322,900 research outputs found
On BV Supermanifolds and the Super Atiyah Class
No full textWe study global and local geometry of forms on odd symplectic BV
supermanifolds, constructed from the total space of the bundle of 1-forms on a
base supermanifold. We show that globally 1-forms are an extension of vector
bundles defined on the base supermanifold. In the holomorphic category, we
prove that this extension is split if and only if the super Atiyah class of the
base supermanifold vanishes. This is equivalent to the existence of a
holomorphic superconnection: we show how this condition is related to the
characteristic non-split geometry of complex supermanifolds. From a local point
of view, we prove that the deformed de Rham double complex naturally arises as
a de-quantization of the de Rham/Spencer double complex of the base
supermanifold. Following \v{S}evera, we show that the associated spectral
sequence yields semidensities on the BV supermanifold, together with their
differential in the form of a super BV Laplacian.Comment: 23 pages, major changes and more material added. To appear in
European Journal of Mathematic
On the Geometry of Forms on Supermanifolds
No full textThis paper provides a rigorous account on the geometry of forms on
supermanifolds, with a focus on its algebraic-geometric aspects. First, we
introduce the de Rham complex of differential forms and we compute its
cohomology. We then discuss three intrinsic definitions of the Berezinian sheaf
of a supermanifold - as a quotient sheaf, via cohomology of the super Koszul
complex or via cohomology of the total de Rham complex. Further, we study the
properties of the Berezinian sheaf, showing in particular that it defines a
right -module. Then we introduce integral forms and their complex
and we compute their cohomology, by providing a suitable Poincar\'e lemma. We
show that the complex of differential forms and integral forms are
quasi-isomorphic and their cohomology computes the de Rham cohomology of the
reduced space of the supermanifold. The notion of Berezin integral is then
introduced and put to the good use to prove the superanalog of Stokes' theorem
and Poincar\'e duality, which relates differential and integral forms on
supermanifolds. Finally, a different point of view is discussed by introducing
the total tangent supermanifold and (integrable) pseudoforms in a new way. In
this context, it is shown that a particular class of integrable pseudoforms
having a distributional dependence supported at a point on the fibers are
isomorphic to integral forms. Within the general overview several new proofs of
results are scattered.Comment: 60 pages, matches the published version. Thanks to the anonymous
referee for useful comments and remark
Non-Projected Supermanifolds and Embeddings in Super Grassmannians
No full textIn this paper we give a brief account of the relations between non-projected supermanifolds and projectivity in supergeometry. Following the general results (L. Sergio et al., 2018), we study an explicit example of non-projected and non-projective supermanifold over the projective plane and show how to embed it into a super Grassmannian. The geometry of super Grassmannians is also reviewed in detail
Supergeometry of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>Π</mml:mi></mml:math>-projective spaces
No full textIn this paper we prove that Π-projective spaces PΠn arise naturally in supergeometry upon considering a non-projected thickening of Pn related to the cotangent sheaf ΩPjavax.xml.bind.JAXBElement@2499360f1. In particular, we prove that for n⩾2 the Π-projective space PΠn can be constructed as the non-projected supermanifold determined by three elements (Pn,ΩPjavax.xml.bind.JAXBElement@14aca00a1,λ), where Pn is the ordinary complex projective space, ΩPjavax.xml.bind.JAXBElement@1bab69961 is its cotangent sheaf and λ is a non-zero complex number, representative of the fundamental obstruction class ω∈H1(TPjavax.xml.bind.JAXBElement@53a5c4eb⊗⋀2ΩPjavax.xml.bind.JAXBElement@3ee1b9de1)≅C. Likewise, in the case n=1 the Π-projective line PΠ1 is the split supermanifold determined by the pair (P1,ΩPjavax.xml.bind.JAXBElement@4f3cf2f81≅OPjavax.xml.bind.JAXBElement@f9bd550(−2)). Moreover we show that in any dimension Π-projective spaces are Calabi–Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also Π-Grassmannians can be constructed the same way using the cotangent sheaf of their underlying reduced Grassmannians, provided that also higher, possibly fermionic, obstruction classes are taken into account. This suggests that this unexpected connection with the cotangent sheaf is characteristic of Π-geometry
Projective superspaces in practice
No full textThis paper is devoted to the study of supergeometry of complex projective superspaces Pn|m. First, we provide formulas for the cohomology of invertible sheaves of the form OPn|m(l), that are pullbacks of ordinary invertible sheaves on the reduced variety Pn. Next, by studying the even Picard group Pic0(Pn|m), classifying invertible sheaves of rank 1|0, we show that the sheaves OPn|m(l) are not the only invertible sheaves on Pn|m, but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the Π-Picard group PicΠ(Pn|m), classifying Π-invertible sheaves of rank 1|1, proving that there are also non-split Π-invertible sheaves on supercurves P1|m. Further, we investigate infinitesimal automorphisms and first order deformations of Pn|m, by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special attention is paid to the meaningful case of supercurves P1|mand of Calabi–Yau's Pn|n+1. Last, with an eye to applications to physics, we show in full detail how to endow P1|2with the structure of N=2 super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra osp(2|2). A particular effort has been devoted to keep the exposition as concrete and explicit as possible
A note on super Koszul complex and the Berezinian
No full textWe construct the super Koszul complex of a free supercommutative A-module V of rank p|q and prove that its homology is concentrated in a single degree and it yields an exact resolution of A. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to Πp+qAΠp+qA, with ΠΠ the parity changing functor. Finally, we show that, given an automorphism of V, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of V
Non projected calabi-yau supermanifolds over P2
No full textWe start a systematic study of non-projected supermanifolds, concentrating on supermanifolds with fermionic dimension 2 and with the reduced manifold a complex projective space. We show that all the non-projected supermanifolds of dimension 2j2 over P2 are completely characterised by a non-zero cohomology class 2 H1(TP2 (3)) and by a locally free sheaf FM of rank 0j2, satisfying Sym2FM = KP2 . Denoting such supermanifolds with P2 (FM ), we show that all of them are Calabi-Yau supermanifolds and, when 6= 0, they are non-projective, that is they cannot be embedded into any projective superspace Pnjm. Instead, we show that every non-projected supermanifold over P2 admits an embedding into a super Grassmannian. By contrast, we give an example of a supermanifold P2 (FM ) that cannot be embedded in any of the projective superspaces Pn introduced by Manin and Deligne. However, we also show that when FM is the cotangent bundle over P2, then the non-projected P2 (FM ) and the -projective plane P2 do coincide
THE UNIVERSAL DE RHAM / SPENCER DOUBLE COMPLEX ON A SUPERMANIFOLD
No full textThe universal Spencer and de Rham complexes of sheaves over a smooth or analytical manifold are well known to play a basic role in the theory of D-modules. In this article we consider a double complex of sheaves generalizing both complexes for an arbitrary supermanifold, and we use it to unify the notions of differential and integral forms on real, complex and algebraic supermanifolds. The associated spectral sequences give the de Rham complex of differential forms and the complex of integral forms at page one. For real and complex supermanifolds both spectral sequences converge at page two to the locally constant sheaf. We use this fact to show that the cohomology of differential forms is isomorphic to the cohomology of integral forms, and they both compute the de Rham cohomology of the reduced manifold. Furthermore, we show that, in contrast with the case of ordinary complex manifolds, the Hodge-to-de Rham (or Frolicher) spectral sequence of supermanifolds with Kahler reduced manifold does not converge in general at page one
Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces
No full textWe study Chevalley-Eilenb erg cohomology of physically relevant Lie superalgebras related to supersymmetric theories, providing explicit expressions for their cocycles in terms of their Maurer-Cartan forms. We include integral forms in the picture by defining the notions of constant densities and integral forms related to a Lie superalgebra. We develop a suitable generalization of Chevalley-Eilenb erg cohomology extended to integral forms and we prove that it is isomorphic via a Poincare duality-type pairing to the ordinary Chevalley-Eilenb erg cohomology of the Lie superalgebra. Next, we study equivariant Chevalley-Eilenb erg cohomology for coset superspaces, which play a crucial role in supergravity and superstring models. Again, we treat explicitly several examples, providing cocycles' expressions and revealing a characteristic infinite -dimensional cohomology
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