Quiver Gauge Theory and Noncommutative Vortices

TL;DR

This paper constructs explicit solutions to Yang-Mills equations on noncommutative spaces with G-symmetry, reducing to vortex equations in quiver gauge theories, and interprets these as D-brane configurations.

Contribution

It provides a method to obtain G-equivariant solutions on noncommutative spaces and links them to vortex equations in quiver gauge theories, including Seiberg-Witten monopoles.

Findings

Explicit noncommutative BPS and non-BPS solutions constructed.

Reduction of Donaldson-Uhlenbeck-Yau equations to vortex-type equations.

Interpretation of solutions as D0-branes in brane-antibrane systems.

Abstract

We construct explicit BPS and non-BPS solutions of the Yang-Mills equations on noncommutative spaces R^{2n}_theta x G/H which are manifestly G-symmetric. Given a G-representation, by twisting with a particular bundle over G/H, we obtain a G-equivariant U(k) bundle with a G-equivariant connection over R^{2n}_theta x G/H. The U(k) Donaldson-Uhlenbeck-Yau equations on these spaces reduce to vortex-type equations in a particular quiver gauge theory on R^{2n}_theta. Seiberg-Witten monopole equations are particular examples. The noncommutative BPS configurations are formulated with partial isometries, which are obtained from an equivariant Atiyah-Bott-Shapiro construction. They can be interpreted as D0-branes inside a space-filling brane-antibrane system.