Some edits in the superspace appendix.

1 files changed, 19 insertions, 15 deletions

diff --git a/marginal.tex b/marginal.tex
index 038f214..3a59487 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1593,12 +1593,11 @@ x\,d\bar{\pmb\eta}\,d\pmb\eta\,\frac{d\hat{\mathbf x}}{(2\pi)^N}$. Besides some
 deep connections to the physics of BRST, this compact notation dramatically
 simplifies the analytical treatment of the problem. The energy of stationary points can also be fixed using this notation, by writing
 \begin{equation}
-  \int d\pmb\phi\,\frac{d\hat\beta}{2\pi}\,e^{\hat\beta E+\int d1\,(1-\hat\beta\bar\theta_1\theta_1)H(\pmb\phi(1))}
+  \int d\pmb\phi\,d\hat\beta\,e^{\hat\beta E+\int d1\,(1-\hat\beta\bar\theta_1\theta_1)H(\pmb\phi(1))}
 \end{equation}
 which a small calculation confirms results in the same expression as \eqref{eq:delta.energy}.
 
-The reason why this simplification is
-possible is because there are a large variety of superspace algebraic and
+The reason why this transformation is a simplification is because there are a large variety of superspace algebraic and
 integral operations with direct corollaries to their ordinary real
 counterparts. For instance, consider a super linear operator $M(1,2)$, which
 like the super vector $\pmb\phi$ is made up of a linear combination of $N\times
@@ -1613,7 +1612,7 @@ The identity supermatrix is given by
 \end{equation}
 Integrals involving superfields contracted into such operators result in schematically familiar expressions, like that of the standard Gaussian:
 \begin{equation}
-  \int d\pmb\phi\,e^{\int\,d1\,d2\,\pmb\phi(1)^TM(1,2)\pmb\phi(2)}
+  \int d\pmb\phi\,e^{-\frac12\int\,d1\,d2\,\pmb\phi(1)^TM(1,2)\pmb\phi(2)}
   =(\operatorname{sdet}M)^{-1/2}
 \end{equation}
 where the usual role of the determinant is replaced by the superdeterminant.
@@ -1642,19 +1641,23 @@ superdeterminant of $M$ is given by
   \operatorname{sdet}M=\det(A-BD^{-1}C)\det(D)^{-1}
 \end{equation}
 which is the same for the normal equation for the determinant of a block matrix
-save for the inverse of $\det D$. The same method can be used to calculate the
-superdeterminant in arbitrary superspaces, where for $\mathbb R^{N|2D}$ each
+save for the inverse of $\det D$. Likewise, the supertrace of $M$ is is given by
+\begin{equation}
+  \operatorname{sTr}M=\operatorname{Tr}A-\operatorname{Tr}D
+\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 $\mathbf f(1,2)=\{\bar\theta_1,\theta_1,\bar\theta_2,\theta_2,\bar\theta_1\theta_1\bar\theta_2,\bar\theta_2\theta_2\theta_1,\bar\theta_1\theta_1\theta_2,\bar\theta_2\theta_2\theta_1\}$.
 
 \section{BRST symmetry}
 \label{sec:brst}
 
-The superspace representation is also helpful because it can make manifest an
-unusual symmetry in the dominant complexity of minima that would otherwise be
-obfuscated. This arises from considering the Kac--Rice formula as a kind of
-gauge fixing procedure \cite{Zinn-Justin_2002_Quantum}. Around each stationary
-point consider making the coordinate transformation $\mathbf u=\nabla H(\mathbf
-x)$. Then in the absence of fixing the trace, the Kac--Rice measure becomes
+When the trace $\mu$ is not fixed, there is an unusual symmetry in the dominant
+complexity of minima \cite{Annibale_2004_Coexistence, Kent-Dobias_2023_How}.
+This arises from considering the Kac--Rice formula as a kind of gauge fixing
+procedure \cite{Zinn-Justin_2002_Quantum}. Around each stationary point
+consider making the coordinate transformation $\mathbf u=\nabla H(\mathbf x)$.
+Then in the absence of fixing the trace, the Kac--Rice measure becomes
 \begin{equation}
     \int d\nu(\mathbf x,\pmb\omega\mid E)
     =\int\sum_\sigma d\mathbf u\,\delta(\mathbf u)\,
@@ -1679,8 +1682,7 @@ symmetry of the measure can then be written
 where $\delta\epsilon=-\pmb\eta^T\delta\mathbf u$ is a Grassmann number. This
 establishes that $\delta\mathbf x=\bar{\pmb\eta}\delta\epsilon$, now linear. The rest of
 the transformation can be built by requiring that the action is invariant after
-expansion in $\delta\epsilon$. Ignoring for a moment the piece of the measure
-fixing the trace of the Hessian, this gives
+expansion in $\delta\epsilon$. This gives
 \begin{align}
   \delta\mathbf x=\bar{\pmb\eta}\,\delta\epsilon &&
   \delta\hat{\mathbf x}=-i\hat\beta\bar{\pmb\eta}\,\delta\epsilon &&
@@ -1693,7 +1695,9 @@ so that the differential form of the symmetry is
   -i\hat\beta\bar{\pmb\eta}\cdot\frac\partial{\partial\hat{\mathbf x}}
   -i\hat{\mathbf x}\cdot\frac\partial{\partial\pmb\eta}
 \end{equation}
-The Ward identities associated with this symmetry give rise to relationships among the order parameters. These identities are
+The Ward identities associated with this symmetry give rise to relationships
+among the order parameters. These identities come from applying the
+differential symmetry to Grassmann-valued order parameters, and are
 \begin{align}
   \begin{aligned}
     0&=\frac1N\mathcal D\langle\mathbf x_a\cdot\pmb\eta_b\rangle