Differential operators and their distributional adjoints Let v : M

Distributional adjoints of natural operators 245

3. Differential operators and their distributional adjoints Let v : M

→ TM be a smooth vector field. If α : M → ∧ r T ∗ M is a smooth ordinary current and β : M → ∧ m −r T ∗ M is a test current, then hv.α, βi = −hα, v.βi , where v. denotes the Lie derivative operator. This formula can be taken as the definition of v.α for an arbitrary current α , and it turns out that the mapping α 7→ v.α is a continuous linear operator in DM, ∧ r T ∗ M . A similar construction can be carried out for scalar generalized half-densities [4]. In the domain of a coordinate chart, in particular, on has the partial derivatives of a scalar current or generalized half-density. Consider now E-valued currents the following argument is quite similar for E- valued generalized half-densities. Given a frame of E , the components of any such current are scalar currents, whose partial derivatives with respect to the base coor- dinates are well-defined. If D is a differential operator acting on smooth currents, which can be locally expressed in terms of partial derivatives, then D has a unique continuous extension to the whole space of currents, and this extension maintains the same coordinate expression. More specifically, one considers a polynomial deriva- tion operator of order k ∈ N, which means that its coordinate expression is of the type Dα i = P |p|≤k c i p j ∂ p α j ; here, p = { p 1 , .., p m } is a multi-index of order | p| := P a p a , and c i p j : M → C are smooth functions. Because the operator D can be expressed in terms of Lie derivatives, it yields a distributional adjoint operator D ′ , fulfilling D ′ φ, θ = hφ, Dθi ; this acts in the dual space, is still a polynomial operator of degree k, and extends to the whole, appropriate space of generalized sections. In practice, its coordinate expression is usually found by assuming φ to be C k , expressing hφ, Dθi as an integral and applying integration by parts boundary terms disappear. In the case of scalar-valued currents one has some important instances. Let v : M → TM be a smooth vector field, and ω : M → ∧ T ∗ M a smooth form; indicating by |θ| the degree of the current θ, by simple exterior algebra calculations and taking into account ∂M = ∅ one obtains hv|α, βi = −1 |α|+1 hα, v|βi , |β| = m − |α| + 1 . hω ∧ α, βi = −1 |α|·|ω| hα, ω ∧ βi , |β| = m − |α| − |ω| . hdα, βi = −1 |α|+1 hα, dβi , |β| = m − |α| − 1 . hdv|α, βi = −hα, v|dβi , |β| = m − |α| . hv|dα, βi = −hα, dv|βi , |β| = m − |α| . hv.α, βi = −hα, v.βi , |β| = m − |α| . h∗α, βi = −1 |α|m+1 hα, ∗βi , |β| = |α| . hδα, βi = −1 |α| hα, δβi , |β| = m − |α| + 1 . hdδα, βi = hα, δdβi , |β| = m − |α| . hdδ + δdα, βi = hα, dδ + δdβi , |β| = m − |α| . 246 D. Canarutto In the four last formulas, ∗ stands for the Hodge isomorphism relatively to a given non- degenerate pseudometric on M. The operators δ : = ∗d∗ and dδ + δd are, up to sign, the usual codifferential and Laplacian of scalar-valued currents; in general, the latter is different from the connection-induced Laplacian defined on E-valued currents §6. 4. Operators induced by the Fr¨olicher-Nijenhuis bracket Let N be a classical manifold. Let us recall that the Fr¨olicher-Nijenhuis bracket [7, 9] of two differentiable tangent-valued forms F : N → ∧ p T ∗ N ⊗ N TN, G : N → ∧ r T ∗ N ⊗ N TN, is a tangent-valued form [F, G] : N → ∧ p +r T ∗ N ⊗ N TN . This operation can be carried on fibred manifolds; in particular consider the case when N ≡ E, the total space of our vector bundle. A smooth current α : M → ∧ r T ∗ M ⊗ M E can also be seen as a tangent-valued r -form on E because of the natural inclusions T ∗ M ⊂ T ∗ E and E ⊂ E ⊗ M E ≡ VE ⊂ TE. Then we can consider the F-N bracket [ω, α] , whenever ω is a smooth tangent-valued form on E. In practice, one is interested mainly in basic forms ω : M → ∧ p T ∗ M ⊗ E TE which are projectable over TM-valued forms. Taking fibred coordinates x a , y i : E X → R m ×R n , the projectability condition for ω becomes ∂ j ω b a 1 ... a p = 0 , and one obtains the local expression [ω, α] = ω b a 1 ... a p ∂ b α i a p +1 ... a p +r + −1 pr r ∂ a r ω b a r +1 ... a r + p α i a 1 ... a r −1 b + + −1 pr +1 ∂ j ω i a r +1 ... a r + p α j a 1 ... a r d x a 1 ∧ . . . ∧ d x a p +r ⊗ ∂ y i . In order that [ω, α] be still a current, then, one has to assume that ω is a linear morphism over M, which in coordinates reads ω i a 1 ... a p = y j ω i j a 1 ... a p with ω i j a 1 ... a p : M → R . Now it is clear that the mapping α 7→ [ω, α] extends naturally to a continuous operator acting on the distributional space of E-valued currents. The coordinate expression of its distributional adjoint is readily written. We see that, in general, its geometrical in- terpretation involves viewing α as a T ∗ M ⊗ M E-valued r −1 current, leading to a rather involved situation. However, the really interesting cases for field theory applications are simpler: essentially, when ω is either a vertical-valued form or a connection. Consider the first instance, namely ω : M → ∧ p T ∗ M ⊗ M VE, so that ω b a 1 ... a p = 0 . We obtain the distributional adjoint formula h[ω, α], βi = −1 pr +1 Z M α j a 1 ... a r ω i j a r +1 ... a r + p β i a r + p+1 ... a m d x a 1 ∧ . . . ∧ d x a m ⊗ d y j = −1 pr +1 α, [ω ∗ , β ] , where β : M → ∧ m −p−r T ∗ M ⊗ M E ∗ and ω ∗ : E ∗ → ∧ p T ∗ M ⊗ M E ∗ is the transpose morphism over M. Distributional adjoints of natural operators 247 A similar result holds for a linear connection on the bundle E ֌ M; this can be defined [9] as a section γ : E