# Definition of Givental $J$-function of cotangent bundle of flag variety

I would like to know the definition of Givental $J$-function of cotangent bundle of flag variety. To state my question more precisely, let us briefly recall the definition of the Givental $J$-function of a compact Kahler variety $X$. Let $T_0=1,T_1, \cdots, T_m$ be the basis of the cohomology group $H^*(X,\mathbb{Z})$, and $T_1, \cdots, T_r$ be the basis of the second cohomology group $H^2(X,\mathbb{Z})$. We define the matrix $g_{ij}=\int_{X}T_i\cup T_j$, and its inverse matrix $g^{ij}=(g_{ij})^{-1}$, which provide the dual basis $$T^a=\sum_{b=1}^mg^{ab}T_b~,$$ so that $\int_X T^i\cup T_j=\delta^i_j$. We denote by $\overline{\mathcal{M}}_{g,n}(X,\beta)$ the moduli space of stable maps from connected genus $g$ curves with $n$-marked points to $X$ representing the class $\beta\in H_2(X,\mathbb{Z})$. Let $\mathcal{L}_1,\cdots,\mathcal{L}_n$ be the corresponding tautological line bundles over $\overline{\mathcal{M}}_{g,n}(X,\beta)$. For $\gamma_1,\cdots,\gamma_n\in H^*(X,\mathbb{Z})$ and non-negative integers $d_i$, the gravitational correlation function is defined $$\left\langle \tau_{d_1} \gamma_1, \cdots, \tau_{d_n} \gamma_n \right\rangle_{g,\beta}=\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\textrm{vir}}} \prod_{i=1}^n c_1(\mathcal{L}_i) ^{d_i}\cup\textrm{ev}^*(\gamma_i)~.$$ The $J$-function of $X$ is defined by using the psi class $\psi=c_1\mathcal{L}_1)$ $$J[X]=e^{\delta/\hbar}\left(1+\sum_{\beta\in H_2(X,\mathbb{Z})}\sum_{a=1}^m q^\beta \left\langle\frac{T_a}{\hbar-\psi},1 \right\rangle_{0,\beta}~T^a\right)~,$$ where $\delta=\sum_{i=1}^r t_i T_i$ and $q^\beta=e^{\int_\beta \delta}$.

Recently, I have conjectured in this paper the expression of hypergeometric type for $J$-function of cotangent bundle of complete flag variety $\mathrm{Fl}_N=SL(N,\mathbb{C})/B$ by using the supersymmetric partition function on $S^2$: \begin{eqnarray} J[T^*\mathrm{Fl}_N]&=&\sum_{\vec{k}^{(I)}} \prod_{I=1}^{N-1}z_I^{\vert k^{(I)}\vert } \prod_{I=2}^{N-1}\prod_{s\neq t}^{I} \tfrac{(1+\hbar^{-1} H_{st}^{(I)}+\hbar^{-1}m)_{k_s^{(I)}-k_t^{(I)}}}{(\hbar^{-1} H_{st}^{(I)})_{k_s^{(I)}-k_t^{(I)}}} \cr && \prod_{I=1}^{N-2} \prod_{s=1}^{I}\prod_{t=1}^{{I+1}} \tfrac{(\hbar^{-1} H_{s}^{(I)} -\hbar^{-1} H_{t}^{(I+1)}-\hbar^{-1}m)_{k^{(I)}_s - k^{(I+1)}_t}}{(1+\hbar^{-1} H_{s}^{(I)} -\hbar^{-1} H_{t}^{(I+1)})_{k^{(I)}_s - k^{(I+1)}_t}}\cr && \prod_{s=1}^{N-1}\prod_{t=1}^{N} \tfrac{(\hbar^{-1} H_{s}^{(N-1)}-\hbar^{-1} H_{t}^{(N)}-\hbar^{-1}m)_{k^{(N-1)}_s}}{(1+\hbar^{-1} H_{s}^{(N-1)}-\hbar^{-1} H_{t}^{(N)})_{k^{(N-1)}_s}}~, \end{eqnarray} where we identify $H_{s}^{(I)}$ $(s = 1,...,I)$ with Chern roots to the duals of the universal bundles $\mathcal{S}_I$: $$0\subset \mathcal{S}_1 \subset \mathcal{S}_2 \subset \dots \subset \mathcal{S}_{N-1} \subset \mathcal{S}_{N} = {\mathbb{C}}^N \otimes {\cal O}_{\mathrm{Fl}_N}~.$$ I have checked that, after multiplying an appropriate factor, it becomes an eigenfunction of trigonometric Calgero-Moser Hamiltonian $$H=\hbar^2\sum_{i}\frac{\partial^2}{\partial t_i^2}-2m(m+\hbar)\sum_{\alpha\in \Delta^+} \frac 1{(e^{\langle t,\alpha\rangle/2}-e^{-\langle t,\alpha\rangle/2})^2}~.$$ This is consistent with theorem 3.2 of Braverman-Maulik-Okounkov.

Question: The formula of $J[T^*\mathrm{Fl}_N]$ certainly contains the parameter $m$ that does not appear in the definition of $J$-function of a compact Kahler variety. It appears to me that one has to introduce the parameter $m$ since the cotangent bundle of the complete flag variety is non-compact. I would like to know the definition of $J[T^*\mathrm{Fl}_N]$ and how it incorporates the parameter $m$.

The parameter $m$ appears if you study everything equivariantly with respect to the ${\mathbb C}^*$-action which acts in the standard way on the fibers of the cotangent bundle.