CYT and SKT Metrics on Compact Semi-Simple Lie Groups

TL;DR

This paper investigates the existence of special Hermitian metrics, specifically CYT and SKT, on compact semi-simple Lie groups with Samelson complex structures, revealing conditions under which such metrics must be Bismut flat.

Contribution

It characterizes when left-invariant CYT and SKT metrics on these Lie groups are necessarily Bismut flat, under symmetry assumptions.

Findings

CYT and SKT metrics coincide with Bismut flat metrics under certain invariance conditions.

Existence of invariant SKT and CYT metrics is constrained by the group structure and complex structure.

The paper provides conditions for the uniqueness of such metrics on compact semi-simple Lie groups.

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 K\"ahler 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.