I have been thinking a little bit recently about \(L_{\infty}\)-algebroids. So what is such an object?

Let us work with super-stuff from the start. I will be lax about signs etc, so this should not course any real confusion. First we need a little background.

Heuristically, a supermanifold is a “manifold” in which the coordinates have an underlying \(\mathbb{Z}_{2}\)-grading. Morphisms between charts are smooth and respect this grading. In more physical language, we have bosonic coordinates and even coordinates. The bosonic coordinates commute as where the fermionic coordinates anticommute. I will refer to bosonic coordinates as even parity and fermionic as odd parity. To set this up properly one needs the theory of locally ringed spaces. However, we will not need this here.

A graded manifold is a supermanifold, in which the coordinates are assigned an additional weight in \(\mathbb{Z}^{n}\) and the changes of coordinates respect the parity as well as the additional weight. In general the parity and weight are completely independent.

A Q-manifold is a supermanifold (or a graded manifold ) that comes equipped with a homological vector field, usually denoted by Q. That is we have an odd parity vector field that “self-commutes” under the Lie bracket,

\([Q,Q] = 0\).

Note that as the homological vector field is odd, this is a non-trivial condition. Sometimes, if the supermanifold is a graded manifold then conditions on the weight of Q can be imposed.

Now we can describe \(L_{\infty}\)-algebroids. The best way to describe them is as follows:

Definition:
A vector bundle \(E \rightarrow M\) is said to have an \(L_{\infty}\)-algebroid structure if there exists a homological vector field, denote as \(Q\) on the total space of \(\Pi E\), thought of as a graded manifold.

That is the pair \((\Pi E, Q)\) is a Q-manifold. We call this pair an \(L_{\infty}\)-algebroid.

The weight, in this case just in \(\mathbb{Z}\) is assigned by equipping the base coordinates of \(E \) with weight zero and the fibre coordinates with weight one (or some other integer). The “\(\Pi \)” is the parity reversion functor. It shifts the parity of the fibre coordinates. So, a coordinate that is originally even\odd get replaced by a coordinate that is odd\even. It does nothing to the weight. Note that this shift is fundamental here and not just for convenience.

It is very easy to see that the original vector bundle \(E \rightarrow M\) is equivalent to the graded manifold \(\Pi E\). Stronger than this, the equivalence is functorial. That is we have equivalent categories.

Further note that there is no condition on the weight of the homological vector bundle in this definition, nor is there any “compatibility condition” with the vector bundle (or graded) structure.

Definition:
An \(L_{\infty}\)-algebroid is said to be strict if the restriction of Q to the “base manifold” \(M \subset \Pi E \) is a genuine homological vector field on \(M \).

This does not sound very invariant at first, but simply put restriction of Q to the weight zero “part” of \(\Pi E\) should still be homological.

For those that know Lie algebroids and \(L_{\infty}\)-algebras, the question is why call them \(L_{\infty}\)-algebroids? An \(L_{\infty}\)-algebroid is to an \(L_{\infty}\)-algebra what a Lie algebroid is to a Lie algebra.

It also turns out that some of the main constructions relating to Lie algebroids carry over to \(L_{\infty}\)-algeboids, see [1]. (I may say more another time.) This may also be of use for \(L_{\infty}\)-algebras. I am currently also pondering this.

So maybe I should end for now on a little motivation as to why such things are interesting. First, if we insist on the homological vector field being of weight one we recover Lie algebroids. If we insist on the vector bundle being over a point we recover \(L_{\infty}\)-algebras. (I should post on these later) Also, very similar things appear in quantum field theory via the BV and BFV formalisms (again I should post on these another time). However, at the moment it is not exactly clear how \(L_{\infty}\)-algebroids fit in here. One “barrier” is that Q is inhomogenous in weight, in the BV and BFV formulations the homological vector field is homogeneous in “ghost number”. It would also be interesting to see if these structures can be used in the BV-AKSZ formalism.

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References

[1] Andrew James Bruce, From \(L_{\infty}\)-algebroids to higher Schouten\Poisson structures. Submitted for publication, available as arXiv:1007.1389v2 [math-ph]