Shifted quantum groups and matter multiplets in supersymmetric gauge theories

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 $\ddot{U}_{q_1,q_2}^\boldsymbol{\mu}(\mathfrak{gl}(1))$ and $\dot{U}_q^\boldsymbol{\mu}(\mathfrak{sl}(2))$). It defines several new representations, including finite dimensional highest $\ell$-weight representations for the toroidal algebra, and a vertex representation of $\dot{U}_q^\boldsymbol{\mu}(\mathfrak{sl}(2))$ acting on Hall-Littlewood polynomials. It also explores the relations between representations of $\dot{U}_q^\boldsymbol{\mu}(\mathfrak{sl}(2))$ and $\ddot{U}_{q_1,q_2}^\boldsymbol{\mu}(\mathfrak{gl}(1))$ 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.