Conjecture on quasi-Hamiltonian structures of multiplicative bow varieties

Let $V_{i-1}$ and $V_i$ be the vector spaces associated with adjacent vertices in a bow, and let $A_{i-1}$, $B_{i-1}$, $B'_i$, $a_i$, and $b_{i-1}$ satisfy conditions (i), (a), (S1), and (S2) from the bow construction. The associated multiplicative bow variety is obtained from these spaces by the stated quasi-Hamiltonian reduction, and the Hanany–Witten transition and the isomorphism with the multiplicative quiver variety are the maps described below. Conjecture on quasi-Hamiltonian structures. (1) The space consisting of $A_{i-1}$, $B_{i-1}$, $B'_i$, $a_i$, and $b_{i-1}$ is a quasi-Hamiltonian $\operatorname{GL}(V_{i-1})\times\operatorname{GL}(V_i)\times\mathbb{C}^*$-space with group-valued moment map components $B_{i-1}^{-1}$, $B'_i$, and $\det B_{i-1}\det(B'_i)^{-1}$. Consequently, a multiplicative bow variety is a quasi-Hamiltonian reduction. (2) The Hanany–Witten transition is an isomorphism of quasi-Hamiltonian $\operatorname{GL}(V_1)\times\operatorname{GL}(V_3)\times\mathbb{C}^*$-spaces. (3) The isomorphism of the cobalanced case with the multiplicative quiver variety is an isomorphism of quasi-Hamiltonian $T(W)$-spaces. These assertions would strengthen the previously obtained isomorphism between the relevant Coulomb branch and multiplicative quiver variety to an isomorphism of Poisson varieties, while also asserting quasi-Hamiltonian compatibility for the bow construction and Hanany–Witten transition.