'(1) + -i\frac12n\omega_1^*\hat\omega_1r_df'(1) -\frac12\sum_{ab}^n \hat\omega_1^2f(C_{ab}) +\frac12\log\det - \left(\frac{-i\hat\omega_1}{\omega_1r_d}C+I\right) + \left(\frac{-i\hat\omega_1}{\omega_1^*r_d}C+I\right) \Bigg\} \end{aligned} \end{equation} -If we redefine $\hat\beta=-i\hat\omega_1$ and $\tilde r_d=\omega_1 r_d$, we find +where $\omega_1^*$ is a constant set by satisfying the extremal equations for $D$. +If we redefine $\hat\beta=-i\hat\omega_1$ and $\tilde r_d=\omega_1^*r_d$, we find \begin{equation} \begin{aligned} \overline{\log\chi(\Omega)} |