Bismut Ricci flat generalized metrics on compact homogeneous spaces

Abstract

A generalized metric on a manifold $M$, i.e., a pair $(g,H)$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, is a fixed point of the generalized Ricci flow if and only if $(g,H)$ is Bismut Ricci flat: $H$ is $g$-harmonic and $Ric(g)=\frac{1}{4} H_g^2$. On any homogeneous space $M=G/K$, where $G=G_1\times G_2$ is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat $G$-invariant generalized metric, which is proved to be unique among a $4$-parameter space of metrics in many cases, including when $K$ is neither abelian nor semisimple. On the other hand, if $K$ is simple and the standard metric is Einstein on both $G_1/\pi_1(K)$ and $G_2/\pi_2(K)$, we give a one-parameter family of Bismut Ricci flat $G$-invariant generalized metrics on $G/K$ and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form $M=G\times G/\Delta K$ and for $M=SO(8)\times SU(4)/SU(3)$.Comment: 24 pages. Formula for the Ricci curvature in Prop 3.2, (iv) corrected. New (and much simpler) statement for Thm 5.1, (i). New main result on curves in Thm 5.17. Remarks 5.2, 5.9, 5.18 and 5.19 adde