Let $n,N$ be two positive integers. Consider the total space of the line bundle $\mathcal O(-n)$ on $\mathbb C\mathbb P^N$. This is an algebraic variety with an action of $G=GL(N+1,\mathbb C)\times \mathbb C^*$.
$\mathbf {Question:}$ Is the $G$-equivariant quantum cohomology of this variety known? I would appreciate any reference. Same question for (negative) line bundles on flag variety (instead of $\mathbb C\mathbb P^N$).
I have done a simple calculation for $G$-equivariant quantum K-ring for $\mathcal{O}(-n)$ bundle over $\mathbb{CP}^N$ by using 3d $\mathcal{N}=2$ $U(1)$ gauge theory with $N+1$ flavors of charge 1 and one flavor of charge $-n$. I have followed the method of supersymmetric localization in this paper, and the Higgs branch of the theory is indeed $\mathcal{O}(-n)$ bundle over $\mathbb{CP}^N$. The result is as follows: \begin{equation} QK^\bullet_G(\mathcal{O}(-n)\to \mathbb{CP}^N)\simeq \mathbb{C}\left[p^{\pm1},\tau^{\pm},\eta^{\pm},\mu_i^{\pm}\right]/\mathcal{I}\,,\quad i=1,\dots, N+1\,, \end{equation} where the ideal $\mathcal{I}$ is given by \begin{equation} \prod_{i=1}^{N+1}(p-\mu_i)=\tau (p-\eta)^n~. \end{equation} Note that $\mu_i$ are the equivariant parameters of $GL(N+1,\mathbb{C})$ and $\eta$ is that of $\mathbb{C}^*$. I have also computed the K-theoretic Givental $J$-function: \begin{equation} J[\mathcal{O}(-n)\to \mathbb{CP}^N]=\sum_{k\ge0}\tau^k\frac{(q\eta;q)_{nk}}{\prod_{j=1}^{N+1}(q\mu_j/\mu_i;q)_k}~. \end{equation}
