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A direct calculation of super-Teichmüller space. (English) Zbl 0627.58004
The notion of “supermanifolds” arose in theoretical physics [see 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 L. Crane and J. M. Rabin, “Super Riemann surfaces: Uniformization and Teichmüller theory”, Commun. Math. Phys. (1988)].
Reviewer: B.Zimmermann

58A50 Supermanifolds and graded manifolds
58C50 Analysis on supermanifolds or graded manifolds
32G99 Deformations of analytic structures
30F99 Riemann surfaces
57M99 General low-dimensional topology
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