论文标题
超对称仪表理论中移动的量子组和物质多重
Shifted quantum groups and matter multiplets in supersymmetric gauge theories
论文作者
论文摘要
转移量子组的概念最近在代数几何形状中发挥了重要作用。对原始定义的这种微妙的修改在量子群的表示理论中具有更大的灵活性。本文的第一部分介绍了转移的量子环形$ \ mathfrak {gl}（1）$和量子仿射$ \ mathfrak {sl}（2）$ algebras（seps。Endoted$ \ ddot {q_2} _ {q_2}^$ boldsymbold（Q_2}^$ boldsymbold） $ \ dot {u} _q^\boldsymbolμ（\ mathfrak {sl}（2））$）。它定义了几个新表示形式，包括圆环代数的有限尺寸最高$ \ ell $ - 加权表示形式，以及$ \ dot {u} _q^\boldsymbolμ（\boldsymbolμ（\ mathfrak {sl}（2）（2））的顶点表示。它还探讨了$ \ dot {u} _q^\boldsymbolμ（\ Mathfrak {sl}（2））$的表示之间的关系修复），并介绍了几个新的互换者的构建。这些结果在第二部分中用于构造5D $ \ MATHCAL {N} = 1 $和3D $ \ MATHCAL {N} = 2 $量规理论的BPS可观察结果。特别是，可以使用移位表示，可以在代数工程框架中引入5D超强和3D手性多重，并从此角度重新审视Higgsing过程。
The notion of shifted quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal $\mathfrak{gl}(1)$ and quantum affine $\mathfrak{sl}(2)$ algebras (resp. denoted $\ddot{U}_{q_1,q_2}^\boldsymbolμ(\mathfrak{gl}(1))$ and $\dot{U}_q^\boldsymbolμ(\mathfrak{sl}(2))$). It defines several new representations, including finite dimensional highest $\ell$-weight representations for the toroidal algebra, and a vertex representation of $\dot{U}_q^\boldsymbolμ(\mathfrak{sl}(2))$ acting on Hall-Littlewood polynomials. It also explores the relations between representations of $\dot{U}_q^\boldsymbolμ(\mathfrak{sl}(2))$ and $\ddot{U}_{q_1,q_2}^\boldsymbolμ(\mathfrak{gl}(1))$ in the limit $q_1\to\infty$ ($q_2$ fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d $\mathcal{N}=1$ and 3d $\mathcal{N}=2$ gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective.
