nLab shifted tangent bundle

Idea

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

Definition

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

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

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

For more details see at geometry of physics – supergeometry.

Alternative characterizations

As an NQ-supermanifold

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

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

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

As the tangent Lie algebroid

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

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

As an internal hom object

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

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

is the internal hom object in the category of supermanifolds of maps from $\mathbb{R}^{0|1}$ to $X$. 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 $\mathbb{R}^{0|1}$ and $X$, is itself representable and is represented by $\Pi T X$.

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

References

• Denis Kochan, Pavol Ševera, §3.1 in: Differential gorms, differential worms, Mathematical Physics, World Scientific (2005) 128-130 [arXiv:math/0307303, doi:10.1142/9789812701862_0034]

• Henning Hohnhold, Matthias Kreck, Stephan Stolz, Peter Teichner: Ex. 2.1 & Prop. 3.1 in: Differential forms and 0-dimensional supersymmetric field theories, Quantum Topology 2 1 (2011) 1–41 [doi:10.4171/QT/12, pdf]

• David Carchedi, Dmitry Roytenberg: Ex. 5.3 in: Homological Algebra for Superalgebras of Differentiable Functions [arXiv:1212.3745]

• Urs Schreiber: Super mapping spaces , section of: geometry of physics – supergeometry (2016)

• Simone Noja: §3 in: On the Geometry of Forms on Supermanifolds [arXiv:2111.12841]