# Graded Geometry, $Q$-Manifolds, and Microformal Geometry

## TL;DR

This paper provides an exposition of graded and microformal geometry, focusing on $Q$-manifolds, their algebraic structures, and introduces microformal morphisms, including new results and quantum aspects related to Batalin-Vilkovisky geometry.

## Contribution

It introduces the concept of microformal geometry and morphisms, linking them to $Q$-manifolds and homotopy Poisson structures, with some novel results and quantum extensions.

## Key findings

- Microformal morphisms generalize smooth maps with nonlinear pullbacks.
- Poisson thick morphisms induce $L_{
abla}$-morphisms of homotopy Poisson brackets.
- Quantum versions involve Fourier integral operators applicable to Batalin-Vilkovisky geometry.

## Abstract

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 "thick" or "microformal" morphisms, which generalize ordinary smooth maps, but whose crucial feature is that the corresponding pullbacks of functions are nonlinear. In particular, "Poisson thick morphisms" of homotopy Poisson supermanifolds induce $L_{\infty}$-morphisms of homotopy Poisson brackets. There is a quantum version based on special type Fourier integral operators and applicable to Batalin-Vilkovisky geometry. Though the text is mainly expository, some results are new or not published previously.