diff options
| -rw-r--r-- | marginal.tex | 31 |
1 files changed, 29 insertions, 2 deletions
diff --git a/marginal.tex b/marginal.tex index fb212db..59c9bc8 100644 --- a/marginal.tex +++ b/marginal.tex @@ -71,18 +71,45 @@ associated with the minimal eigenvalue $\lambda_\mathrm{min}$. \end{aligned} \end{equation} +\begin{equation} + \beta^2f''(1)\sum A_{ab}^2+\hat x^2f''(1)A_{11}^2+\beta\hat xf''(1)\sum_a A_{1a}+\hat\beta^2f(C_{ab})+(2\hat\beta(R_{ab}-F_{ab})-D_{ab})f'(C_{a-b})+(R_{ab}^2-F_{ab}^2)f''(C_{ab}) + +\log\det\begin{bmatrix}C&iR\\iR&D\end{bmatrix}-\log\det F +\end{equation} + \section{Superfield formalism} \begin{equation} - \pmb\phi=\pmb\sigma+\bar\theta\pmb\eta+\bar{\pmb\eta}\theta+\hat{\pmb\sigma}\bar\theta\theta+\mathbf x\bar\vartheta\theta+\mathbf x\bar\theta\vartheta + \pmb\phi_a(1)=\pmb\sigma_a+\bar\theta(1)\pmb\eta_a+\bar{\pmb\eta}_a\theta(1)+\hat{\pmb\sigma}_a\bar\theta(1)\theta(1) \end{equation} \begin{equation} - \int d\theta\,d\bar\theta\,d\vartheta\,d\bar\vartheta\,(\bar\vartheta\vartheta+\beta+\hat\beta\bar\vartheta\vartheta\bar\theta\theta)H(\pmb\phi) + \pmb\xi_b(1)=\pmb\sigma_1+\mathbf x_b\bar\vartheta(1)\theta(1)+\mathbf x_b\bar\theta(1)\vartheta(1) +\end{equation} +\begin{equation} + \int d\theta\,d\bar\theta\,\left[ + (1+\hat\beta\bar\theta\theta)H(\pmb\phi) + +\int d\vartheta\,d\bar\vartheta\,H(\pmb\xi) + \right] =\hat{\pmb\sigma}^T\partial H(\pmb\sigma) +\pmb\eta^T\partial\partial H(\pmb\sigma)\pmb\eta +\beta\mathbf x^T\partial\partial H(\pmb\sigma)\mathbf x +\hat\beta H(\pmb\sigma) \end{equation} +\begin{equation} + \int d1\,d2\,(1+\hat\beta\bar\theta(1)\theta(1))(1+\hat\beta\bar\theta(2)\theta(2)) + f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right) + +\int d1\,(1+\hat\beta\bar\theta(1)\theta(1))f\left(\frac{\pmb\phi_a(1)\cdot\pmb\xi_b(2)}N\right) +\end{equation} + +\section{Twin spherical model} + +$\Omega=S^{N-1}\times S^{N-1}$ +\begin{equation} + H(\pmb\sigma)=H_1(\pmb\sigma^{(1)})+H_2(\pmb\sigma^{(2)})+\epsilon\pmb\sigma^{(1)}\cdot\pmb\sigma^{(2)} +\end{equation} +\begin{equation} + \overline{H_s(\pmb\sigma_1)H_s(\pmb\sigma_2)} + =Nf_s\left(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\right) +\end{equation} \section{Multi-species spherical model} |