Generalized geometry finds many applications in the mathematical description of some aspects of string theory. In a nutshell, it explores various structures on a generalized tangent bundle associated to a given manifold. In particular, several integrability conditions can be formulated in terms of a canonical Dorfman bracket, an example of Courant algebroid. On the other hand, smooth manifolds can be generalized to involve functions of $\mathbb{Z}$-graded variables which do not necessarily commute. This leads to a mathematical theory of graded manifolds. It is only natural to combine the two theories by exploring the structures on a generalized tangent bundle associated to a given graded manifold.
After recalling elementary graded geometry, graded Courant algebroids on graded vector bundles are introduced. We show that there is a canonical bracket on a generalized tangent bundle associated to a graded manifold. Graded analogues of Dirac structures and generalized complex structures are explored. We introduce differential graded Courant algebroids which can be viewed as a generalization of Q-manifolds. A definition and examples of graded Lie bialgebroids are given.
We define the notion of the Ricci tensor for NQ symplectic manifolds of degree 2 and show that it corresponds to the standard generalized Ricci tensor on Courant algebroids. We use an appropriate notion of connections compatible with the generalized metric on the graded manifold.
A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded variables which can either commute or anticommute, according to their degree. To obtain a consistent global description of graded manifolds, one resorts to sheaves of graded commutative associative algebras on second countable Hausdorff topological spaces, locally isomorphic to a suitable "model space". This paper aims to build robust mathematical foundations of…
In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of classical differential geometric constructions in the context can be conveniently described using the language of Q-structures, and thus Q-structure preserving integrators are potentially of great use in mechanics. We give some hints how the latter can be constructed, and formulate some open problems. Since the text is intended both to mathematics and mechan…
In recent years, a close connection between supergravity, string effective actions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded geometry, introducing an approach based on deformations of graded Poisson structures and derive the corresponding gravity actions. We consider in particular natural deformations of the $2$-graded symplectic manifold $T^{*}[2]T[1]M$ that are based on a metric $g$, a closed Neveu-Schwarz $3$-form $H$ (locally expressed in terms of a Kalb-Ramond 2-for…
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…
We give a generalization of the theory of $\mathbb{Z}_2$-graded manifolds to a theory of $\mathcal{I}$-graded manifolds, where $\mathcal{I}$ is a commutative semi-ring with some additional properties. We prove Batchelor's theorem in this generalized setting. To our knowledge, such a proof is still missing except for some special cases.
We show how the theory of $\mathbb{Z}_2^n$ -manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed.
We extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.
Instituto de Matemática Pura e Aplicada（巴西国家数学研究所）；Institut für Mathematik, Universität Zürich-Irchel（苏黎世大学）；Department of Mathematical Sciences, Smith College（史密斯学院）；KU Leuven, Department of Mathematics（荷语鲁汶大学）
We generalize the Donagi and Witten construction of a first obstruction class for splitting of a supermanifold via differential operators using the theory of $n$-fold vector bundles and graded manifolds. Applying the generalized Donagi--Witten construction we obtain a family of embeddings of the category of supermanifolds into the category of $n$-fold vector bundles and into the category of graded manifolds. This leads to a realization of any non-split supermanifold in terms of a collection of vector bundles and some morphism between them. Further we study the images of these embeddings into t…
On one hand, together with Pelle Steffens, we recently characterized the infinity category of derived manifolds up to equivalence by a universal property. On the other hand, it is shown in recent work of Behrend-Liao-Xu that the category of differential graded manifolds admits a homotopy theory. In this paper, we prove that the associated infinity category of dg-manifolds is equivalent to that of derived manifolds. This allows the well developed theory of dg-geometry to live in symbiosis with infinity categorical approach to derived differential geometry. As a consequence, we prove that a dg-C…
The complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry aspect of a superspace(-time) $\widehat{X}$ in Sec.\,1 of D(14.1) (arXiv:1808.05011 [math.DG]) together with the Spin-Statistics Theorem in Quantum Field Theory, which requires fermionic components of a superfield be anticommuting, lead us to the notion of towered superspace(-time) $\widehat{X}^{\widehat{\boxplus}}$ and the built-in purely even physics sector $X^{\mbox{physics}}$ from $\widehat{X}^{\widehat{\boxplus}}$. We use this to reproduce the $d=3+1$, $N=1$ Wess-Zumino model and the $d=3+1$, $N=1$ supersymmetric $U(1)$ ga…
Forty-six years after the birth of supersymmetry in 1973 from works of Julius Wess and Bruno Zumino, the standard quantum-field-theorists and particle physicists' language of `superspaces', `supersymmetry', and `supersymmetric action functionals in superspace formulation' as given in Chapters IV, V, VI, VII, XXII of the classic on supersymmetry and supergravity: Julius Wess & Jonathan Bagger: Supersymmetry and Supergravity (2nd ed.), is finally polished, with only minimal mathematical patches added for consistency and accuracy in dealing with nilpotent objects from the Grassmann algebra involv…
