Appendix: Signs: de Rham and integral forms complexes

P. Deligne

1. Some standard conventions for ordinary manifolds don’t sit well with the principle of treating p-forms as objects of parity p. Here are troublesome conventions:

(a) for a p-form α_{1}∧ … ∧α_{p}, and vector fields X_{1}, …, X_{p},

α_{1} ∧ … ∧ α_{p}(X_{1}, …, X_{p}) = det(α_{i}(X_{j})),

with typical term α_{i}(X_{1}) … α_{p}(X_{p}), despite the fact that X_{i} passed over α_{j} for i < j.

(b) for f a function and X a vector field,

Xf = df(X) = i_{X}df = X⊥df.

In the middle equality, df and X are permuted without sign consequence.

(c) In ℝ^{n}, the density corresponding to the n-form dx_{1}∧ … ∧dx_{n}, when ℝ^{n} is oriented by (e_{1}, …, e_{n}), is positive.

The bigraded points of view, by keeping separate the “cohomological” grading, used above, and the parity, allows to keep those formuli while treating parity consistently.

2. Let M be a supermanifold. The vector fields on M are the derivations of the structural sheaf 𝒪:

D(fg) = Df.g + (−1)^{p(D)p(f)}fDg

The sheaf of 1-forms is the dual, the duality being defined by

T ⊗ Ω^{1} → 𝒪: D ⊗ α ⟼ i_{D}α.

One defines

α(D) = (−1)^{p(D)p(α)}i_{D}α.

To the extent possible, it is best to consider only vector fields and forms of even parity, considering instead of an odd D (resp. α) the product εD (resp. εα) for ε an auxiliary odd constant. If we work over a basis B, as we should, odd constant means odd function on B.

The bilinearity of i_{D}α and of α(D) is as dictated by the sign rule: with (uD)(f) = u(Df), we have

i_{fD}α = fi_{D}α ,  i_{D}fα = (−1)^{p(D)p(f)}fi_{D}α
(fα)(D) = f(α(D)) ,  α(fD) = (−1)^{p(α)p(f)}fα(D).

The de Rham complex Ω^{∗} is ∧Ω^{1}, with Ω^{p} = ∧^{p} Ω^{1} in cohomological degree p, and with a differential d characterized by the following identities

(1) d is an even derivation of cohomological degree one;

(2) from 𝒪 to Ω^{1}, d is defined by i_{D}df = Df;

(3) d^{2} = 0.

One may prefer to say that Ω^{∗} is the symmetric algebra in the bigraded sense on Ω^{1}, put in cohomological degree 1. The defining formula (2) can be rewritten df(D) = (−1)^{p(f)p(D)}Df.

3. An even derivation D generate (at least locally) a 1-parameter group of automorphisms exp(tD), with

(3.1)  Df = ∂_{t}(exp(tD)^{∗}f) at t = 0.

By transport of structures, this group acts on all kinds of tensor fields, and the Lie derivative ℒ_{D} is defined by the right side of (3.1). If D is odd, one defines ℒ_{D} by

(3.2)  εℒ_{D} = ℒ_{εD}

for ε an odd constant.

One should beware that f ↦ exp(tD)^{∗}f is the inverse of the action of exp(tD) on functions by transport of structures. Because of this,if derivations are viewed as forming the Lie algebra of a group of diffeomorphisms, with D ↦ infinitesimal diffeomorphism exp(εD), i.e. D generating the one-parameter subgroup exp(tD), the bracket in this Lie algebra is the opposite of the bracket of vector fields.

Instead of considering exp(tD) for t ∈ ℝ, one can work mod t^{2}: exp(tD) is then the family of automorphisms parametrized by the scheme with coordinate ring ℝ[t]/(t^{2}) defined by

(3.3)  exp(tD)^{∗}f := f + tDf  (for t^{2} = 0)

and the Lie derivative of any tensor g is given by

(3.4)  exp(tD)^{∗}g = g + tℒ_{D}g  (for t^{2} = 0)

For D an odd derivation, the same formuli hold when t is taken to be odd.

A vector field D defines a derivation i_{D} of Ω^{∗}, of cohomological degree −1 and parity that of D, characterized as extending the already defined i_{D} on Ω^{1}. Cartan’s formula for the Lie derivative acting on Ω^{∗} is

(3.5)  ℒ_{D} = i_{D}d + di_{D}.

As d is of parity 0 and i_{D} and d are of odd cohomological degree, the second member is the bracket [i_{D}, d] in the bigraded sense.

4. Let V be a vector space of dimension (0, 1), with generator D. The free Lie aglebra generated by V, i.e. by D, has D, D^{2} = \frac{1}{2}[D, D] as basis. The corresponding Lie group has as points over a base B the exp(tD^{2} + θD), t and θ being, respectively, even and odd functions on B, with the group law given by the Campbell-Hausdorff formula:

(4.1)  (t_{1}, θ_{1}) ∗ (t_{2}, θ_{2}) = (t_{1} + t_{2} − θ_{1}θ_{2}, θ_{1} + θ_{2}).

The minus sign comes from \frac{1}{2}[θ_{1}D, θ_{2}D] = −\frac{1}{2}θ_{1}θ_{2}[DD] = −θ_{1}θ_{2}D^{2}.

An odd vector field on a supermanifold M generates an action of that group. More precisely, if D^{2} is the vector field defined by DDf = D^{2}f, i.e. D^{2} = \frac{1}{2}[D, D], the automorphisms exp(tD^{2} + θD) (at least locally defined) form a group, with

(4.2)  exp(t_{1}D^{2} + θ_{1}D) exp(t_{2}D^{2} + θ_{2}D) = exp((t_{1} + t_{2} + θ_{1}θ_{2})D^{2} + (θ_{1} + θ_{2})D).

The reason for the change of sign from (4.1) is explained in 3.

5. A coordinate system x^{1}, …, x^{n} (first p even, last n−p odd) defines derivations ∂_{i}, with ∂_{i}x^{j} = δ_{i}^{j}. Our conventions so far force the unusual looking

(5.1)  df = ∑ dx^{i}∂_{i}f.

6. The bundle of densities is defined by the property that its sections are in duality with the space of functions with compact support. The duality pairing is written ∫ μf. Accordingly, the 𝒪-module structure is given by

(6.1)  ∫ aμ.f = (−1)^{p(a)p(μ)} ∫ μ.af

According to the sign rule,

(6.2)  ∫ fμ := (−1)^{p(f)p(μ)} ∫ μf,

and one also has ∫ f(aμ) = ∫ (fa)μ.

By “density”, we mean here C^{∞}-density: if, in a local coordinate system (x_{1}, …, x_{p}, θ_{1}, …, θ_{q}), f is expanded as ∑ θ^{I}f_{I}, we consider only the linear forms f ↦ ∫ μf which are linear combinations of the ∫ f_{I}(n_{1}, …, n_{p})φ(x_{1}, …, x_{p})dx_{1} … dx_{p} for C^{∞} functions φ.

A local coordinate system x_{1}, …, x_{p}, θ_{1}, …, θ_{q} defines a density [x_{1}, …, x_{p}, θ_{1}, …, θ_{q}] on its domain of definition: for f_{[1,q]} the (post) coefficient of θ_{1} … θ_{q}, as above, the integral

(6.3)  ∫ f_{[1,q]}(x_{1}, …, x_{p})dx_{1} … dx_{p}.

It is the integral, for the density dx_{1} … dx_{p}, of the restriction to θ_{1} = … = θ_{q} = 0 of ∂_{θ_{q}} … ∂_{θ_{1}}f. It is of parity (−1)^{q} (and cohomological degree 0). It is a basis of densities, which is hence of dimension (1, 0) for q even and (0, 1) for q odd.

Warning. Densities μ_{1} on M_{1} and μ_{2} on M_{2} give a density μ_{1} ⊠ μ_{2} on M_{1} × M_{2} by

∫_{M_{1}×M_{2}} μ_{1} ⊠ μ_{2}f_{1} ⊠ f_{2} = (−1)^{p(f_{1})p(μ_{2})} ∫ μ_{1}f_{1} ∫ μ_{2}f_{2}.

One has, in local coordinates,

\begin{matrix}[x_{1}, …, & x_{p(1)}, θ_{1}, …, θ_{q(1)}] ⊠ [y_{1}, …, y_{p(2)}, η_{1}, …, η_{q(2)}] = \\ (−1)^{q(2)q(1)}[x_{1}, …, x_{p(1)}, y(1), …, y_{p(2)}, θ_{1}, …, θ_{q(1)}, η_{1}, …, η_{q(2)}]. \end{matrix}

The densities given by two coordinate systems are related by the Berezinian of a jacobian matrix, and a sign: + if the orientations given by x_{1}, …, x_{p} and y_{1}, …, y_{p} on the reduced space agree, − otherwise. This is best expressed by telling that the bundle of densities is canonically isomorphic to the tensor product Ber⊗or of the Berezinian of Ω^{1} (cohomological degree p, parity q) by the orientation line bundle (cohomological degree −p).

7. The generalized functions (distributions) on a supermanifold can be defined as the dual of densities with compact support, or as a suitable completion of the space of functions. It makes sense to take the pull back f(g) of a generalized function f by a submersion g. Example. Let Y be the function on ℝ with value 1 for x < 0 and 0 for x > 0. On ℝ^{1,2}, with coordinates (t, θ_{1}, θ_{2}), we have

(7.1)  Y (t + θ_{1}θ_{2}) = Y (t) + (\frac{d}{dt} Y ) (t).θ_{1}θ_{2} = Y (t) − δ(t).θ_{1}θ_{2}.

The integration up to 0 of a density μ with support bounded below is

∫_{−∞}^{0} μ := ∫ Y (t)μ.

The formula (7.1) shows that it is not invariant by the automorphism α: (t, θ_{1}, θ_{2}) ↦ (t − θ_{1}θ_{2}, θ_{1}, θ_{2}) of ℝ^{1,1}, despite α being the identity on the reduced space: integration on an open set requires the choice of a codimension (1, 0) boundary submanifold.

8. Integration of a density μ with compact support is defined, hence invariant by diffeomorphisms. It follows that for any vector field v, one has

(8.1)  ∫ ℒ_{v}(μ) = 0

Applying (8.1) to a product μf, we get

(8.2)  ∫ ℒ_{v}(μ).f = −(−1)^{p(v)p(μ)} ∫ μ.vf ,

from which results that for any g

(8.3)  ℒ_{vg}(μ) = ℒ_{v}(gμ)

(with vg = (−1)^{p(v)p(g)}gv). The map v ⊗ μ ↦ ℒ_{v}μ hence induces a morphism

(8.4)  d: (tangent bundle) ⊗ densities → densities

with

(8.5)  ∫ dα = 0

if α has compact support.

9. For an ordinary manifold of dimension n, one could construct Ω^{∗} not from the bottom up, defining p-forms as wedge of 1-forms, but from the top down, defining a p-form as obtained by contracting a top form with (n − p) tangent vectors. This corresponds to two descriptions of Ω^{∗} as a Clifford module over the sum of the tangent and cotangent bundle: the usual description is being generated by 𝒪, which is killed by the i_{X}, the other is being generated by Ω^{n}, which is killed by the α∧. In the second description, d can be defined (as was done in 8.) by its relation (3.4) with the Lie derivative. Instead of Ω^{n}, one could use its twist by the orientation bundle, i.e. the bundle of densities. In the case of super varieties, one can similarly define an integral forms complex I^{n} (n ≤ 0) starting from the line bundle of densities. It differs (more than by a twist) from the de Rham complex, but one can arrange the sign conventions so that it obeys the same formalism. One disposes of contractions i_{v}, exterior products α∧ for α a 1-form and exterior differential d, of cohomological degree −1, 1 and 1, as usual. Iterated contractions are linear and induce an isomorphism

∧^{p} T ⊗ I^{0} → I^{−p}.

Exterior products are linear,

[i_{v}, α ∧] = i_{v}(α)

and iterated products give a module structure over Ω^{∗}. One has

[d, α ∧] = (dα) ∧ and
[d, i_{v}] = ℒ_{v}

for v a vector field.

10. The basic property of integral (−p)-forms is that they can be integrated on (p, 0)- codimensional cooriented subvarieties. If the orientation of the normal bundle to W is defined by the basis dual to dt_{1}, …, dt_{p} for t_{1}, …, t_{p} a system of equations for W, our convention is that for any function with compact support on V, one has

(10.1)  ∫_{W} α|W.f = ∫_{V} δ(t_{1})dt_{1} ∧ … ∧ δ(t_{n})dt_{n} ∧ α.f

The rule (10.1) has the virtue that if t: V → ℝ is submersive above 0 and 1, one has for a codimension 1-form with compact support α

(10.2)  ∫_{0 ≤ t≤1} dα = ∫_{t=1} α − ∫_{t=0} α

if χ is the characteristic function of the interval [0, 1] of ℝ, the left side of (10.2) is by definition ∫ χ(t)dα (cf. 7.) and an integration by part using (8.5), shows it equals ∫ −d(χ(t))α = ∫ (δ(t − 1)dt)α − ∫ δ(t)dtα, as claimed by (10.2).