Consider the quantum group \(SU_{q}(2)\) with \(0<q<1\). It is the quantum group corresponding to the Hopf \(*\)-algebra \(\mathcal{A}(SU_{q}(2))\). Associated to this Hopf algebra, one has a Hopf algebra projection \(\pi: \mathcal{A}(SU_{q}(2)) \twoheadrightarrow \mathcal{A}(U(1))\), whose space of coinvariants \(\mathcal{A}(SU_{q}(2))^{co\, \pi}\) gives the non-commutative coordinate algebra \(\mathcal{A}(S_{q}^{2})\) of the standard Podleś sphere. Thus, the algebra inclusion \(\mathcal{A}(S_{q}^{2}) \hookrightarrow \mathcal{A}(SU_{q}(2))\) can be seen as a quantum principle bundle. As the quantum sphere \(S_{q}^{2}\) is endowed with a natural complex structure, one might use the convention to treat it as a quantum projective line \(\mathbb{C}\text{P}^{1}_{q}\).
Let \(M\) be a smooth manifold and denote by \(\underline{M} = \mathbb{C}\text{P}^{1}_{q} \times M \) the quantum space given by the family of quantum projective lines \(\mathbb{C}\text{P}^{1}_{q}\times \{p\} \simeq \mathbb{C}\text{P}^{1}_{q}\) parametrized by points \(p\in M\). Then the algebra of \(\underline{M}\) is given by \(\mathcal{A}(\underline{M}) = \mathcal{A}(\mathbb{C}\text{P}^{1}_{q}) \otimes \mathcal{A}(M)\). Using connections on the quantum principal bundle over \(\mathbb{C}\text{P}^{1}_{q}\), the authors construct invariant connections on \(SU_{q}(2)\)-equivariant modules over the algebra \(\mathcal{A}(\underline{M})\) and describe their dimensional reduction over \(\mathbb{C}\text{P}^{1}_{q}\). As a main theorem a reduction of Yang-Mills gauge theory on \(\mathcal{A}(\underline{M})\) to a type of Yang-Mills-Higgs theory on the manifold \(M\) is obtained. These results can be seen as a continuation of the work [L. Álvarez-Cónsul and O. García-Prada, “Dimensional reduction, SL\((2,\mathbb{C})\)-equivariant bundles and stable holomorphic chains”, Int. J. Math. 12, No. 2, 159–201 (2001; Zbl 1110.32305)], where the case \(q=1\) is treated. There an \(SU(2)\)-equivariant dimensional reduction over the product \(\mathbb{C}\text{P}^{1} \times M \) of the complex projective lines \(\mathbb{C}\text{P}^{1}\) with a Kähler manifold \(M\) was studied.
The paper ends with the description of explicit examples and a comparison with analogous results in the literature for the case \(q=1\), showing that the \(q\)-deformation generically improves the geometrical structure of the associated moduli spaces. The authors also analyse moduli spaces of \(q\)-vortices on Riemann surfaces, which give new examples of non-Abelian vortices, and show that the \(q\)-deformations on instations on Kähler surfaces are analogous to those of some previous non-commutative deformations of the self-duality equations.