Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle $T^*Fl_n$ of the complete flag variety? Negut mentioned in the first page of the paper that the partition function $Z(m)$ in the paper is closely related to the $J$-function of $T^*Fl_n$. Does it mean that the $J$-function of $T^*Fl_n$ is an eigenfunction of the Calogero-Sutherland Hamiltonian $L(m)$ written in p.5 of the paper? Or does it satisfy some integrable differential equations which is closely related to Calogero-Sutherland?

I have done simple calculation for the case of $n=2$. For $Fl_2=\mathbb{P}^1$, the $J$-function is written as \begin{equation} J(\mathbb{P}^1;\hbar)=e^{\frac{tx}{\hbar}}\sum_{d\ge0} \frac{e^{td}}{\prod_{k=1}^d(x+k\hbar)^2}~. \end{equation} It is easy to check that \begin{equation} \left[\hbar^2\frac{\partial^2}{\partial t^2}-e^t\right]J(\mathbb{P}^1;\hbar)=0~. \end{equation} On the other hand, the $J$-function of $T^*\mathbb{P}^1$ takes the form \begin{equation} J(T^*\mathbb{P}^1;\hbar,m)\propto e^{\frac{tx}{\hbar}}\sum_{d\ge0} \frac{e^{td}\prod_{k=0}^{d-1}(x+m+k\hbar)^2}{m^{2d}\prod_{k=1}^d(x+k\hbar)^2}~, \end{equation} where we introduce $m$ in such a way that $J(T^*\mathbb{P}^1;\hbar,m) \to J(\mathbb{P}^1;\hbar)$ as $m\to\infty$. Essentially, $J(T^*\mathbb{P}^1;\hbar,m)$ satisfy the Gauss hypergeometric differential equation since it is of ${}_2F_1$ form. However, I cannot see that $J(T^*\mathbb{P}^1;\hbar,m)$ (up to a certain factor) is an eigenfunction of the Calogero-Sutherland Hamiltonian $L(m)$. Is there any relation between the Calogero-Sutherland Hamiltonian of $A_1$-type and the Gauss hypergeometric differential equation? Or is $J(T^*\mathbb{P}^1;\hbar,m)$ NOT an eigenfunction of the Calogero-Sutherland Hamiltonian?

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    $\begingroup$ Have you tried arxiv.org/abs/1001.0056 ? $\endgroup$ – David Ben-Zvi Mar 31 '14 at 16:39
  • $\begingroup$ Thanks for your comments. Yes, I have tried it although I, as a physicist, cannot understand everything. Does the quantum (KZ) connection in this paper becomes a Hamiltonian of or annihilates the J-function of $T^*Fl_n$? The J-function involves gravitational descendent while it seems to me that the paper consider only quantum cohomology of springer resolutions. So I am not sure that this paper can be directly applied to the J-function. $\endgroup$ – Satoshi Nawata Mar 31 '14 at 17:05
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    $\begingroup$ If I understand correctly, the point of the quantum D-module is that the J-function is a solution of it.. (the $\hbar$ in the quantum D-module counts descendants). That paper identifies the equivariant quantum D-module of $T^*Fl$ with the Calogero-Moser system for the dual group... (ie the quantized Seiberg Witten integrable system for the N=2* theory as you might expect) $\endgroup$ – David Ben-Zvi Mar 31 '14 at 17:31
  • $\begingroup$ Thank you very much! I will read the paper more in detail. $\endgroup$ – Satoshi Nawata Mar 31 '14 at 17:37

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