nLab
NQ-supermanifold

Context

Super-Geometry

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

Another term for dg-manifold .

Definition

Definition (NQP-supermanifold)

An N-[supermanifold] is a supermanifold equipped with a lift of the 2\mathbb{Z}_2-grading to a \mathbb{N}-grading through the standard homomorphism even/odd: 2even/odd : \mathbb{N} \to \mathbb{Z}_2.

A Q-[supermanifold] is a supermanifold equipped with an odd-graded vector field QQ (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[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( 01)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)