Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 028, 15 pages      arXiv:2212.07915      https://doi.org/10.3842/SIGMA.2023.028
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

CYT and SKT Metrics on Compact Semi-Simple Lie Groups

Anna Fino ab and Gueo Grantcharov b
a) Dipartimento di Matematica ''G. Peano'', Università degli studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
b) Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA

Received January 02, 2023, in final form May 11, 2023; Published online May 25, 2023

Abstract
A Hermitian metric on a complex manifold $(M, I)$ 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 ${\rm SU}(n)$ and it is called strong Kähler with torsion (SKT) or pluriclosed if the associated fundamental form $F$ is $\partial \overline \partial$-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 $I$. In particular, we show that if $I$ is determined by some maximal torus $T$ and $g$ is a left-invariant Hermitian metric, which is also invariant under the right action of the torus $T$, and is both CYT and SKT, then $g$ has to be Bismut flat.

Key words: Bismut connection; Hermitian metric.

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