Rank two quiver gauge theory, graded connections and noncommutative vortices

Abstract

We consider equivariant dimensional reduction of Yang-Mills theory on Kähler manifolds of the form M ×CP 1 ×CP 1 . This induces a rank two quiver gauge theory on M which can be formulated as a Yang-Mills theory of graded connections on M . The reduction of the Yang-Mills equations on M ×CP 1 ×CP 1 induces quiver gauge theory equations on M and quiver vortex equations in the BPS sector. When M is the noncommutative space R 2n θ both BPS and non-BPS solutions are obtained, and interpreted as states of D-branes. Using the graded connection formalism, we assign D0-brane charges in equivariant K-theory to the quiver vortex configurations. Some categorical properties of these quiver brane configurations are also described in terms of the corresponding quiver representations.

SU(3)-equivariant quiver gauge theories and nonabelian vortices

“…In the former case the coset space is the complex projective plane CP 2 . We will find that many aspects of the induced rank two quiver gauge theory on M 2d in this case are qualitatively similar to that obtained from the symmetric space CP 1 × CP 1 [13]. However, the technical aspects are much more involved and some new features emerge from the nonabelian SU(2) instanton degrees of freedom which now reside at the vertices of the quiver.…”

“…Note that the energy density is independent of the degeneracy label n. The energy of the V -spin charge is down by 1 4 due to the area of the embedded two-cycle CP 1 dual to the (1, 1)-form γ 1 ∧γ 1 on Q 3 (see (3.46)). All of this is qualitatively similar to the quiver energies associated to the symmetric space CP 1 × CP 1 [13], which also carry abelian vertex charges but only two arrows per vertex. For the present solutions the noncommutative vortex number is now 52) and the verification of the Yang-Mills equations on X = C d θ × Q 3 proceeds exactly as outlined before in the basis generated by the canonical one-forms γ a andγ a on Q 3 .…”

“…However, the technical aspects are much more involved and some new features emerge from the nonabelian SU(2) instanton degrees of freedom which now reside at the vertices of the quiver. In the latter case the coset space is the six-dimensional homogeneous manifold Q 3 , and the qualitative features of the rank two quiver gauge theories are much different in this case, even though the quiver vertex degrees of freedom involve two U(1) monopole charges as in the CP 1 × CP 1 case [13]. This space has appeared before in a variety of different physical contexts, such as in connection with the dynamics of M-theory on a manifold of G 2 holonomy which develops a conical singularity [15], and for the dimensional reduction of ten-dimensional supersymmetric gauge theories over six-dimensional coset spaces in much the same spirit as our more general reductions [16].…”

“…Since u is an abelian algebra, the arrows (2.19) generate commutative quiver diagrams with quadratic holomorphic relations R around any elementary square of the quiver. These graphs are formally the same as the rectangular lattice quiver diagrams associated to the symmetric space CP 1 × CP 1 [10,13], which are generically a product of two chains. For a fixed pair of non-negative integers (k, l), the irreducible SU(3) representation C k,l determines a finite quiver vertex set Q 0 (k, l) as follows.…”

“…Note that in contrast to the SU(2)-equivariant quiver gauge theories based on the symmetric space CP 1 [4,13], the ansatz (5.28) automatically yields a finite Yang-Mills energy, without the need of multiplying them by suitable coefficient functions on Q 0 (k, l).…”