The aim of this work is the construction of a "supermanifold of morphisms $X \rightarrow Y$", given two finite-dimensional supermanifolds $X$ and $Y$. More precisely, we will define an object $\underline{SC}^\infty(X,Y)$ in the category of supermanifolds proposed by Molotkov and Sachse. Initially, it is given by the set-valued functor characterised by the adjunction formula $\mathrm{Hom}(P \times X,Y) \cong \mathrm{Hom}(P,\underline{SC}^\infty(X,Y))$ where $P$ ranges over all superpoints. We determine the structure of this functor in purely geometric terms: We show that it takes values in the set of certain differential operators and establish a bijective correspondence to the set of sections in certain vector bundles associated to $X$ and $Y$. Equipping these spaces of sections with infinite-dimensional manifold structures using the convenient setting by Kriegl and Michor, we obtain at a supersmooth structure on $\underline{SC}^\infty(X,Y)$, i.e. a supermanifold of all morphisms $X \rightarrow Y$.
We recall the notion of a differential operator over a smooth map (in linear and non-linear settings) and consider its versions such as formal $\hbar$-differential operators over a map. We study constructions and examples of such operators, which include pullbacks by thick morphisms and quantization of symplectic micromorphisms.
"Thick" or "microformal" morphisms of supermanifolds generalize ordinary maps. They were discovered as a tool for homotopy algebras. Namely, the corresponding pullbacks provide $L_{\infty}$-morphisms for $S_{\infty}$ or Batalin--Vilkovisky algebras. It was clear from the start that constructions used for thick morphisms closely resemble some fundamental notions in quantum mechanics and their classical limits (such as action, Schrödinger and Hamilton--Jacobi equations, etc.) There was also a natural question about any connection of thick morphisms with spinor representation. We answer both ques…
In \cite{Covolo:2016}, \cite{Covolo:2012} and \cite{Poncin:2016}, we introduced the category of colored supermanifolds ($\mathbb{Z}_2^n$-super\-ma\-ni\-folds or just $\mathbb{Z}_2^n$-manifolds ($\mathbb{Z}_2^n=\mathbb{Z}_2\times\ldots\times\mathbb{Z}_2$ ($n$ times))), explicitly described the corresponding $\mathbb{Z}_2^n$-Berezinian and gave first insights into $\mathbb{Z}_2^n$-integration theory. The present paper contains a detailed account of parts of the $\mathbb{Z}_2^n$-differential calculus and of the $\mathbb{Z}_2^n$-variants of the trilogy of local theorems, which consists of the inve…
We give an exposition of graded and microformal geometry, and the language of $Q$-manifolds. $Q$-manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non-linear analogue of Lie algebras (in parallel with even and odd Poisson manifolds), a basis of "non-linear homological algebra", and a powerful tool for describing algebraic and geometric structures. This language goes together with that of graded manifolds, which are supermanifolds with an extra $\mathbb{Z}$-grading in the structure sheaf. "Microformal geometry" is a new notion referring to "thic…
It is a classical fact in Poisson geometry that the cotangent bundle of a Poisson manifold has the structure of a Lie algebroid. Manifestations of this structure are the Lichnerowicz differential on multivector fields (calculating Poisson cohomology) and the Koszul bracket of differential forms. "Raising indices" by the Poisson tensor maps the de Rham differential to the Lichnerowicz differential and the Koszul bracket to the Schouten bracket. In this paper, we present a homotopy analog of the above results. When an ordinary Poisson structure is replaced by a homotopy one, instead of a single …
We generalize to the homotopy case a result of K. Mackenzie and P. Xu on relation between Lie bialgebroids and Poisson geometry. For a homotopy Poisson structure on a supermanifold $M$, we show that $(TM, T^*M)$ has a canonical structure of an $L_{\infty}$-bialgebroid. (Higher Koszul brackets on forms introduced earlier by H. Khudaverdian and the author are part of one of its manifestations.) The underlying general construction is that of a "(quasi)triangular" $L_{\infty}$-bialgebroid, which is a specialization of a "(quasi)triangular" homotopy Poisson structure. We define both here.
We define and make an initial study of (even) Riemannian supermanifolds equipped with a homological vector field that is also a Killing vector field. We refer to such supermanifolds as Riemannian Q-manifolds. We show that such Q-manifolds are unimodular, i.e., come equipped with a Q-invariant Berezin volume.
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 c…
A new construction of homogeneous quaternionic manifolds and related geometric structures
设V是签名为（p，q）的伪欧几里德向量空间，p>2，W是偶Clifford代数Cl ^0（V）上的模.对任意自旋（V）-等变线性映射\wedge^2 W \to V，构造了一个齐次四元数流形（M，Q）.若斜对称向量值双线性型是非退化的，则（M，Q）被赋予一个正则伪黎曼度量g，使得（M，Q，g）是一个齐次四元数伪Kähler流形.该结构被证明在超流形范畴中有一个自然镜像。事实上，对于任意的自旋（V）-等变线性映射n：Sym^2 W \to V，构造了一个齐次四元数超流形（M，Q），而且，如果对称向量值双线性型n是非退化的，则构造了一个齐次四元数伪Kähler超流形（M，Q，g）。
The purpose of this work is to study Lie superalgebroid structures on the space of superdifferential $1$-forms over the supermanifolds whose superfunctions are the differential forms on its underlying manifold. These superalgbroids are constructed from graded Poisson structures defined on the latter superalgebra.
In this paper we construct two groupoids from morphisms of groupoids, with one from a categorical viewpoint and the other from a geometric viewpoint. We show that for each pair of groupoids, the two kinds of groupoids of morphisms are equivalent. Then we study the automorphism groupoid of a groupoid.
