# 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} }

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…

## 9 Citations

Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

- MathematicsCommunications in Mathematical Physics
- 2022

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

- MathematicsJournal of Geometry and Physics
- 2020

L-infinity bialgebroids and homotopy Poisson structures on supermanifolds

- Mathematics
- 2019

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

- Mathematics
- 2022

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

- MathematicsFortschritte der Physik
- 2019

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

- Mathematics
- 2022

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

- Mathematics
- 2020

Higher Koszul Brackets on the Cotangent Complex

- MathematicsInternational Mathematics Research Notices
- 2022

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 ,…

## References

SHOWING 1-10 OF 106 REFERENCES

The "nonlinear pullback" of functions and a formal category extending the category of supermanifolds

- Mathematics
- 2014

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

- Mathematics
- 1997

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

- Mathematics
- 2012

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…

Characteristic classes associated to Q-bundles

- Mathematics
- 2007

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

- Mathematics
- 2012

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

- Mathematics
- 2002

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

- Mathematics
- 2012

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

- Mathematics
- 2016

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

- Mathematics
- 2018

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…