Appendix: the 2-form on a space of extremals

P. Deligne

In his lectures, Bernstein considered lagrangian densities ℒ(ϕ) on V, for ϕ in the space ℳ of maps V → M, or more generally for ϕ a section of a fiber bundle M_{V} over V. The case of maps to M correspond to M_{V} = M × V. He considered the case where the density ℒ(ϕ) at v ∈ V depends only on ϕ and dϕ at v, and explained how to construct a closed 2-form ω on the space ℰ of extremals. The construction depended on a cohomology class of hypersurfaces Γ with an oriented normal bundle: Γ has a “past” side and a “future” side. In practice, Γ is a “space-like” hypersurface. The construction uses integration on Γ, hence requires some control at the infinity of Γ, and a similar control on homologies among Γ’s. We will ignore this by pretending that Γ is compact and that cohomology is taken with compact support.

In this appendix, our aim is to explain a small modification of the construction which works as well when ℒ(ϕ) depends on higher derivatives of ϕ. One can also take ϕ to be for instance a bundle with connection, rather than a map to some M. If V is a supermanifold, the new construction makes clear that the 2-form obtained is the same whether one works in term of superfields ϕ: V → M and of superdensities ℒ(ϕ) on V, or in term of “components” of ϕ and of the Lagrangian density \overline{ℒ}(ϕ) on V_{red} deduced from ℒ(ϕ) by “integrating out” the odd variables, for some projection V → V_{red}.

What we don’t get are 2-forms ω_{Γ}, depending on Γ, on the space ℳ of all ϕ, and inducing ω on the space ℰ of extremals.

The basic construction is local around a space-like hypersurface Γ and, to explain it, we will assume that V = Γ × ℝ, the projection to ℝ being called “time”.

For a 2-parameter family ϕ[a_{1}, a_{2}] of extremals, we now define the value at (a_{1}, a_{2}) = (0, 0) of the pull back of ω by (a_{1}, a_{2}) ↦ ϕ[a_{1}, a_{2}]. We choose a family ϕ_{1}[a_{,}a_{2}] of maps to M, deforming ϕ[a_{1}, 0], agreeing with ϕ[a_{1}, a_{2}] in the future, and agreeing with ϕ[a_{1}, 0] in the past:

ϕ_{1}[a_{1}, 0] = ϕ[a_{1}, 0],
ϕ_{1}[a_{1}, a_{2}] = ϕ[a_{1}, a_{2}]  in the (far) future,
ϕ_{1}[a_{1}, a_{2}] = ϕ[a_{1}, 0]  in the (far) past.

We similarly choose ϕ_{2}, having the same properties with the roles a_{1} and a_{2} permuted. Definition:

ω(∂_{1}, ∂_{2}) = ∫_{V} (∂_{1}∂_{2}ℒ(ϕ_{1}) − ∂_{2}∂_{1}ℒ(ϕ_{2})) at (a_{1}, a_{2}) = (0, 0).

The integrand vanishes in the future, where ϕ_{1} = ϕ_{2}, as well as in the past, where both terms vanish. This makes the integral defined.

Using Γ, with equation t = 0, we now express ω as an exterior derivative dα_{Γ} and show independence of the choices of ϕ_{1} and ϕ_{2}. Let Y be the function on ℝ which is 1 for x < 0 and 0 for x > 0: \frac{d}{dx} Y = −δ(x). We will have to integrate on V up to t = 0 some smooth density μ vanishing in the past. In the supervariety case, this means integrating Y (t)μ (integration on a supervariety with boundary Γ). This depends on Γ, not just on Γ_{red}.

The 1-form α_{Γ} is defined only at extremals, but makes sense on tangent vectors to the space ℳ of all ϕ. Fix an extremal ϕ[0] and consider a family ϕ[a] deforming it. We now define α_{Γ} at the corresponding tangent vector. Let ϕ_{1} be a deformation of ϕ[0] agreeing with ϕ[a] for t > −ε, and with ϕ[0] in the past. Definition:

α_{Γ}(∂_{a}) = ∫ Y (t)∂_{a}ℒ(ϕ_{1}[a]) at a = 0

The integrand vanishes in the future, because Y (t) does, and in the past, where ϕ_{1}[a] is constant. This makes the integral defined. Because ϕ[0] is an extremal, it does not depend on the choice of ϕ_{1}: the density ∂_{a}ℒ(ϕ_{1}[a]) is the variation of the density ℒ for a variation δ_{1}ϕ of ϕ[0], and if we consider another deformation ϕ_{2}, δ_{1}ϕ − δ_{2}ϕ has support in a time interval [−A, −ε]. It is the tangent vector for a deformation ϕ_{3} with a support in t < 0 and

\begin{matrix}∫ Y (t) & [∂_{a}ℒ(ϕ_{1}[a]) − ∂_{a}ℒ(ϕ_{2}[a])] (at a = 0) \\ = ∫ Y (t)∂_{a}ℒ(ϕ_{3}[a]) (at a = 0) = ∫ ∂_{a}ℒ(ϕ_{3}[a]) (at a = 0) = 0 \end{matrix}

To see that ω = dα_{Γ}, one uses the formula that for commuting vector fields X and Y, one has

(dα)(X, Y) = X.α(Y) − Y.α(X) .

For the pull back of α_{Γ} by (a_{1}, a_{2}) ↦ ϕ[a_{1}, a_{2}], this gives, provided that ϕ_{1}[a_{1}, a_{2}] = ϕ[a_{1}a_{2}] = ϕ[a_{1}a_{2}] for t > −ε:

dα_{Γ}(∂_{1}, ∂_{2}) = ∂_{1} ∫ Y (t)∂_{2}ℒ(ϕ_{1}) − ∂_{2} ∫ Y (t)∂_{1}ℒ(ϕ_{2})
= ∫ Y (t)(∂_{1}∂_{2}ℒ(ϕ_{q}) − ∂_{2}∂_{1}ℒ(ϕ_{2}))
= ∫ ∂_{1}∂_{2}ℒ(ϕ_{1}) − ∂_{2}∂_{1}ℒ(ϕ_{2})

at (a_{1}, a_{2}) = (0, 0): the definition of ω.

In the case treated by Bernstein, integration by parts shows that the form α_{Γ} is the restriction to ℰ of the one he considers, on ℳ.

The modification we presented does not require the tangent space of ℳ at ϕ to be the space of sections of a vector bundle on V. Basically, it requires only that it be the space of sections of a soft sheave of ℝ-vector spaces on V. One has to be able to deform ϕ in a prescribed way in different regions of V. For V a supermanifold, this does not preclude imposing suitable constraints on ϕ.