Riemannian supergeometry

Abstract

Motivated by Zirnbauer in J Math Phys 37(10): 4986-5018 (1996), we develop a theory of Riemannian supermanifolds up to a definition of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian and the Lie theoretical viewpoint are introduced, e.g., geodesics, isometry groups and invariant metrics on Lie supergroups and homogeneous superspaces.

Killing vector fields and harmonic superfield theories

“…which is the definition of the pullback in [Goe08]. The next lemma will be needed in calculations below.…”

“…We shall occasionally abuse notation and write M instead of (M, O M ). Modern monographs on the general theory of supermanifolds include [Var04] and [CCF11] while aspects of Riemannian supergeometry are studied in [Goe08]. References for more specialised topics will be given at suitable positions throughout the text.…”

“…If (N, g) is a semi-Riemannian supermanifold, there is a unique connection which is (graded) metric and torsion-free, called the Levi-Civita connection [Goe08]. In general, for a connection ∇ on N , there is a canonical pullback connection ∇ Φ : SΦ → ϕ * S * M ⊗ ϕ * O M SΦ (see [GW12]), that, in the following, we shall denote simply by ∇.…”

“…4. There are several equivalent definitions for a Killing vector field on a semi-Riemannian manifold, most of which have been generalised to supergeometry ([MSV96], [Goe08], [ACDS97]), while a homogeneous treatment of the subject is yet missing. Another aim of this article is to fill this gap.…”

Quasi-riemannian structures on supermanifolds and characteristic classes

“…It appears that when going to supermanifolds, it is not possible to satisfy simultaneously both properties (i) and (ii). The standard definition of the supermetric on the Riemannian supermanifold [1,2] insists on the preservation of the symmetry property when the condition (i) is replaced by the condition of supersymmetry of the tensor g under the permutation of indices; in this case, condition (ii) for the positive definiteness becomes senseless and is replaced by the requirement of nondegeneracy of the tensor g . Unfortunately, the combination of these requirements imposes severe restrictions on the supermanifold structure.…”