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

Giovanni Landi, Richard Joseph Szabo

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish
Pages (from-to)365-413
JournalCommunications in Mathematical Physics
Volume308
Issue number2
DOIs
Publication statusPublished - Dec 2011

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Q-deformation
Dimensional Reduction
Vortex
Quiver
Instantons
Gauge Theory
Equivariant
Moduli Space
Bundle
Product Space
Yang-Mills Theory
Quantum Groups
Riemann Surface
Stability Condition
Existence of Solutions
Gauge
Quotient
Vacuum
Invariant
Demonstrate

Cite this

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abstract = "We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a K{\"a}hler 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-K{\"a}hler quotient construction.",
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Dimensional reduction over the quantum sphere and non-abelian q-vortices. / Landi, Giovanni; Szabo, Richard Joseph.

In: Communications in Mathematical Physics, Vol. 308, No. 2, 12.2011, p. 365-413.

Research output: Contribution to journalArticle

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AU - Szabo, Richard Joseph

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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.

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