nLab NQ-supermanifold

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 (Kontsevich 97), amplified by Pavol Ševera (Ševera 07) 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.

• NQP-supermanifolds are hence symplectic Lie n-algebroids

Related concepts

• dg-manifold, derived smooth manifold

• For the analogous phenomenon relating super Lie algebras to dg-Lie algebras see there.

References

The “Q-manifold”-terminology is due to

• M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997 (arXiv:hep-th/9502010)

• Maxim Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216, [arXiv:q-alg/9709040, doi:10.1023/B:MATH.0000027508.00421.bf]

• Pavol Ševera, $L_\infty$‐algebras as first approximations, Geometrical Methods in Physics 956 (2007), 199–204 [doi:10.1063/1.2820968]