Abstract
We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY’s having reduced manifold equal to {mathrm{mathbb{P}}}^1 , namely the projective super space {mathrm{mathbb{P}}}^{left.1right|2} and the weighted projective super space mathbb{W}{mathrm{mathbb{P}}}_{(2)}^{left.1right|1} . Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces {mathrm{mathbb{P}}}^{left.nright|m} . We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of {mathrm{mathbb{P}}}^{left.1right|2} , whose automorphism group turns out to be larger than the projective super group. By considering the cohomology of the super tangent sheaf, we compute the deformations of {mathrm{mathbb{P}}}^{left.1right|m} , discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that {mathrm{mathbb{P}}}^{left.1right|2} is self-mirror, whereas mathbb{W}{mathrm{mathbb{P}}}_{(2)}^{left.1right|1} has a zero dimensional mirror. Also, the mirror map for {mathrm{mathbb{P}}}^{left.1right|2} naturally endows it with a structure of N = 2 super Riemann surface.
Highlights
Rham cohomology for generic projective super spaces Pn|m
We will not dwell into a detailed exposition, and we address the interested reader e.g. to [11] and [12] for a mathematically thorough treatment of supergeometry
In the present paper we have investigated some basic questions about super Calabi-Yau varieties (SCY’s)
Summary
The mathematical basic notion that lies on the very basis of any physical supersymmetric theory is the one of Z2-grading: algebraic constructions such as rings, vector spaces, algebras and so on, have their Z2-graded analogues, usually called in physics super rings, super vector spaces and super algebras, respectively. Of the standard description of manifolds in differential geometry Even if this is feasible [49], due to the presence of nilpotent elements, in supergeometry it is preferable to adopt an algebraic geometric oriented point of view, in which a supermanifold is conceived as a locally ringed space [11, 12, 14, 54]. Within this global point of view, we define a super space M to be a Z2-graded locally ringed space, that is a pair (|M |, OM ), consisting of a topological space |M | and a sheaf of super algebras OM over |M |, such that the stalks OM ,x at every point x ∈ |M | are local rings Notice that this makes sense as a requirement, for the odd elements are nilpotent and this reduces to ask that the even component of the stalk is a usual local commutative ring. This bears a nice geometric view of split supermanifolds: they can be globally regarded as a vector bundle E → Mred on the reduced manifold having purely odd fibers, as the above definition of supermanifold suggests by itself
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