Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences

Abstract

We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group G. We first study the case when G = O(V ) or Sp(V ), interpreting them as orthogonal (resp. symplectic) bundle representations of the symmetric quivers introduced by Derksen-Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. Finally, we discuss Hitchin-Kobayashi correspondences for these objects.this can be summarized as follows.Proposition 1.1. Let Q be a quiver. A twisting for Q is equivalent to a principal GL(α)-bundle F C together with vector spaces M α . A twisted representation of Q with dimension vector is equivalent to the prescription of a principal GL(i)-bundle E C together with a finite-dimensional representation space Rep(Q, V ), and a section φ ∈ Ω 0 ((E C × F C ) × ρ Rep(Q, V )).From our notation, it is already obvious that we are interested in passing to maximal compacts, by endowing our vector bundles with hermitian metrics. Indeed, a hermitian metric on V i corresponds to a reduction h i ∈ A 0 (E C i (GL i /U i )) of the structure group of E C i to the unitary group U i := U(V i ).

Harmonic metrics for the Hull-Strominger system and stability

“…The relation between slope stability and the Hermitian-Einstein equation (2.11) for compatible Hermitian metrics is provided by the following version of the Donaldson-Uhlenbeck-Yau Theorem (see e.g. [3,28]): Theorem 2.9. Let X be a compact complex manifold.…”