nLab
shifted tangent bundle

Contents

Idea

The shifted tangent bundle ΠTX\Pi T X of a manifold XX is an incarnation of the ordinary tangent bundle of XX as a supermanifold with the underlying manifold in even degree and the tangent vectors in odd degree.

Definition

For XX a smooth manifold the supermanifold ΠTX\Pi T X is defined to be the one specified by the fact that its superalgebra of functions is the graded-commutative exterior algebra of differential forms

C (ΠTX):=Ω (X)= C (X) Γ(T *X). C^\infty(\Pi T X) := \Omega^\bullet(X) = \wedge^\bullet_{C^\infty(X)} \Gamma(T^* X) \,.

More precisely, for each open UXU \subset X the value of the structure sheaf of ΠTX\Pi T X is Ω (U)\Omega^\bullet(U).

Alternative characterizations

As an NQ-supermanifold

In the context of supergeometry the algebra Ω (X)\Omega^\bullet(X) is regarded as a 2\mathbb{Z}_2-graded algebra, but of course this 2\mathbb{Z}_2-grading lifts to an \mathbb{N}-grading in the obvious way.

Moreover, there is a canonical odd vector field v dv_d on ΠTX\Pi T X, which as an odd derivation on the function algebra Ω (X)\Omega^\bullet(X) is just the deRham differential.

Equipped with this structure ΠTX\Pi T X is naturally an NQ-supermanifold.

As the tangent Lie algebroid

The dg-algebra (Ω (X),d dR)(\Omega^\bullet(X), d_{dR}) may also be regarded as the Chevalley-Eilenberg algebra of the tangent Lie algebroid of XX, which identifies the shifted tangent bundle in its refinement to an NQ-supermanifold with the tangent Lie algebroid of XX.

From this perspective, the fact that the vectors are regarded as being in degree one in ΠTX\Pi T X corresponds to the fact that these are the tangents to the 1-morphisms of the fundamental groupoid of XX. (Which is denoted Π(X)\Pi(X) but with the “Π\Pi” of completely different meaning than the “Π\Pi” as used here, which just indicates degree shift).

As an internal hom object

With 01\mathbb{R}^{0|1} denoting the odd line, i.e. the supermanifold with function algebra the algebra of dual numbers, one finds that

ΠTX=[ 01,X] \Pi T X = [\mathbb{R}^{0|1}, X]

is the internal hom object in the category of supermanifolds of maps from 01\mathbb{R}^{0|1} to XX. More precisely this means that the internal hom which exists in the closed monoidal structure on presheaves on the category of supermanifolds, between the presheaves represented by 01\mathbb{R}^{0|1} and XX, is itself representable and is represented by ΠTX\Pi T X.

The existence of the structure of an NQ-supermanifold is from this point of view a consequence of the fact that [ 01,X][\mathbb{R}^{0|1},X] naturally carries an action of the endomorphism object End( 01)End(\mathbb{R}^{0|1}). For more on this see NQ-supermanifold.

Revised on November 24, 2009 21:14:22 by Toby Bartels (173.60.119.197)