superalgebra and (synthetic ) supergeometry
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.
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
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.
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.
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).
With $\mathbb{R}^{0|1}$ denoting the odd line, i.e. the supermanifold with function algebra the algebra of dual numbers, one finds that
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.
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]
Last revised on September 6, 2024 at 15:20:15. See the history of this page for a list of all contributions to it.