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Our goal is to describe superconformal structures on super Riemann surfaces (SRSs) based on data assigned to a fatgraph. We start from the complex structures on punctured (1 | 1) -supermanifolds, characterizing the corresponding moduli and the deformations using Strebel differentials and certain Čech cocycles for a specific covering, which we reproduce from fatgraph data, consisting of U (1) -graph connection and odd parameters at the vertices. Then, we consider dual (1 | 1) -supermanifolds and related superconformal structures for N = 2 super Riemann surfaces. The superconformal structures, N = 1 SRS, are computed as the fixed points of involution on the supermoduli space of N = 2 SRS. [ABSTRACT FROM AUTHOR]

We formulate minimal SuperGeometric Quantum Field Theories (SG-QFTs) that allow for scalar-fermion field transformations in a manifestly reparameterisation covariant manner. First, we discuss the issue of uniqueness in defining the field-space supermetric of the underlying supermanifold, and clarify the fact that different supermetric definitions can lead to distinct theories in the off-shell kinematic region. By adopting natural choices for the field-space supermetric, we then show that scalar fields alone cannot induce a non-trivial field-space Riemannian curvature in the fermionic sector, beyond the one originating from the scalar part of the theory. We present for the first time minimal SG-QFT models that feature non-zero fermionic curvature both in two and four spacetime dimensions. Physical applications of SG-QFTs are discussed.

We develop the theory of Free Integro-Differential Algebras (FIDA) extending the powerful technique of Free Differential Algebras constructed by D. Sullivan. We extend the analysis beyond the superforms to integral- and pseudo-forms used in supergeometry. It is shown that there are novel structures that might open the road to a deeper understanding of the geometry of supergravity. We apply the technique to some models as an illustration and we provide a complete analysis for D = 11 supergravity. There, it is shown how the Hodge star operator for supermanifolds can be used to analyze the set of cocycles and to build the corresponding FIDA. A new integral form emerges which plays the role of the truly dual to 4-form F (4) and we propose a new variational principle on supermanifolds. [ABSTRACT FROM AUTHOR]

We give the superalgebra of N = 2 chiral (and antichiral) quantum superfields realized as a subalgebra of the quantum supergroup SL q (4|2). The multiplication law in the quantum supergroup induces a coaction on the set of chiral superfields. We also realize the quantum deformation of the chiral Minkowski superspace as a quantum principal bundle. [ABSTRACT FROM AUTHOR]

Our goal is to describe superconformal structures on super Riemann surfaces (SRSs) based on data assigned to a fatgraph. We start from the complex structures on punctured (1|1)-supermanifolds, characterizing the corresponding moduli and the deformations using Strebel differentials and certain Čech cocycles for a specific covering, which we reproduce from fatgraph data, consisting of U(1)-graph connection and odd parameters at the vertices. Then, we consider dual (1|1)-supermanifolds and related superconformal structures for N=2 super Riemann surfaces. The superconformal structures, N=1 SRS, are computed as the fixed points of involution on the supermoduli space of N=2 SRS.

We develop the theory of Free Integro-Differential Algebras (FIDA) extending the powerful technique of Free Differential Algebras constructed by D. Sullivan. We extend the analysis beyond the superforms to integral- and pseudo-forms used in supergeometry. It is shown that there are novel structures that might open the road to a deeper understanding of the geometry of supergravity. We apply the technique to some models as an illustration and we provide a complete analysis for D = 11 supergravity. There, it is shown how the Hodge star operator for supermanifolds can be used to analyze the set of cocycles and to build the corresponding FIDA. A new integral form emerges which plays the role of the truly dual to 4-form F(4) and we propose a new variational principle on supermanifolds.

This paper is devoted to an exposition of the Koszul complex of a supermodule and its Berezinian from an intrinsic and as general as possible point of view. As an application, an analogue to Bott's formula in the supercommutative setting is given, computing the cohomology of twisted differential p -forms on the projective superspace. [ABSTRACT FROM AUTHOR]

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 Π p + q A , with Π 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. [ABSTRACT FROM AUTHOR]

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 S U (2 | 1) 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 S U (1 , 1 | 1). [ABSTRACT FROM AUTHOR]

We generalize Artin's three main algebraicity theorems to the setting of supergeometry: approximation [2] , algebraization of formal deformations [3] , and algebraization of stacks [4]. [ABSTRACT FROM AUTHOR]

It is known that the supermultiplet of beta-deformations of N = 4 supersymmetric Yang–Mills theory can be described in terms of the exterior product of two adjoint representations of the superconformal algebra. We present a supergeometrical interpretation of this fact, by evaluating the deforming operator on some special coherent states in the space of supersingletons. We also discuss generalization of this approach to other finite-dimensional deformations of the N = 4 supersymmetric Yang–Mills theory. [ABSTRACT FROM AUTHOR]

We consider various A∞-algebras of differential (super)forms, which are related to gauge theories and demonstrate explicitly how certain reformulations of gauge theories lead to the transfer of the corresponding A∞-structures. In addition, for N=2 3D space, we construct the homotopic counterpart of the de Rham complex, which is related to the superfield formulation of the N=2 Chern-Simons theory. [ABSTRACT FROM AUTHOR]

We formulate minimal SuperGeometric Quantum Field Theories (SG-QFTs) that allow for scalar-fermion field transformations in a manifestly reparameterisation covariant manner. First, we discuss the issue of uniqueness in defining the field-space supermetric of the underlying supermanifold, and clarify the fact that different supermetric definitions can lead to distinct theories in the off-shell kinematic region. By adopting natural choices for the field-space supermetric, we then show that scalar fields alone cannot induce a non-trivial field-space Riemannian curvature in the fermionic sector, beyond the one originating from the scalar part of the theory. We present for the first time minimal SG-QFT models that feature non-zero fermionic curvature both in two and four spacetime dimensions. Physical applications of SG-QFTs are discussed. [ABSTRACT FROM AUTHOR]

We interpret superfields in a functorial formalism that explains the properties that are assumed for them in the physical applications. We study the non-trivial relation of scalar superfields with the defining sheaf of the supermanifold of super spacetime. We also investigate in the present work some constraints that are imposed on the superfields, which allow for non-trivial solutions. They give rise to superschemes that, generically, are not regular, that is they do not define a standard supermanifold.

This paper is devoted to the study of supergeometry of complex projective superspaces P n | m . First, we provide formulas for the cohomology of invertible sheaves of the form O P n | m ( ℓ ) , that are pullbacks of ordinary invertible sheaves on the reduced variety P n . Next, by studying the even Picard group Pic 0 ( P n | m ) , classifying invertible sheaves of rank 1 | 0 , we show that the sheaves O P n | m ( ℓ ) are not the only invertible sheaves on P n | 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 Pic Π ( P n | m ) , classifying Π -invertible sheaves of rank 1 | 1 , proving that there are also non-split Π -invertible sheaves on supercurves P 1 | m . Further, we investigate infinitesimal automorphisms and first order deformations of P n | 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 P 1 | m and of Calabi–Yau’s P n | n + 1 . Last, with an eye to applications to physics, we show in full detail how to endow P 1 | 2 with 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 o s p ( 2 | 2 ) . A particular effort has been devoted to keep the exposition as concrete and explicit as possible. [ABSTRACT FROM AUTHOR]

In this paper we prove that Π -projective spaces P Π n arise naturally in supergeometry upon considering a non-projected thickening of P n related to the cotangent sheaf Ω P n 1 . In particular, we prove that for n ⩾ 2 the Π -projective space P Π n can be constructed as the non-projected supermanifold determined by three elements ( P n , Ω P n 1 , λ ) , where P n is the ordinary complex projective space, Ω P n 1 is its cotangent sheaf and λ is a non-zero complex number, representative of the fundamental obstruction class ω ∈ H 1 ( T P n ⊗ ⋀ 2 Ω P n 1 ) ≅ C . Likewise, in the case n = 1 the Π -projective line P Π 1 is the split supermanifold determined by the pair ( P 1 , Ω P 1 1 ≅ O P 1 ( − 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. [ABSTRACT FROM AUTHOR]

For an arc on a bordered surface with marked points, we associate a holonomy matrix using a product of elements of the supergroup OSp (1 | 2) , which defines a flat OSp (1 | 2) -connection on the surface. We show that our matrix formula of an arc yields its super λ -length in Penner-Zeitlin's decorated super Teichmüller space. This generalizes the matrix formula of Fock-Goncharov and Musiker-Williams. We also prove that our matrix formulas agree with the combinatorial formulas given in the authors' previous works. As an application, we use our matrix formula in the case of an annulus to obtain new results on super Fibonacci numbers. [ABSTRACT FROM AUTHOR]

In a attempt to treat a supergravity as a tensor representation, the four-dimensional N-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. These quaternionic superspaces have 4 + k(N - k) even-quaternionic coordinates and 4N odd-quaternionic coordinates, where each coordinate is a quaternion composed by four C-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 N-extended quaternionic superspaces. The role of tensorial central charges for any N even USp(N) = Sp(N,HC) ∩ U(N,HC) is elucidated from this theoretical context.context. [ABSTRACT FROM AUTHOR]

A mathematical framework of cohomological field theories (CohFTs) is formulated in the language of bigraded manifolds. Algebraic properties of operators in CohFTs are studied. Methods of constructing CohFTs, with or without gauge symmetries, are discussed. In particular, a generalization of the Mathai-Quillen formalism is given. Examples such as topological quantum mechanics, topological sigma model, topological M-theory, and topological Yang-Mills theory can be obtained uniformly using this new formalism. [ABSTRACT FROM AUTHOR]

Smooth Z 2 n -supermanifolds have been introduced and studied recently. The corresponding sign rule is given by the ‘scalar product’ of the involved Z 2 n -degrees. It exhibits interesting changes in comparison with the sign rule using the parity of the total degree. With the new rule, nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. The classical Batchelor–Gawȩdzki theorem says that any smooth supermanifold is diffeomorphic to the ‘superization’ Π E of a vector bundle E . It is also known that this result fails in the complex analytic category. Hence, it is natural to ask whether an analogous statement goes through in the category of Z 2 n -supermanifolds with its local model made of formal power series. We give a positive answer to this question. [ABSTRACT FROM AUTHOR]

In this paper we give a brief account of the main aspects of the theory of associated and principal super bundles. As an application, we review the Borel-Weil-Bott Theorem in the super setting, and some results on projective embeddings of homogeneous spaces.

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 in the field of quaternions . 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 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 Ref. 12 where the coset 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 and ; ) in the twistor one. [ABSTRACT FROM AUTHOR]

We study a generalization of the results in [4] to the case of SU(1|1) interpreted as the supercircle S. 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. [ABSTRACT FROM AUTHOR]

Using the fact that Π -invertible sheaves can be interpreted as locally free sheaves of modules for the super skew field D , we give a new construction of the Π -projective superspace P Π , B n over affine k superschemes B , k an algebraically closed field. We characterize morphisms into P Π , B n and give a new interpretation of the composition of Π -invertible sheaves in terms of the algebra of D . [ABSTRACT FROM AUTHOR]

This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Such a theory is called a super Fermat theory. Any category of superspaces and smooth functions has an associated such theory. This includes both real and complex supermanifolds, as well as algebraic superschemes. In particular, there is a super Fermat theory of C1-superalgebras. C1-superalgebras are the appropriate notion of supercommutative algebras in the world of C1-rings, the latter being of central importance both to synthetic differential geometry and to all existing models of derived smooth manifolds. A super Fermat theory is a natural generalization of the concept of a Fermat theory introduced by E. Dubuc and A. Kock. We show that any Fermat theory admits a canonical superization, however not every super Fermat theory arises in this way. For a fixed super Fermat theory, we go on to study a special subcategory of algebras called near-point determined algebras, and derive many of their algebraic properties. [ABSTRACT FROM AUTHOR]

We study the functor of points and different local functors of points 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. [ABSTRACT FROM AUTHOR]