1
$\begingroup$

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 $\mathcal{K}$.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.