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We develop and study the notion of a spherical supervariety, which is a generalization of the classical notion of a spherical variety in algebraic geometry. Spherical supervarieties are supervarieties admitting an action of a quasi-reductive group with an open orbit of a hyperborel subgroup. Three characterizations of spherical supervarieties are given: one which generalizes the Vinberg-Kimelfeld characterization of affine spherical varieties, another that extends the ideas of the affine case to the quasi-projective case, and finally one in terms of invariant rational functions which applies to any supervariety. Our characterization of affine spherical supervarieties leads to (non-constructive) existence theorems for finite-dimensional highest weight representations admitting certain coinvariants under spherical quasireductive subgroups.Several interesting examples of spherical supervarieties are given. We present a classification of indecomposable spherical representations (for certain supergroups) and for each the description of its algebra of functions. Adjoint orbits of odd self-commuting elements are shown to be spherical in many cases, in particular for basic simple Lie superalgebras. We study group-graded supergroups and their spherical homogeneous supervarieties, showing in particular that the algebra of functions on an affine homogeneous supervariety is almost never completely reducible for such supergroups.Finally we study the case of symmetric supervarieties and show that, despite their not always being spherical (in contrast to the classical case), we may under some circumstances guarantee the existence of an Iwasawa decomposition, which implies sphericity. The fixed points of automorphisms of generalized root systems coming from supersymmetric pairs are studied along the way. We use the Iwasawa decomposition to gain partial understanding of the structure of the space of functions as a representation. Finally, the case of a supergroup as a symmetric supe

We develop and study the notion of a spherical supervariety, which is a generalization of the classical notion of a spherical variety in algebraic geometry. Spherical supervarieties are supervarieties admitting an action of a quasi-reductive group with an open orbit of a hyperborel subgroup. Three characterizations of spherical supervarieties are given: one which generalizes the Vinberg-Kimelfeld characterization of affine spherical varieties, another that extends the ideas of the affine case to the quasi-projective case, and finally one in terms of invariant rational functions which applies to any supervariety. Our characterization of affine spherical supervarieties leads to (non-constructive) existence theorems for finite-dimensional highest weight representations admitting certain coinvariants under spherical quasireductive subgroups.Several interesting examples of spherical supervarieties are given. We present a classification of indecomposable spherical representations (for certain supergroups) and for each the description of its algebra of functions. Adjoint orbits of odd self-commuting elements are shown to be spherical in many cases, in particular for basic simple Lie superalgebras. We study group-graded supergroups and their spherical homogeneous supervarieties, showing in particular that the algebra of functions on an affine homogeneous supervariety is almost never completely reducible for such supergroups.Finally we study the case of symmetric supervarieties and show that, despite their not always being spherical (in contrast to the classical case), we may under some circumstances guarantee the existence of an Iwasawa decomposition, which implies sphericity. The fixed points of automorphisms of generalized root systems coming from supersymmetric pairs are studied along the way. We use the Iwasawa decomposition to gain partial understanding of the structure of the space of functions as a representation. Finally, the case of a supergroup as a symmetric supe

These notes are based on a series of lectures given by the first author at the school of `Poisson 2010', held at IMPA, Rio de Janeiro. They contain an exposition of the theory of super- and graded manifolds, cohomological vector fields, graded symplectic structures, reduction and the AKSZ-formalism.

These notes are based on a series of lectures given by the first author at the school of `Poisson 2010', held at IMPA, Rio de Janeiro. They contain an exposition of the theory of super- and graded manifolds, cohomological vector fields, graded symplectic structures, reduction and the AKSZ-formalism.

We generalize to the supersymmetric case the representation of the KP hierarchy as a set of conservation laws for the generating series of the conserved densities. We show that the hierarchy so obtained is isomorphic to the JSKP of Mulase and Rabin. We identify its "bosonic content" with the so-called Darboux-KP hierarchy, which geometrically encompasses the theory of Darboux-Bäcklund transformations, and is an extension both of the KP theory and of the modified KP theory. Finally, we show how the hierarchy can be linearized and how the supersymmetric counterpart of a wide class of rational solutions can be quite explicitly worked out. © 2000 Elsevier Science B.V.

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.

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.

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.