Dimensional reduction and vacuum structure of quiver gauge theory

Abstract

We describe the structure of the vacuum states of quiver gauge theories obtained via dimensional reduction over homogeneous spaces, in the explicit example of SU(3)-equivariant dimensional reduction of Yang-Mills-Dirac theory on manifolds of the form M × CP 2 . We pay particular attention to the role of topology of background gauge fields on the internal coset spaces, in this case U(1) magnetic monopoles and SU(2) instantons on CP 2 . The reduction of Yang-Mills theory induces a quiver gauge theory involving coupled Yang-Mills-Higgs systems on M with a Higgs potential leading to dynamical symmetry breaking. The criterion for a ground state of the Higgs potential can be written as the vanishing of a non-abelian Yang-Mills flux on the quiver diagram, regarded as a lattice with group elements attached to the links. The reduction of SU(3)-symmetric fermions yields Dirac fermions on M transforming under the low-energy gauge group with Yukawa couplings. The fermionic zero modes on CP 2 yield exactly massless chiral fermions on M , though there is a unique choice of spin c structure on CP 2 for which some of the zero modes can acquire masses through Yukawa interactions. We work out the spontaneous symmetry breaking patterns and determine the complete physical particle spectrum in a number of explicit examples, some of which possess quantum number assignments qualitatively analogous to the manner in which vector bosons, quarks and leptons acquire masses in the standard model.

Equivariant dimensional reduction and quiver gauge theories

“…Including spinor fields, coupling to background equivariant fluxes, can give rise to chiral theories on M 4 . One expects zero modes of the Dirac operator on S/R to manifest themselves as massless chiral fermions in M 4 but, as we shall see, Yukawa couplings are induced and the dimensional reduction can give masses to some zero modes [13,14].…”

“…2.3 A more general example: complex projective plane As a more general example consider CP 2 SU (3)/U (2) (for details see [12] and [14]). Label the irreducible representations of The root diagram for SU (3) is…”

“…The full spectrum of the twisted Dirac operator is complicated but for equivariant dimensional reduction we only need the zero modes. For fermions coupling to an equivariant monopole of magnetic charge m and an equivariant instanton of topological charge n, the index of the Dirac operator on CP 2 is [14] Index D (n,m) = 1 8…”

Non-Abelian Vortices, Super Yang–Mills Theory and Spin(7)-Instantons

“…12 Note that Φ i := Re φ i and Φ i+3 := Im φ i take values in u(k) and belong to the fundamental representation of SO(6) ∼ =SU(4). For transition to the SU(4) notation…”

“…The Seiberg-Witten monopole equations [6] for n=2 and the ordinary vortex equations [7] for n=1 are particular instances of quiver vortex equations. Recently, the formalism of G-equivariant reductions has been applied in a variety of contexts [8]- [12].…”