A direct calculation of super-Teichmüller space
The notion of ``supermanifolds'' arose in theoretical physics [see \textit{B. De Witt}, Supermanifolds (Cambridge 1984; Zbl 0551.53002); (reprint 1987)]. A supermanifold of dimension (m,n) is locally modelled on \(B_ L^{m,n}=(B_{L,odd})^ m\times (B_{L,even})^ n,\) where \(B_{L,odd}\) resp. \(B_{L,even}\) is the odd- resp. even-dimensional part of the Grassmann algebra of \({\mathbb{R}}^ L\) or \({\mathbb{C}}^ L\), \(L\in {\mathbb{N}}\), with an appropriate generalization of differentiability. Especially, an (m,n)-dimensional supermanifold M is also a \(2^{L- 1}\cdot (m+n)\)-dimensional manifold, which possesses some bundle structure over an m-dimensional manifold, called the ``body'' of M. In the present note, results about the super-Teichmüller space of a super-Riemann surface are announced (and sketches of proofs are given): for genus \(g>1\), it is a complex analytic supermanifold of dimension (3g- 3, 2g-2), and its body is the manifold of pairs (marked Riemann surface, spin structure) in genus g [see also \textit{L. Crane} and \textit{J. M. Rabin}, ``Super Riemann surfaces: Uniformization and Teichmüller theory'', Commun. Math. Phys. (1988)].
scientific article; zbMATH DE number 4020030