标题： $Q$-算子，$q$-操作符，以及5d$\mathcal{N}=1$规范理论中的R矩阵 显示英文标题

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

摘要： 我们通过五维 $\mathcal{N}=1$ 超对称规范理论在 $\Omega$-背景下的视角，研究乘法Higgs丛模空间的量化。 我们将先前工作中建立的四维 $\mathcal{N}=2$ 规范理论构造的关键几何和表示论结构扩展到五维提升。 我们构建并分析与规范共维二缺陷相关的$Q$算子和$q$操作子：$Q$算子通过缺陷的插入定义，而$q$操作子则作为其存在下的$q$差分共形环方程出现。 $q$操作子差分方程进一步被识别为由量子仿射代数类型${\mathfrak{gl}}(n)$的双无限评价模的张量积构造的 XXZ 自旋链的 Baxter TQ 方程。 我们定义了在单色性余维数二缺陷分划函数空间上的$q$-差分模结构，并表明由单色性缺陷构造的$Q$-算子的本征态同时对角化 XXZ 自旋链的量子哈密顿量。 Fourier 变换交换了与彼此双谱对偶的两个 XXZ 自旋链相关的$Q$-算子。 最后，我们将这些构造与来自 5d 理论的 BPS 引擎的量子簇代数相关联，并用簇变量重新表达了 R 矩阵。 显示英文摘要

摘要： 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.