Graded Geometry, Q‐Manifolds, and Microformal Geometry

@article{Voronov2019GradedGQ,
  title={Graded Geometry, Q‐Manifolds, and Microformal Geometry},
  author={Theodore Th. Voronov},
  journal={Fortschritte der Physik},
  year={2019},
  volume={67}
}
  • T. Voronov
  • Published 7 March 2019
  • Mathematics
  • Fortschritte der Physik
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 Z… 
View on Wiley
arxiv.org
9 Citations
Background Citations
2

9 Citations

Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras
This paper is devoted to the study of the relation between ‘formal exponential maps,’ the Atiyah class, and Kapranov L∞[1] algebras associated with dg manifolds in the C ∞ context. We prove that, for…
Thick morphisms of supermanifolds, quantum mechanics, and spinor representation
L-infinity bialgebroids and homotopy Poisson structures on supermanifolds
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…
Almost Commutative Manifolds and Their Modular Classes
An almost commutative algebra, or a ρ-commutative algebra, is an algebra which is graded by an abelian group and whose commutativity is controlled by a function called a commutation factor. The same…
Higher Structures in M‐Theory
The key open problem of string theory remains its non‐perturbative completion to M‐theory. A decisive hint to its inner workings comes from numerous appearances of higher structures in the limits of…
The BRST Double Complex for the Coupling of Gravity to Gauge Theories
We consider (effective) Quantum General Relativity coupled to the Standard Model (QGR-SM) and study the corresponding BRST double complex. This double complex is generated by infinitesimal…
On differential operators over a map, thick morphisms of supermanifolds, and symplectic micromorphisms
Higher Koszul Brackets on the Cotangent Complex
Let $A=\boldsymbol{k}[x_1,x_2,\dots ,x_n]/I$ be a commutative algebra where $\boldsymbol{k}$ is a field, $\operatorname{char}(\boldsymbol{k})=0$, and $I\subseteq S:=\boldsymbol{k}[x_1,x_2,\dots ,…
Higher spin gravities and presymplectic AKSZ models

References

SHOWING 1-10 OF 106 REFERENCES
The "nonlinear pullback" of functions and a formal category extending the category of supermanifolds
We introduce mappings between function spaces on smooth (super)manifolds, which are generally nonlinear and which generalize the pullbacks with respect to smooth maps. The construction uses canonical…
The Geometry of the Master Equation and Topological Quantum Field Theory
In Batalin–Vilkovisky formalism, a classical mechanical system is specified by means of a solution to the classical master equation. Geometrically, such a solution can be considered as a QP-manifold,…
Homological Algebra for Superalgebras of Differentiable Functions
This is the second 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…
“Nonlinear pullbacks” of functions and L∞-morphisms for homotopy Poisson structures
Characteristic classes associated to Q-bundles
A Q-manifold is a graded manifold endowed with a vector field of degree 1 squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each…
Deformation theory and rational homotopy type
We regard the classification of rational homotopy types as a problem in algebraic deformation theory: any space with given cohomology is a perturbation, or deformation, of the "formal" space with…
On the structure of graded symplectic supermanifolds and Courant algebroids
This paper is devoted to a study of geometric structures expressible in terms of graded symplectic supermanifolds. We extend the classical BRST formalism to arbitrary pseudo-Euclidean vector bundles…
On Theories of Superalgebras of Differentiable Functions
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…
Thick morphisms of supermanifolds and oscillatory integral operators
We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of 'quantum thick morphisms' defined here as particular…
Thick morphisms, higher Koszul brackets, and $L_{\infty}$-algebroids
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…
...
...