$Q$-operators, $q$-opers, and R-matrices in 5d $\mathcal{N}=1$ gauge theory

We study the quantization of the moduli space of multiplicative Higgs bundles through the lens of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories in $\Omega$-background. We extend the 4d $\mathcal{N}=2$ gauge theoretical construction of key geometric and representation-theoretic structures, established in earlier works, to the five-dimensional uplift. We construct and analyze the $Q$-operators and $q$-opers associated with the canonical codimension-two defect: the $Q$-operators are defined via the insertion of the defect, while the $q$-opers arise as the $q$-difference chiral ring equations in its presence. The $q$-oper difference equations are further identified with the Baxter TQ equations for XXZ spin chains constructed from tensor products of bi-infinite evaluation modules over quantum affine algebras of type ${\mathfrak{gl}}(n)$. We define a $q$-difference module structure on the space of monodromy codimension-two defect partition functions and show that the eigenstates of the $Q$-operators, constructed from monodromy defects, simultaneously diagonalize the quantum Hamiltonians of the XXZ spin chain. A Fourier transformation exchanges the $Q$-operators associated with two XXZ spin chains bispectral dual to each other. Finally, we relate these constructions to the quantum cluster algebra arising from the BPS quiver of the 5d theory, and re-express the R-matrices in terms of the cluster variables.