In an earlier post, here, I showed that the homological condition on an odd vector field \(Q \in Vect(M)\), on a supermanifold \(M\), that is \(2Q^{2}= [Q,Q]=0\), is precisely the condition that \(\gamma^{*}x^{A} = x^{A}(\tau)\), where \(\gamma \in \underline{Map}(\mathbb{R}^{0|1}, M)\), be an integral curve of \(Q\).

A very natural question to answer is what is the geometric interpretation of a pair of mutually commuting homological vector fields?

Suppose we have two odd vector fields \(Q_{1}\) and \(Q_{2}\) on a supermanifold \(M\). Then we insist that any linear combination of the two also be a homological vector field, say \(Q = a Q_{1} + b Q_{2}\), where \(a,b \in \mathbb{R}\). It is easy to verify that this forces the conditions

\([Q_{1}, Q_{1}]= 0 \), $latex[Q_{2}, Q_{2}]= 0 $ and \([Q_{1}, Q_{2}]= 0 \).

That is, both our original odd vector fields must be homological and they mutually commute. Such a pair of homological vector fields are said to be compatible. So far this is all algebraic.

Applications of pairs, and indeed larger sets of compatible vector fields, include the description of n-fold Lie algebroids [1,3] and Q-algebroids [2].

The geometric interpretation
Based on the earlier discussion about integrability of odd flows, a pair of compatible homological vector fields should have something to do with an odd flow. We would like to interpret the compatibility of a pair of homological vector fields as the integrability of the flow of \(\tau = \tau_{1} + \tau_{2}\). Indeed this is the case;

Consider \(\gamma^{*}_{\tau_{1} + \tau_{2}}(x^{A}) = x^{A}(\tau_{1} + \tau_{2}) = x^{A}(\tau_{1}, \tau_{2})\), remembering that we define the flow via a Taylor expansion in the “odd time”. Expanding this out we get

\( x^{A}(\tau_{1}, \tau_{2}) = x^{A} + \tau_{1}\psi_{1}^{A} + \tau_{2}\psi_{2}^{A} + \tau_{1} \tau_{2}X^{A}\).

Now we examine the flow equations with respect to each “odd time”. We do not assume any conditions on the odd vector fields \(Q_{1}\) and \(Q_{2}\) at this stage.

\(\frac{\partial x^{A}}{\partial \tau_{1}} = \psi_{1}^{A} + \tau_{2}X^{A} = Q_{1}^{A}(x(\tau_{1}, \tau_{2}))\)
\(= Q_{1}^{A}(x) + \tau_{1}\psi^{B}_{1} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}} + \tau_{2}\psi^{B}_{2} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}} + \tau_{1}\tau_{2}X^{B} \frac{\partial Q^{A}_{1}(x)}{\partial x^{B}}\),

and
\(\frac{\partial x^{A}}{\partial \tau_{2}} = \psi_{2}^{A} {-} \tau_{1}X^{A} = Q_{2}^{A}(x(\tau_{1}, \tau_{2}))\)
\(= Q_{2}^{A}(x) + \tau_{1}\psi^{B}_{1} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}} + \tau_{2}\psi^{B}_{2} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}} + \tau_{1}\tau_{2}X^{B} \frac{\partial Q^{A}_{2}(x)}{\partial x^{B}}\).

Then equating coefficients in order of \(\tau_{1}\) and \(\tau_{2}\) we arrive at three types of equations

i) \(\psi_{1}^{A} = Q_{1}^{A}\), \(\psi^{B}_{1} \frac{\partial Q_{1}^{A}}{\partial x^{B}}=0\) and \(\psi_{2}^{A} = Q_{2}^{A}\), \(\psi^{B}_{2} \frac{\partial Q_{2}^{A}}{\partial x^{B}}=0\).

ii) \(X^{A} = \psi^{B}_{2} \frac{\partial Q_{1}^{A}}{\partial x^{B}}\) and \(X^{A} = {-}\psi^{B}_{1} \frac{\partial Q_{2}^{A}}{\partial x^{B}}\).

iii) \(X^{B} \frac{\partial Q_{1}^{A}}{\partial x^{B}} =0\) and \(X^{B} \frac{\partial Q_{2}^{A}}{\partial x^{B}} =0\).

It is now easy to see that;

i) implies that \([Q_{1}, Q_{1}] =0 \) and \([Q_{2}, Q_{2}] =0 \) meaning we have a pair of homological vector fields.

ii) implies that \([Q_{1}, Q_{2}]=0\), that is they are mutually commuting, or in other words compatible.

iii) is rather redundant and follows from the first two conditions.

Thus our geometric interpretation was right.

References
[1] Janusz Grabowski and Mikolaj Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59(2009), 1285-1305.

[2]Rajan Amit Mehta, Q-algebroids and their cohomology, Journal of Symplectic Geometry 7 (2009), no. 3, 263-293.

[3] Theodore Th. Voronov, Q-Manifolds and Mackenzie Theory, Commun. Math. Phys. 2012; 315:279-310.

On a supermanifold one has not just even vector fields but also odd vector fields. Importantly, the Lie bracket of an odd vector field with itself does not automatically vanish.

This is in stark contrast to the even vector fields on a supermanifold and indeed all vector fields on a classical manifold. Odd vector fields that self-commute under Lie bracket are known as homological vector fields and a supermanifold equipped with such a vector field is known as a Q-manifold.

In the literature one is often interested in homological vector fields from an algebraic perspective. Indeed, the nomenclature “homological” refers to the fact that on a Q-manifold one has a cochain complex on the algebra of functions on the supermanifold. You should have in mind the de Rham differential and the differential forms on a manifold in mind here.

In fact, if we think of differential forms as functions on the supermanifold \(\Pi TM\), then the pair \((\Pi TM, d)\) is a Q-manifold.

But can we understand the geometric meaning of a homological vector field?

Odd curves and maps between supermanifolds
Consider a map \(\gamma : \mathbb{R}^{0|1} \longrightarrow M\), for any supermanifold \(M\). We need to be a little careful here as we take $latex\gamma \in \underline{Map}(\mathbb{R}^{0|1}, M)$, that is we include odd maps here. Informally, we will use odd parameters at our free disposal. More formally, we need the inner homs, which requires the use of the functor of points, but we will skip all that.

Let us employ local coordinates $latex(x^{A})$ on \(M\) and \(\tau \) on \(\mathbb{R}^{0|1}\). Then

\(\gamma^{*}(x^{A}) = x^{A}(\tau) = x^{A} + \tau \; \; v^{A}\),

where $latex\widetilde{v^{A}} = \widetilde{A}+1$. This is why we need to include odd variables in our description. Note that as \(\tau\) is odd, functions of this variable can be at most linear.

Aside One can now show that \(\Pi TM = \underline{Map}(\mathbb{R}^{0|1}, M)\). Basically we have local coordinates \((x^{A}, v^{A})\) noting the shift in parity of the second factor. One can show we have the right transformation rules here directly.

Odd Flows
Now consider the flow on odd vector field, that is the differential equation

\(\frac{d x^{A}(\tau)}{d \tau} = X^{A}(x(\tau)) \),

where in local coordinates \(X = X^{A}(x)\frac{\partial}{\partial x^{A}}\).

From our previous considerations the flow equation becomes

\(v^{A} = X^{A}(x) + \tau v^{B} \frac{\partial X^{A}}{\partial x^{B}}\).

Thus equating the coefficients in order of \(\tau\) shows that

\(X^{A}(x) = v^{A}\) and \(v^{B} \frac{\partial X^{A}}{\partial x^{B}}=0\).

Then we conclude that

\(X^{B} \frac{\partial X^{A}}{\partial x^{B}}=0\), which implies that \([X,X]=0\) and thus we have a homological vector field.

Conclusion
The homological condition is the necessary and sufficient condition for the integrability of an odd vector field. Note that in the classical case there are no integrability conditions on vector fields.