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Moduli spaces of real algebraic \(N=2\) supercurves. (English. Russian original) Zbl 0895.58005

Funct. Anal. Appl. 30, No. 4, 237-245 (1996); translation from Funkts. Anal. Prilozh. 30, No. 4, 19-30 (1996).
In this article, the moduli space of real \(N=2\) supercurves is studied. \(N=2\) complex supercurves are obtained by factoring out a super-Fuchsian group from the \(N=2\) super-half-plane. Real supercurves are given by a complex supercurve with an antiholomorphic involution. The real points of the curve are the fixed points of the involution. A complete set of topological invariants of the curve is given by the genus, the number of ovals, and by connectedness (resp. non-connectedness) of the complex curve with the real points removed. The connected components of the moduli space are explicitly described and a complete set of topological invariants for the components are given. Each of the components is represented as \(\mathbb R^{(m| n)}/\text{Mod}\) of a linear superspace by a discrete group \(\text{Mod}\). The dimensions \(m\) and \(n\) are given. Depending on whether the two Arf functions coincide or not, one obtains \((m| n)=(4g-3| 4g-4)\), resp. \((4g-4| 4g-4)\). Real \(N=2\) supercurves can be mapped to real spinor bundles of rank 2 over real algebraic curves. Hence a description of the moduli space of spinor bundles of rank 2 over a real curve is obtained.

MSC:

58A50 Supermanifolds and graded manifolds
32C11 Complex supergeometry
14M30 Supervarieties
14H15 Families, moduli of curves (analytic)
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References:

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