# One-point partition

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

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