×

zbMATH — the first resource for mathematics

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

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] MooreG., NelsonP., and PolchinskiJ., ?Strings and Supermoduli?, Phys. Lett. 169B, 47-53 (1986).
[2] de Witt, B. S., Supermanifolds, Cambridge, 1984.
[3] SeifertH. and ThrelfallW., Lehrbuch der Topologie, 1934 (reprinted) Chelsea, New York, n.d.).
[4] Crane, L. and Rabin, J. M., ?Super Riemann Surfaces: Uniformization and Teichmüller Theory?, University of Chicago, preprint No. EFI 86-25 (1986). · Zbl 0659.30039
[5] Hodgkin, L., ?Non-Abelian Cohomology and Super-Teichmüller Theory? (in preparation). · Zbl 0647.32022
[6] Gunning, R. C., Lectures on Riemann Surfaces-Jacobi Varieties, Princeton, 1972. · Zbl 0387.32008
[7] HarveyW. J. (ed.), Discrete Groups and Automorphic Functions, Academic Press, London, 1977.
[8] RogersA., ?A Global Theory of Supermanifolds?, J. Math. Phys. 21, 1352-1365 (1980). · Zbl 0447.58003
[9] SerreJ.-P., Cohomologie galoisienne, Lecture Notes in Mathematics No. 5, Springer-Verlag, London, Heidelberg, New York, 1965.
[10] Gunning, R. C. Lectures on Riemann Surfaces, Princeton, 1965. · Zbl 0175.36801
[11] BelavinA. A. and KnizhnikV. G., ?Algebraic Geometry and the Geometry of Quantum Strings?, Phys. Lett. 168B, 201-206 (1986).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.