diff options
Diffstat (limited to 'marginal.tex')
| -rw-r--r-- | marginal.tex | 30 |
1 files changed, 22 insertions, 8 deletions
diff --git a/marginal.tex b/marginal.tex index 508b674..8efa79f 100644 --- a/marginal.tex +++ b/marginal.tex @@ -1772,21 +1772,30 @@ Integrals involving superfields contracted into such operators result in schemat \end{equation} where the usual role of the determinant is replaced by the superdeterminant. The superdeterminant can be defined using the ordinary determinant by writing a -block version of the matrix $M$: if $\mathbf e(1)=\{1,\bar\theta_1\theta_1\}$ is +block version of the matrix $M$. If $\mathbf e(1)=\{1,i\bar\theta_1\theta_1\}$ is the basis vector of the even subspace of the superspace and $\mathbf -f(1)=\{\bar\theta_1,\theta_1\}$ is that of the odd subspace, then we can form a +f(1)=\{i\bar\theta_1,i\theta_1\}$ is that of the odd subspace, dual bases $\mathbf e^\dagger(1)=\{i\bar\theta_1\theta_1,1\}$ and $\mathbf f^\dagger(1)=\{\theta_1,-\bar\theta_1\}$ can be defined by the requirement that +\begin{align} + \int d1\,\mathbf e(1)\mathbf e^\dagger(1)=iI + && + \int d1\,\mathbf f(1)\mathbf f^\dagger(1)=iI \\ + \int d1\,\mathbf e(1)\mathbf f^\dagger(1)=0 + && + \int d1\,\mathbf f(1)\mathbf e^\dagger(1)=0 +\end{align} +With such bases and dual bases defined, we can form a block representation of $M$ in analogy to the matrix form of an operator in quantum mechanics by \begin{equation} \int d1\,d2\,\begin{bmatrix} - \mathbf e(1)M(1,2)\mathbf e(2)^T + \mathbf e(1)M(1,2)\mathbf e^\dagger(2) & - \mathbf e(1)M(1,2)\mathbf f(2)^T + \mathbf e(1)M(1,2)\mathbf f^\dagger(2) \\ - \mathbf f(1)M(1,2)\mathbf e(2)^T + \mathbf f(1)M(1,2)\mathbf e^\dagger(2) & - \mathbf f(1)M(1,2)\mathbf f(2)^T + \mathbf f(1)M(1,2)\mathbf f^\dagger(2) \end{bmatrix} - =\begin{bmatrix} + =i\begin{bmatrix} A & B \\ C & D \end{bmatrix} \end{equation} @@ -1802,7 +1811,12 @@ save for the inverse of $\det D$. Likewise, the supertrace of $M$ is is given by \end{equation} The same method can be used to calculate the superdeterminant and supertrace in arbitrary superspaces, where for $\mathbb R^{N|2D}$ each -basis has $2^{2D-1}$ elements. For instance, for $\mathbb R^{N|4}$ we have $\mathbf e(1,2)=\{1,\bar\theta_1\theta_1,\bar\theta_2\theta_2,\bar\theta_1\theta_2,\bar\theta_2\theta_1,\bar\theta_1\bar\theta_2,\theta_1\theta_2,\bar\theta_1\theta_1\bar\theta_2\theta_2\}$ and $\mathbf f(1,2)=\{\bar\theta_1,\theta_1,\bar\theta_2,\theta_2,\bar\theta_1\theta_1\bar\theta_2,\bar\theta_2\theta_2\theta_1,\bar\theta_1\theta_1\theta_2,\bar\theta_2\theta_2\theta_1\}$. +basis has $2^{2D-1}$ elements. For instance, for $\mathbb R^{N|4}$ we have +\begin{align} + &\mathbf e(1,2)=\{1,i\bar\theta_1\theta_1,i\bar\theta_2\theta_2,i\bar\theta_1\theta_2,i\bar\theta_2\theta_1,i\bar\theta_1\bar\theta_2,i\theta_1\theta_2,\bar\theta_1\theta_1\bar\theta_2\theta_2\}\notag \\ + &\mathbf f(1,2)=\{i\bar\theta_1,i\theta_1,i\bar\theta_2,i\theta_2,\bar\theta_1\theta_1\bar\theta_2,\bar\theta_2\theta_2\theta_1,\bar\theta_1\theta_1\theta_2,\bar\theta_2\theta_2\theta_1\} +\end{align} +with the dual bases defined analogously to those above. \section{BRST symmetry} \label{sec:brst} |