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

Abstract

We consider a complex vector bundle E endowed with a connection A over the eight-dimensional manifold R 2 × G/H, where G/H = SU(3)/U(1)×U(1) is a homogeneous space provided with a never integrable almost complex structure and a family of SU(3)-structures. We establish an equivalence between G-invariant solutions A of the Spin(7)-instanton equations on R 2 × G/H and general solutions of non-Abelian coupled vortex equations on R 2 . These vortices are BPS solitons in a d = 4 gauge theory obtained from N = 1 supersymmetric Yang-Mills theory in ten dimensions compactified on the coset space G/H with an SU(3)-structure. The novelty of the obtained vortex equations lies in the fact that Higgs fields, defining morphisms of vector bundles over R 2 , are not holomorphic in the generic case. Finally, we introduce BPS vortex equations in N = 4 super Yang-Mills theory and show that they have the same feature.

Yang-Mills instantons and dyons on homogeneous G 2-manifolds

“…Writing the equations of motion (4.7) as 9) we see that they describe the motion of a particle on C 3 under the influence of the inverted quartic potential −V , where 10) or, alternatively, the dynamics of three identical particles on the complex plane, with an external potential given by the (negative of) the first line in (4.10) and two-and three-body interactions in the second line.…”

“…The potential on this subspace becomes V ( φ, φ, ± φ) = 6 φ 2 ∓ (κ−3)(2 φ − 1) φ ∓ 1 2 , (A 9) and its critical zeros on the positive real axis are ( φ 1 , φ 2 , φ 3 ; κ) = (+1, +1, +1; any) and (+1, +1, −1; −3) (A.10)…”

“…The Jacobi elliptic functions arise from the inversion of the elliptic integral of the first kind, where K(k) is the complete elliptic integral of the first kind, 9) In the following we sometimes drop the parameter k, i.e. write sn[u; k] = sn(u) etc.…”

“…Such equations in d>4 dimensions were first introduced in [2] and further considered e.g. in [3]- [9]. Some of their solutions were found e.g.…”

Deformations of Nearly Kähler Instantons

“…Although instantons were first introduced in dimension four, the study of instantons on manifolds of dimension greater than four has a long history [1,6,7,13,14,16,17,[19][20][21]30,33,34,[36][37][38]40,41,43,[46][47][48][49][50][51][52][53][54][55][56][57][58][59] (not to mention the long and fruitful study of Hermitian-Yang-Mills connections). In favourable circumstances instantons solve the Yang-Mills equation; for this reason and others the study of instantons informs string theory, supergravity, and theoretical physics.…”