Let following be the partition function in infinitely many variables $x_i$ the linear coordinates on the vector space $V$. $$ \mathcal{Z}=exp\Big(\sum_{\substack{g\geq 0\\n\geq 1}}\frac{h^{g-1}}{n!}\sum_{i_1 ,\ldots , i_n}F_{g,n}(i_1 ,\ldots , i_n)x_{i_1}\cdots x_{i_n} \Big) $$ where $F_{g,n}(i_1 ,\ldots , i_n)$ are scalars.

Let $L_{i}\in W_{V}^{h}$ are differential equations that annihilates $\mathcal{Z}$ that is $$ L_{i}\ \mathcal{Z}=0.$$ where $W_{V}^{h}=\mathbb{C}[\hbar]\langle (x_{i},\partial_i )\rangle/\langle [\partial_i ,x_i]=\hbar\rangle$

Let $$ \mathcal{K}=\sum_{\substack{g\geq 0\\}}h^{g-1}\sum_{i_1 }F_{g,1}(i_1)x_{i_1} $$ where $F_{g,1}(i_1)$ are scalars and are the same scalar appearing in $\mathcal{Z}$

I was wondering how I can derive $\mathcal{K}$ from $\mathcal{Z}$ by some operation. The one I had in mind is the following, I choose a parameter $s$ and replace

$$x_i\rightarrow sx_i.$$ Then $$\frac{\partial}{\partial s}\mathcal{Z}|_{s=0}= \mathcal{K}$$ (I don't know if I am making any mistake above calculation)

My question is if we can say something about the annihilator of $\mathcal{K}$ in terms of $L_i$ in general.

Say $L_i$ has this particular form

\begin{equation} L_{i}:= \hbar\partial^{i}-\frac12 \sum_{jk}A^{ijk}x_{j}x_{k}-\sum_{jk}\hbar B_{k}^{ij}\partial^{k}x_{j}-\frac12 \sum_{jk}\hbar^{2}C_{jk}^{i}\partial^{j}\partial^{k}-\hbar D^{i}, \ 1\leq i \leq dim\ V \end{equation} where $A,B,C,D$ are scalars. then something can be said about the annihilator for $\mathcal{K}$.


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