$\sum_{i=1}^s\gamma_i\otimes\gamma^i$.

### Brackets and double brackets

Let $\psi_i$ be the first Chern class of the canonical line bundle on ${\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}$ with fiber the cotangent line to the domain curve at the $i^\mathrm{th}$ marking. For any stability parameter $0+\leq \ke\leq\infty$, the descendant invariants of $\ke$-quasimap theory are defined by $$\label{e-invariants}
\langle \delta_1\psi_1^{a_1},\dots,\delta_k\psi_k^{a_k}
\rangle^\ke_{g,k,\beta}=\int_{[{\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}]^{\mathrm{vir}}} \prod_{i=1}^k ev_i^*(\delta_i)\psi_i^{a_i},$$ where $\delta_i\in H^*({W/\!\!/{{\bf G}}},\QQ)$, $ev_i:{\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}\lra{W/\!\!/{{\bf G}}}$ are the evaluation maps at the markings, and $g,k,a_1,\dots a_k\geq 0$ are integers such that $2g-2+k+\ke\beta(L_\theta)> 0$.

In Gromov-Witten theory, the name “invariants" for the brackets reflects the fact that they are symplectic invariants of the target ${W/\!\!/{{\bf G}}}$ and in particular do not change under deformations of the target ${W/\!\!/{{\bf G}}}$. The quasimap brackets depend on the [*pair*]{} of stacks $([W/{{\bf G}}],[W^s/{{\bf G}}])$, see Proposition 4.6.1 in [@CKM]. Nevertheless, we will use the same terminology in quasimap theory as well.

When dealing with generating series of invariants, it is convenient to use a double bracket notation $$\label{double bracket}
\begin{split}
&{\langle\!\langle}\delta_1\psi_1^{a_1},\dots,\delta_k\psi_k^{a_k} {\rangle\!\rangle}^\ke_{g,k}={\langle\!\langle}\delta_1\psi_1^{a_1},\dots,\delta_k\psi_k^{a_k} {\rangle\!\rangle}^\ke_{g,k}({\bf t}(\psi)):=\\
&\sum_{m,\beta} \frac{q^\beta}{m!}
\langle \delta_1\psi_1^{a_1},\dots,\delta_k\psi_k^{a_k},
{\bf t}(\psi_{k+1}), \dots ,{\bf t}(\psi_{k+m}) \rangle^\ke_{g,k+m,\beta}.
\end{split}$$ Here, as in the Introduction, ${\bf t}(\psi)=t_0+t_1\psi+t_2\psi^2+t_3\psi^3+\dots$, with $t_j=\sum_i t_{ji}\gamma_i\in H^*({W/\!\!/{{\bf G}}},\QQ)$. The double brackets are formal functions on the [*large phase 