TY - JOUR

T1 - Dimensional reduction over the quantum sphere and non-abelian q-vortices

AU - Landi, Giovanni

AU - Szabo, Richard Joseph

PY - 2011/12

Y1 - 2011/12

N2 - We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a Kähler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU q (2), and also of invariant gauge connections on these bundles. The reduction of Yang–Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-Kähler quotient construction.

AB - We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a Kähler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU q (2), and also of invariant gauge connections on these bundles. The reduction of Yang–Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-Kähler quotient construction.

U2 - 10.1007/s00220-011-1357-z

DO - 10.1007/s00220-011-1357-z

M3 - Article

SN - 0010-3616

VL - 308

SP - 365

EP - 413

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -