A Tannakian approach to dimensional reduction of principal bundles

Abstract

Abstract. Let P be a parabolic subgroup of a connected simply connected complex semisimple Lie group G. Given a compact Kähler manifold X, the dimensional reduction of G-equivariant holomorphic vector bundles over X × G/P was carried out by the first and third authors [2]. This raises the question of dimensional reduction of holomorphic principal bundles over X × G/P . The method used for equivariant vector bundles does not generalize to principal bundles. In this paper, we adapt to equivariant principal bundles the Tannakian approach of Nori, to describe the dimensional reduction of Gequivariant principal bundles over X × G/P , and to establish a Hitchin-Kobayashi type correspondence. In order to be able to apply the Tannakian theory, we need to assume that X is a complex projective manifold. IntroductionDimensional reduction is a very powerful construction in the context of gauge theory, both in physics and geometry. Many important gauge-theoretic equations appear as symmetric solutions of fundamental equations for connections, like the instanton equations on Riemannian 4-manifolds. Examples of these are the Bogomol'nyi equations for magnetic monopoles in 3 dimensions and the vortex equation in 2 dimensions, as well as many other important integrable systems and soliton equations. In the context of model building, some physicists have applied for a long time the method of 'coset-space dimensional reduction' in the construction of gauge unified theories (see, e.g., [15,7,20,26]).In [16,17,11,1] the dimensional reduction techniques were brought into the context of holomorphic vector bundles over Kähler manifolds, to study the dimensional reduction of stable SL(2, C)-equivariant bundles over X × P 1 and the corresponding Hermitian-YangMills equations, where X is a compact Kähler manifold and P 1 is the Riemann sphere. In [2] this construction was generalised to G-equivariant vector bundles on X ×G/P , where G is a connected simply connected complex semisimple Lie group and P ⊂ G is a parabolic subgroup. These gave rise to quiver bundles with relations, that is representations of a quiver with relations in the category of vector bundles, where the quiver with relations (Q, K) is determined by P . In particular when X is a point, one has a description of 2000 Mathematics Subject Classification. 53C07, 14D21, 16G20.

Sasakian quiver gauge theories and instantons on cones over round and squashed seven-spheres

“…The general setup is that of a gauge theory on a product M d × G/H of a d-dimensional Riemannian manifold M d and a homogeneous manifold G/H. The natural objects in geometric considerations of gauge theories are principal fibre bundles, but in this paper we will work with (associated) complex vector bundles; the formulation of equivariant dimensional reduction and the corresponding quiver gauge theories in the setting of principal bundles can be found in [21,22]. Thus let π : E → M d × G/H be a Hermitian vector bundle of rank r, and assume that the group G acts trivially on the Riemannian manifold M d .…”

Real Higgs pairs and Non-abelian Hodge correspondence on a Klein surface

“…where F h is the curvature of the unique connection, on the principal H-bundle E H ⊂ E corresponding to h, that is compatible with the holomorphic structure of E, while Λ denotes the contraction of forms with ω and µ h is a moment map that depends on h; construction of this ω requires fixing a bi-invariant metric B on the Lie algebra h = Lie(H), an H-invariant Hermitian product , on V as well as a Hermitian metric h L on L. Examples of twisted Higgs pairs include: quiver bundles, Higgs bundles and Hodge bundles among other objects (see [1,13,14,21,22] for examples and more on Higgs pairs).…”