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Supersymmetric products of SUSY-curves °
Abstract
The supersymmetric product of a SUSY-curve over a field is constructed with the aid of a theorem of invariants, and the notion of relative superdivisor is introduced. A universal superdivisor is defined in the supersymmetric product by means of Manin's superdiagonal, and it is proven that every superdivisor can be obtained in a unique way as a pullback of the universal superdivisor.
... The results of this paper only in the case of SUSY-curves were stated (without proofs) in [8]. ...
... If X is a SUSY-curve, there exists an isomorphism ψ: X ∼ → X c between X and the supercurve of positive superdivisors of degree 1, as we proved in subsection 4.1. Then we have an isomorphism S g X → S g X c between the supersymmetric product of X and the superscheme S g X c of positive superdivisors of degree g on X, so that the representability theorem now reads (see [8]): ...
The supersymmetric product of a supercurve is constructed with the aid of a theorem of algebraic invariants and the notion of positive relative superdivisor (supervortex) is introduced. A supercurve of positive superdivisors of degree 1 (supervortices of vortex number 1) of the original supercurve is constructed as its supercurve of conjugate fermions, as well as the supervariety of relative positive superdivisors of degre p (supervortices of vortex number p.) A universal superdivisor is defined and it is proved that every positive relative superdivisor can be obtained in a unique way as a pull-back of the universal superdivisor. The case of SUSY-curves is discussed.
We construct local and global moduli spaces of supersymmetric curves with Ramond-Ramond punctures. We assume that the underlying ordinary algebraic curves have a level n structure and build these supermoduli spaces as algebraic superspaces, i.e., quotients of étale equivalence relations between superschemes.
A fine moduli superspace for algebraic super Riemann surfaces with a level-n structure is con- structed as a quotient of the split superscheme of local spin-gravitivo fields by an 6tale equivalence relation. This object is not a superscheme, but still has an interesting structure: it is an algebraic su- perspace, that is, an analytic superspace with sufficiently many meromorphic functions. The moduli of super Riemann surfaces with punctures (fixed points in the supersurface) is also constructed as an algebraic superspace. Moreover, when one only considers ordinary punctures (fixed points in the underlying ordinary curve), it turns out that the moduli is a true superscheme. We prove furthermore that this moduli superscheme is split.
sheaf he on the curve, and the formulation of geometric properties of linear svstems in terms of these sheaves is today one of the cohomological facts of life. Some of the finer properties of linear systems however can really only be expressed by considering X(n), the n-fold symmetric product of the curve. In this paper accordingly we study a vector bundle S,1(S) of rank n on X (n) which is derived from Se by a symmetrization process, and we relate somie of its properties to the linear system. The main geometric result is the determination of rational equivalence class on X (n) of that cycle 2(0) which represents the totality of positive divisors of degree n contained in the system 2. This cycle is, or should have been, a classical object; it enters implicitly into various classical results. To illustrate these notions,
CONTENTSIntroduction Chapter I. Linear algebra in superspaces § 1. Linear superspaces § 2. Modules over superalgebras § 3. Matrix algebra § 4. Free modules § 5. Bilinear forms § 6. The supertrace § 7. The Berezinian (Berezin function) § 8. Tensor algebras § 9. Lie superalgebras and derivations of superalgebras Chapter II. Analysis in superspaces and superdomains § 1. Definition of superspaces and superdomains § 2. Vector fields and Taylor series § 3. The inverse function theorem and the implicit function theorem § 4. Integration in superdomains Chapter III. Supermanifolds § 1. Definition of a supermanifold § 2. Subsupermanifolds § 3. Families Notes References
Multiloop contributions in the Polyakov formulation of the string (resp. superstring) theory are calculated via a measure on the moduli space of curves (resp. supercurves) which equals the modulus squared of the Mumford form (resp. superform). In [2] it is shown that the Mumford form is a horizontal section of a canonical connection. In this paper we extend this proof to superforms.