superalgebra

and

supergeometry

∞-Lie theory

# Contents

## Idea

Another term for dg-manifold .

## Definition

###### Definition (NQP-supermanifold)

An N-[supermanifold] is a supermanifold equipped with a lift of the $\mathbb{Z}_2$-grading to a $\mathbb{N}$-grading through the standard homomorphism $even/odd : \mathbb{N} \to \mathbb{Z}_2$.

A Q-[supermanifold] is a supermanifold equipped with an odd-graded vector field $Q$ (i.e. an odd-graded derivation of the algebra of functions) which is homological, i.e. the super Lie bracket with itself vanishes: $[Q,Q] = 0$.

A P-[supermanifold] is a supermanifold equipped with a graded symplectic structure.

## Remarks

• It is an old observation by Maxim Kontsevich, amplified by Pavol Severa (ref…) that NQ-supermanifolds are precisely those supermanifolds which are equipped with an action of $End(\mathbb{R}^{0|1})$, the endomorphism monoid of the odd line.

• NQ-supermanifolds are an equivalent way of thinking of ∞-Lie algebroids. See the list of references there.

## References

The “Q-manifold”-terminology is due to

Revised on October 17, 2011 13:34:34 by Urs Schreiber (82.113.99.57)