CYT and SKT Metrics on Compact Semi-Simple Lie Groups - Fino, Anna; Grantcharov, Gueo (Symmetry, Integrability and Geometry: Methods and Applications, 2023)
A Hermitian metric on a complex manifold (, ) of complex dimension n is called Calabi-Yau with torsion (CYT) or Bismut-Ricci flat, if the restricted holonomy of the associated Bismut connection is contained in SU(), and it is called strong Kähler with torsion (SKT) or pluriclosed if the associated fundamental form is ∂∂¯-closed. In the paper, we study the existence of left-invariant SKT and CYT metrics on compact semi-simple Lie groups endowed with a Samelson complex structure . In particular, we show that if is determined by some maximal torus and is a left-invariant Hermitian metric, which is also invariant under the right action of the torus , and is both CYT and SKT, then g has to be Bismut flat.