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A new generalization of Grassmannians in supergeometry, called $$\nu $$ -Grassmannians, are constructed by gluing $$\nu $$ -domains. By a $$\nu $$ -domain, we mean a superdomain with an odd involution say $$\nu $$ on its structure sheaf, as morphism of modules. Then we show that $$\nu $$ -Grassmannians are homogeneous superspaces. In addition, in the last section, a supergroup associated to the odd involution $$\nu $$ is introduced.

We 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 $\Pi^{p+q} A$, with $\Pi$ 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 $V$., Comment: 13 pages, reference added

We establish the representability of the general linear ${\mathbb Z}_2^n$-group and use the restricted functor of points - whose test category is the category of ${\mathbb Z}_2^n$-manifolds over a single topological point - to define its smooth linear actions on ${\mathbb Z}_2^n$-graded vector spaces and linear ${\mathbb Z}_2^n$-manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using.

We present the BFV and the BV extension of the Poisson sigma model (PSM) twisted by a closed 3-form H. There exist superfield versions of these functionals such as for the PSM and, more generally, for the AKSZ sigma models. However, in contrast to those theories, here they depend on the Euler vector field of the source manifold and contain terms mixing data from the source and the target manifold. Using an auxiliary connection $\nabla$ on the target manifold M, we obtain alternative, purely geometrical expressions without the use of superfields, which are new also for the ordinary PSM and promise straightforward adaptations to other Lie algebroid based gauge theories: The BV functional, in particular, is the sum of the classical action, the Hamiltonian lift of the (only on-shell-nilpotent) BRST differential, and a term quadratic in the antifields which is essentially the basic curvature and measures the compatibility of $\nabla$ with the Lie algebroid structure on T*M. We finally construct a Diff(M)-equivariant isomorphism between the two BV formulations., 53+1 pages. Version to be published in "Annales Henri Poincar\'e"

In this paper we prove that the etale sheafification of the functor arising from the quotient of an algebraic supergroup by a closed subsupergroup is representable by a smooth superscheme.

We interpret superfields in a functorial formalism that explains the properties that are assumed for them in the physical applications. The starting point of this research was the need to understand in a sound mathematical framework some algebraic constraints imposed on them, but it lead us to revise the very definition of superfield. The constraints that we investigate in the present work give rise to superschemes that, generically, are not regular, that is, they do not define a standard supermanifold., Significant changes, misprints corrected

In this text we combine the notions of supergeometry and supersymmetry. We construct a special class of supermanifolds whose reduced manifolds are (pseudo) Riemannian manifolds. These supermanifolds allow us to treat vector fields on the one hand and spinor fields on the other hand as equivalent geometric objects. This is the starting point of our definition of {supersymmetric Killing structures}. The latter combines subspaces of vector fields and spinor fields, provided they fulfill certain field equations. This naturally leads to a superalgebra that extends the supersymmetry algebra to the case of non-flat reduced space. We examine in detail the additional terms that enter into this structure and we give a lot of examples.

We give an introduction to superalgebra, founded on the difference between even (commuting) and odd (anti-commuting) variables. We give a sketch of Graßmann’s work, and show how derivations of those structures induce various superalgebra structures, Lie superalgebras of Cartan type being obtained with even derivations, while odd derivations induce Jordan-type superalgebras.

Numerous Lie supergroups do not admit superunitary representations except the trivial one, e.g., Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of superunitary representation, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrodinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible superunitary representations and serve as ground to the main result of this paper: a generalized Stone-von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrodinger-like representations to metaplectic supergroups, also fit into this definition of superunitary representations., Comment: 59 pages. v2: section 6 (Proof of Stone-von Neumann theorem) and section 7 (super unitary dual) were corrected and rewritten

The purpose of this paper is to describe the projective linear supergroup, its relation with the automorphisms of the projective superspace and to determine the supergroup of SUSY-preserving automorphisms of P1|1.

This is the author's Master's thesis written under the supervision of Dr. Gregor Weingart at the National Autonomous University of Mexico. The purpose of this study is to rewrite differential supergeometry in terms of classical differential geometry. This rewriting from "first principles" has two main motivations: 1 avoid using local (and usually not very well-defined) odd coordinates; 2 use both the language and the tools (both highly-developed) of classical differential geometry to state and prove results in supergeometry. Although there is work in this direction this work's point of view might be useful to translate from the sheaf-theoretic language to one that is better suited for explicit calculations., Comment: Proof of equivalence completed and corrected

Vector bundles and double vector bundles, or $2$-fold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of super\-geometry and graded manifolds of degree $\leq 2$. Indeed, a link has been established between the super and classical pictures by the geometrization process, leading to an equivalence of the category of graded manifolds of degree $\leq 2$ and the category of (double) vector bundles with additional structures. In this paper we study the geometrization process in the case of $\mathbb Z^r$-graded manifolds of type $\Delta$, where $\Delta$ is a certain weight system and $r$ is the rank of $\Delta$. We establish an equivalence between a subcategory of the category of $n$-fold vector bundles and the category of graded manifolds of type $\Delta$., Comment: 56 pages, this version was accepted for publication in Letters in Mathematical Physics

In this paper we discuss the exponential map in the case of nilpotent superalgebras. This provides global coordinates for nilpotent analytic supergroups, which are useful in the applications.

In this paper, geometries and supergeometries are analyzed from the point of view of the coset coherent states. To this end, before detailing the different factorizations of the simplest cosets, the dynamics and structure of Supercoherent states of the Klauder–Perelomov type (that are defined taking into account the geometry of the coset based on the simplest supergroup [Formula: see text] in the context of extensions of the standard model that were given previously in [A. B. Arbuzov and D. J. Cirilo-Lombardo, Phys. Scr. 94 (2019) 125302] as structural basis of the electroweak sector of the standard model (SM)) are used. The supergeometrical descriptions with such coherent superstates which uses group representation from [Y. Ne’eman, S. Sternberg and D. Fairlie, Phys. Rept. 406 (2005) 303–377] for a beyond SM model, are also defined for the noncompact case [Formula: see text]

We realize the simple Lie superalgebra G ( 3 ) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert–Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G ( 2 ) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the Tanaka–Weisfeiler prolongation of the negatively graded Lie superalgebras associated with two particular choices of parabolics. We discuss non-holonomic superdistributions with growth vector ( 2 | 4 , 1 | 2 , 2 | 0 ) deforming the flat model SHC, and prove that the second Spencer cohomology group gives a binary quadratic form, thereby indicating a “square-root” of Cartan's classical binary quartic invariant for generic rank 2 distributions in a 5-dimensional space. Finally, we obtain super-extensions of Cartan's classical submaximally symmetric models, compute their symmetries and observe a supersymmetry dimension gap phenomenon.

There are two different notions of holonomy in supergeometry, the supergroup introduced by Galaev and our functorial approach motivated by super Wilson loops. Either theory comes with its own version of invariance of vectors and subspaces under holonomy. By our first main result, the Twofold Theorem, these definitions are equivalent. Our proof is based on the Comparison Theorem, our second main result, which characterises Galaev's holonomy algebra as an algebra of coefficients, building on previous results. As an application, we generalise some of Galaev's results to S-points, utilising the holonomy functor. We obtain, in particular, a de Rham-Wu decomposition theorem for semi-Riemannian S-supermanifolds., Comment: 25 pages, in this second version many improvements mainly thanks to suggestions by Alexander Alldridge and Dominik Ostermayr

Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Z2n-manifold, i.e., a Z2n-manifold equipped with a Riemannian metric that may carry non-zero Z2n-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Z2n-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

We study the supergeometry of complex projective superspaces $\mathbb{P}^{n|m}$. First, we provide formulas for the cohomology of invertible sheaves of the form $\mathcal{O}_{\mathbb{P}^{n|m}} (\ell)$, that are pull-back of ordinary invertible sheaves on the reduced variety $\mathbb{P}^n$. Next, by studying the even Picard group $\mbox{Pic}_0 (\mathbb{P}^{n|m})$, classifying invertible sheaves of rank $1|0$, we show that the sheaves $\mathcal{O}_{\mathbb {P}^{n|m}} (\ell)$ are not the only invertible sheaves on $\mathbb{P}^{n|m}$, but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the $\Pi$-Picard group $\mbox{Pic}_\Pi (\mathbb{P}^{n|m})$, classifying $\Pi$-invertible sheaves of rank $1|1$, proving that there are also non-split $\Pi$-invertible sheaves on supercurves $\mathbb{P}^{1|m}$. Further, we investigate infinitesimal automorphisms and first order deformations of $\mathbb{P}^{n|m}$, by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special special attention is paid to the meaningful case of supercurves $\mathbb{P}^{1|m}$ and of Calabi-Yau's $\mathbb{P}^{n|n+1}$. Last, with an eye to applications to physics, we show in full detail how to endow $\mathbb{P}^{1|2}$ with the structure of $\mathcal{N}=2$ super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra $\mathfrak{osp} (2|2)$. A particular effort has been devoted to keep the exposition as concrete and explicit as possible., Comment: 24 pages

In this paper we give a brief account of the relations between non-projected supermanifolds and projectivity in supergeometry. Following the general results of arXiv:1706.01354, we study an explicit example of non-projected and non-projective supermanifold over the projective plane and we show how to embed it into a super Grassmannian. The geometry of super Grassmannians is also reviewed in details., Comment: Contribution to the Special Issue of Universe "Super Geometry for Super Strings". arXiv admin note: text overlap with arXiv:1706.01354

We study a generalization of the results \in \cite{cfk} to the case of $SU(1|1)$ interpreted as the supercircle $S^{1|2}$. We describe all of its finite dimensional complex irreducible representations, we give a reducibility result for representations not containing the trivial character, and we compute explicitly the corresponding matrix elements. In the end we give the Peter-Weyl theorem for $S^{1|2}$.

In this paper we prove that $\Pi$-projective spaces $\mathbb{P}^n_\Pi$ arise naturally in supergeometry upon considering a non-projected thickening of $\mathbb{P}^n$ related to the cotangent sheaf $\Omega^1_{\mathbb{P}^n}$. In particular, we prove that for $n \geq 2$ the $\Pi$-projective space $\mathbb{P}^n_\Pi$ can be constructed as the non-projected supermanifold determined by three elements $(\mathbb{P}^n, \Omega^1_{\mathbb{P}^n}, \lambda)$, where $\mathbb{P}^n$ is the ordinary complex projective space, $\Omega^1_{\mathbb{P}^n}$ is its cotangent sheaf and $\lambda $ is a non-zero complex number, representative of the fundamental obstruction class $\omega \in H^1 (\mathcal{T}_{\mathbb{P}^n} \otimes \bigwedge^2 \Omega^1_{\mathbb{P}^n}) \cong \mathbb{C}.$ Likewise, in the case $n=1$ the $\Pi$-projective line $\mathbb{P}^1_\Pi$ is the split supermanifold determined by the pair $(\mathbb{P}^1, \Omega^1_{\mathbb{P}^1} \cong \mathcal{O}_{\mathbb{P}^1} (-2)).$ Moreover we show that in any dimension $\Pi$-projective spaces are Calabi-Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also $\Pi$-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 $\Pi$-geometry., Comment: 15 pages. Misprints fixed and exposition improved. Some of the main propositions of section 4 got rewritten in a more precise form. Main results are unaffected

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures., 15 pages, to appear in European Journal of Mathematics

In this paper we develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable notion of super wavefront set for superdistributions which generalizes Dencker's polarization sets for vector-valued distributions to supergeometry. In particular, our super wavefront sets detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criteria when two superdistributions may be multiplied. As an application of our framework, we study the singularities of distributional solutions of a supersymmetric field theory., 26 pages, typos corrected, final version published in Journal of Mathematical Physics

In a attempt to treat a supergravity as a tensor representation, the four-dimensional NN-extended quaternionic superspaces are constructed from the (diffeomorphyc) graded extension of the ordinary Penrose-twistor formulation, performed in a previous work of the authors [D. J. Cirilo-Lombardo and V. N. Pervushin, Int. J. Geom. Methods Mod. Phys., doi: http://dx.doi.org/10.1142/S0219887816501139.], with N=p+k.N=p+k. These quaternionic superspaces have 4+k(N−k)4+k(N−k) even-quaternionic coordinates and 4N4N odd-quaternionic coordinates, where each coordinate is a quaternion composed by four Cℂ-fields (bosons and fermions respectively). The fields content as the dimensionality (even and odd sectors) of these superspaces are given and exemplified by selected physical cases. In this case, the number of fields of the supergravity is determined by the number of components of the tensor representation of the four-dimensional NN-extended quaternionic superspaces. The role of tensorial central charges for any NN even USp(N)=Sp(N,HC)∩U(N,HC)USp(N)=Sp(N,ℍℂ)∩U(N,ℍℂ) is elucidated from this theoretical context. Fil: Cirilo, Diego Julio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física del Plasma. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física del Plasma; Argentina. Joint Institute for Nuclear Research; Rusia Fil: Pervushin, Victor N.. Joint Institute for Nuclear Research; Rusia

The first obstruction to splitting a supermanifold S is one of the three components of its super Atiyah class, the two other components being the ordinary Atiyah classes on the reduced space M of the even and odd tangent bundles of S. We evaluate these classes explicitly for the moduli space of super Riemann surfaces ("super moduli space") and its reduced space, the moduli space of spin curves. These classes are interpreted in terms of certain extensions arising from line bundles on the square of the varying (super) Riemann surface. These results are used to give a new proof of the non-projectedness of ${\mathfrak{M}}_{g,1}$, the moduli space of super Riemann surfaces with one puncture.

Associated to any supermanifold is a filtration by spaces, referred to as thickenings. It is the objective of this article to study them up to a certain equivalence and then up to isomorphism in the complex-analytic setting. We study them from two points of view: (1) as structures embedded in supermanifolds and (2) abstractly. Throughout, we will be guided by the goal to clarify and address the question: when does a given thickening embed in a supermanifold? Such a question was, in essence, first studied by Eastwood and LeBrun. In this article we begin with a pedagogical account of their study, after which we further study thickenings in supergeometry and present a classification of thickenings of a given order. As a complement to our study, we comment on the moduli problem for complex supermanifolds and consider the analogous problem for thickenings. Finally, to illustrate the ideas in this article, we conclude by describing some obstructed thickenings of the complex projective plane., 52 pages. Minor revision, references added

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor A : SLoc --> S*Alg to the category of super-*-algebras which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors eA : eSLoc --> eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess-Zumino model in 3|2-dimensions., v1: 54 pages. v2: 56 pages. Minor corrections and modifications. To appear in Communications in Mathematical Physics

Within the framework of complex supergeometry and motivated by two-dimensional genus-zero holomorphic N = 2 superconformal field theory, we define the moduli space of N = 2 super-Riemann spheres with oriented and ordered half-infinite tubes (or equivalently, oriented and ordered punctures, and local superconformal coordinates vanishing at the punctures), modulo N = 2 superconformal equivalence. We develop a formal theory of infinitesimal N = 2 superconformal transformations based on a representation of the N = 2 Neveu–Schwarz algebra in terms of superderivations. In particular, via these infinitesimals we present the Lie supergroup of N = 2 superprojective transformations of the N = 2 super-Riemann sphere. We give a reformulation of the moduli space in terms of these infinitesimals. We introduce generalized N = 2 super-Riemann spheres with tubes and discuss some group structures associated to certain moduli spaces of both generalized and non-generalized N = 2 super-Riemann spheres. We define an action of the symmetric groups on the moduli space. Lastly we discuss the nonhomogeneous (versus homogeneous) coordinate system associated to N = 2 superconformal structures and the corresponding results in this coordinate system.

In this paper we discuss the highest weight $\frak k_r$-finite representations of the pair $(\frak g_r,\frak k_r)$ consisting of $\frak g_r$, a real form of a complex basic Lie superalgebra of classical type $\frak g$ (${\frak g}\neq A(n,n)$), and the maximal compact subalgebra $\frak k_r$ of $\frak g_{r,0}$, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces $G_r/K_r$., This article contains of part of the material originally posted as arXiv:1503.03828 and arXiv:1511.01420. The rest of the material was posted as arXiv:1801.07181 and will also appear in an enlarged version as subsequent posting

Superspace is considered as space of parameters of the supercoherent states defining the basis for oscillator-like unitary irreducible representations of the generalized superconformal group SU(2m,2n/2N) in the field of quaternions H. The specific construction contains naturally the supertwistor one of the previous work by Litov and Pervushin [1] and it is shown that in the case of extended supersymmetry such an approach leads to the separation of a class of superspaces and and its groups of motion. We briefly discuss this particular extension to the domain of quaternionic superspaces as nonlinear realization of some kind of the affine and the superconformal groups with the final end to include also the gravitational field[6] (this last possibility to include gravitation, can be realized on the basis of the reference[12] where the coset ((Sp(8))/(SL(4R)))~((SU(2,2))/(SL(2C)))was used in the non supersymmetric case). It is shown that this quaternionic construction avoid some unconsistencies appearing at the level of the generators of the superalgebras (for specific values of p and q; p+q=N) in the twistor one., Improved version. Accepted in the International Journal of Geometrical Methods in Modern Physics (IJGMMP)12 pages, no figures. In memoriam of Professor Boris Moyseevich Zupnik, pioneer of the development of supersymmetry, group theory and modern mathematical methods in theoretical physics

We investigate (pseudo)differential forms in the framework of supergeometry. Definitions, basic properties and Cartan calculus (DeRham differential, Lie derivative, inner product, Hodge operator) are presented; the symplectic supermechanics (even and odd) is formulated; and the question of quantization is discussed. In the framework of supermechanics, we investigate also classical Hamiltonian systems converting to SUSY-QM after quantization., Comment: 10 pages

We define a new object which we call a \( (k\oplus l |q) \)-dimensional supermanifold or partially formal supermanifold. This manifold has q odd coordinates and k + l even coordinates with l of them taking only nilpotent values. We develop the theory of such supermanifolds using functorial approach. Such a theory is completely analogous to the theory of standard supermanifolds. Therefore the paper can be considered also as an introduction to supergeometry written in the language of functors. We list some cases when \( (k\oplus l |q) \)-dimensional supermanifolds appear naturally in physics and mathematics.

We introduce a notion of Q-algebra that can be considered as a generalization of the notion of Q-manifold (a supermanifold equipped with an odd vector field obeying {Q,Q} =0). We develop the theory of connections on modules over Q-algebras and prove a general duality theorem for gauge theories on such modules. This theorem contains as a simplest case SO(d,d,Z)-duality of gauge theories on noncommutative tori.

A new concept of Loday algebroid (and its pure algebraic version - Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a Loday pseudoalgebra and its natural reduction to a Lie pseudoalgebra is studied. Further, Loday algebroids are interpreted as homological vector fields on a `supercommutative manifold' associated with a shuffle product and the corresponding Cartan calculus is introduced. Several examples, including Courant algebroids, Grassmann-Dorfman and twisted Courant-Dorfman brackets, as well as algebroids induced by Nambu-Poisson structures, are given., Comment: 29 pages

We extend the theory of super Harish-Chandra pairs, originally developed by Kostant and Koszul for smooth Lie supergroups, to algebraic supergroups over a field of characteristic zero. We also review the corresponding complex analytic theory and we give a characterization of the action of an algebraic (resp. complex analytic) super Harish-Chandra pair on a supervariety (resp. complex analytic supermanifold).

Supercurves are a generalization to supergeometry of Riemann surfaces or algebraic curves. They naturally appear in pairs related by a duality. The super Riemann surfaces appearinga s worldsheets in perturbative superstringth eory are precisely the self-dual supercurves. I will review known results and open problems in the geometry of supercurves, with a focus on Abel’s Theorem.

The half-maximal supergravity theories in three dimensions, which have local $SO(8)\xz SO(n)$ and rigid SO(8,n) symmetries, are discussed in a superspace setting starting from the superconformal theory. The on-shell theory is obtained by imposing further constraints; it is essentially a non-linear sigma model that induces a Poincar�� supergeometry. The deformations of the geometry due to gauging are briefly discussed. The possible $p$-form field strengths are studied using supersymmetry and SO(8,n) symmetry. The set of such forms obeying consistent Bianchi identities constitutes a Lie super co-algebra while the demand that these identities admit solutions places a further constraint on the possible representations of SO(8,n) that the forms transform under which can be easily understood using superspace cohomology. The dual Lie superalgebra can then be identified as the positive sector of a Borcherds superalgebra that extends the Lie algebra of the duality group. In addition to the known $p=2,3,4$ forms, which we construct explicitly, there are five-forms that can be non-zero in supergravity, while all forms with $p>5$ vanish. It is shown that some six-forms can have non-trivial contributions at order $\a'$., 30 pages. References added. Some clarification of the text

Witten's superstring field action is shown to be obtained from a purely cubic action by using noncommutative supergeometry.

We extend to the context of Courant algebroids several hierarchies that can be constructed on Poisson-Nijenhuis manifolds. More precisely, we introduce several notions (Poisson-Nijenhuis, deformation-Nijenhuis and Nijenhuis pairs) that extend to Courant algebroids the notion of a Poisson-Nijenhuis manifold, by using the idea that both the Poisson and the Nijenhuis structures, although they seem to be different in nature when considered on manifolds, are just (1, 1)-tensors on the usual Courant algebroid TM ⊕ T ∗M satisfying several constraints. For each of the generalizations mentioned, we show that there are natural hierarchies obtained by successive deformation by one of the (1, 1)-tensor. Introduction The purpose of the present article is to explain how (1, 1)-tensors with vanishing Nijenhuis torsion on a Courant structure naturally give rise to several type of hierarchies and to show it using as much as possible of supergeometric formalism. To start with, we say a few words on, respectively, Courant structures, supergeometric formalism, Leibniz algebroids, Nijenhuis torsion and hierarchies. Having recalled these notions, we explain the purpose of our study. We finish this introduction by a more detailed summary of the content of the present work. 0.1. On Courant structures, Nijenhuis torsion, supergeometry and hierarchies. Courant structures. It has been noticed by Courant [5] that the following bilinear assignment on the space of sections of TM⊕T M , forM a manifold: [(X,α), (Y, β)] := ([X, Y ], LXβ − iY dα) (with X, Y ∈ Γ(TM) = X(M), α, β ∈ Γ(T M) = Ω(M)) still satisfies the Jacobi identity, and that its default of being skew-symmetric is given, for all u, v ∈ Γ(TM ⊕ T M), by: [u, v]− [v, u] = d 〈u, v〉, Received October 10, 2011.

The existence of a natural and projectively invariant quantization in the sense of P. Lecomte [Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132] was proved by M. Bordemann [math.DG/0208171], using the framework of Thomas-Whitehead connections. We extend the problem to the context of supermanifolds and adapt M. Bordemann's method in order to solve it. The obtained quantization appears as the natural globalization of the $\mathfrak{pgl}({n+1|m})$-equivariant quantization on ${\mathbb{R}}^{n|m}$ constructed by P. Mathonet and F. Radoux in [arXiv:1003.3320]. Our quantization is also a prolongation to arbitrary degree symbols of the projectively invariant quantization constructed by J. George in [arXiv:0909.5419] for symbols of degree two.

In our exposition of mathematical SUSY, we use the language of T-points to build supermanifolds up from their foundations in graded linear algebra (superalgebra). Though our treatment is similar to the many pioneering works in supergeometry, however we prove some deeper results related to the Frobenius theorem as well as give a full treatment of superschemes and supergroups.

We study the compactness problem for moduli spaces of holomorphic supercurves which, being motivated by supergeometry, are perturbed such as to allow for transversality. We give an explicit construction of limiting objects for sequences of holomorphic supercurves and prove that, in important cases, every such sequence has a convergent subsequence provided that a suitable extension of the classical energy is uniformly bounded. This is a version of Gromov compactness. Finally, we introduce a topology on the moduli spaces enlarged by the limiting objects which makes these spaces compact and metrisable., Comment: 38 pages

Supercurves are a generalization to supergeometry of Riemann surfaces or algebraic curves. I review the definitions, examples, key results, and open problems in this area.

We study holomorphic supercurves, which are motivated by supergeometry as a natural generalisation of holomorphic curves. We prove that, upon perturbing the defining equations by making them depend on a connection, the corresponding linearised operator is generically surjective. By this transversality result, we show that the resulting moduli spaces are oriented finite dimensional smooth manifolds. Finally, we examine how they depend on the choice of generic data., Comment: 25 pages

We show a correspondence between tensor representations of the super general linear group GL(m|n) and tensor representations of the general linear superalgebra gl(m|n) using a functorial approach.

We study the functor of points and the local functor of points (here called the Weil--Berezin functor) for smooth and holomorphic supermanifolds, providing characterization theorems and fully discussing the representability issues. In the end we examine applications to differential calculus including the transitivity theorems., Comment: arXiv admin note: substantial text overlap with arXiv:0908.1872

Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism supergroup of a compact finite-dimensional supermanifold. The construction provides the diffeomorphism supergroup with the structure of a Frechet supermanifold. In addition, we derive results about the structure of diffeomorphism supergroups., Comment: 39 pages, includes a detailed proof of the supersmoothness of group multiplication

We analyze $S^1$ equivariant cohomology from the supergeometrical point of view. For this purpose we equip the external algebra of given manifold with equivariant even super(pre)symplectic structure, and show, that its Poincare-Cartan invariant defines equivariant Euler classes of surfaces. This allows to derive localization formulae by use of superanalog of Stockes theorem.

We define a superspace over a ring $R$ as a functor on a subcategory of the category of supercommutative $R$-algebras. As an application the notion of a $p$-adic superspace is introduced and used to give a transparent construction of the Frobenius map on $p$-adic cohomology of a smooth projective variety over the ring of $p$-adic integers., 14 pages, expanded introduction, more details

We introduce a notion of $Q$-algebra that can be considered as a generalization of the notion of $Q$-manifold (a supermanifold equipped with an odd vector field obeying $\{Q,Q\} =0$). We develop the theory of connections on modules over $Q$-algebras and prove a general duality theorem for gauge theories on such modules. This theorem containing as a simplest case $SO(d,d,{\bf Z})$-duality of gauge theories on noncommutative tori can be applied also in more complicated situations. We show that $Q$-algebras appear naturally in Fedosov construction of formal deformation of commutative algebras of functions and that similar $Q$-algebras can be constructed also in the case when the deformation parameter is not formal., Comment: Extended version of hep-th/9912212