$ B $ and satisfies the hypothesis of Theorem \[thrm: genchangeofvar\]. It follows that $$\begin{aligned}
\chi (X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}}),\dots ,X_{n}+P_{n}({X_{1},\dots ,X_{n}})|B)=\chi ({X_{1},\dots ,X_{n}}|B) &  & \\
+({{\operatorname{Tr}}}\otimes \tau \otimes \tau )\log |D_{B}F^{\epsilon }({X_{1},\dots ,X_{n}})|. &  & \end{aligned}$$ Hence the derivative in $ \epsilon  $ of $ \chi (X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}}),\dots ,X_{n}+P_{n}({X_{1},\dots ,X_{n}})|B) $ is equal to the derivative of $ (\tau \otimes \tau )\log |D_{B}F^{\epsilon }({X_{1},\dots ,X_{n}})| $. Notice that $$D_{i}F^{\epsilon }_{j}({X_{1},\dots ,X_{n}})=\delta _{ij}+\epsilon D_{j}(P_{i})({X_{1},\dots ,X_{n}}),$$ so that $$D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}})=I+\epsilon M,$$ where $ I $ is the identity matrix and $$M_{ij}=D_{i}P_{j}({X_{1},\dots ,X_{n}}).$$ Hence $$(D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}})^{*}D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}}))=I+\epsilon (M+M^{*}+\epsilon M^{*}M).$$ Since $ \log (1+t) $ has a power series expansion around zero, we have that $$\frac{1}{2}\log (D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}})^{*}D_{B}(F^{\epsilon })({X_{1},\dots ,X_{n}}))=I+\frac{\epsilon }{2}(M+M^{*})+O(\epsilon ^{2}).$$ It follows that the desired derivative is equal to $$\begin{aligned}
(\tau \otimes \tau )\left( \frac{1}{2}[M+M^{*}]\right) =\frac{1}{2}\sum _{i}\tau \otimes \tau ((D_{i}P_{i})({X_{1},\dots ,X_{n}})+(D_{i}P_{i})({X_{1},\dots ,X_{n}})^{*})= &  & \\
\sum _{i}\tau \otimes \tau (D_{i}P_{i})({X_{1},\dots ,X_{n}}), &  & \end{aligned}$$ since $ F_{i} $ maps self-adjoint variables to self-adjoint variables. Hence we have, by the definition of the conjugate variable, that $$\begin{aligned}
\frac{d}{d\epsilon }\chi (X_{1}+\epsilon P_{1}({X_{1},\dots ,X_{n}}),\dots ,X_{n}+P_{n}({X_{1},\dots ,X_{n}})|B)= &  & \\
\sum _{i}\langle J(X_{i}:B\vee W^{*}(X_{1},\dots ,X_{i-1},X_{i+1},\dots ,X_{n}),P_{i}\rangle , & \end{aligned}$$ as 