# Invariant Einstein metrics on basic classical Lie supergroups

## TL;DR

This paper systematically studies invariant Einstein metrics on basic classical Lie supergroups, deriving explicit formulas and showing most admit multiple such metrics, including Ricci flat examples, highlighting unique superalgebra phenomena.

## Contribution

It provides explicit formulas for Einstein metrics on classical Lie supergroups and demonstrates their existence and multiplicity, including Ricci flat cases, which is novel in supergeometry.

## Key findings

- Most classical Lie supergroups admit at least two invariant Einstein metrics.
- Explicit formulas for Levi-Civita connection and Ricci tensor are derived.
- Existence of Ricci flat Einstein metrics in super cases is established.

## Abstract

This paper presents a systematic study of invariant Einstein metrics on basic classical Lie supergroups, whose Lie superalgebras belong to the Kac's classification of finite dimensional classical simple Lie superalgebras over $\mathbb{R}$. We consider a natural family of left invariant metrics parameterized by scaling factors on the simple and Abelian components of the reductive even part, using the canonical bi-invariant bilinear form. Explicit expressions for the Levi-Civita connection and Ricci tensor are derived, and the Einstein condition is reduced to a solvable algebraic system. Our main result shows that, except for the cases of $\mathbf{A}(m,n)$ with $m\neq n$, $\mathbf{F}(4)$, and their real forms, every real basic classical Lie superalgebra admits at least two distinct Einstein metrics. Notably, for $\mathbf{D}(n+1,n)$ and $\mathbf{D}(2,1;\alpha)$, we obtain both Ricci flat and non Ricci flat Einstein metrics, a phenomenon not observed in the non-super setting.