Iterated Differential Forms III: Integral Calculus

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1 Iterated Differential Forms III: Integral Calculus arxiv:math/ v1 [math.dg] 30 Oct 2006 A. M. Vinogradov and L. Vitagliano DMI, Università degli Studi di Salerno and INFN, Gruppo collegato di Salerno, Via Ponte don Melillo, Fisciano (SA), Italy February 13, 2018 Abstract Basic elements of integral calculus over algebras of iterated differential forms Λ k, k <, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms are computed. Various applications and the integral calculus over the algebra Λ will be discussed in subsequent notes. vinograd@unisa.it luca vitagliano@fastwebnet.it 1

2 1 Integral Calculus over Graded Algebras In this note we follow the notation and definitions of [1, 2] (see also [3, 4, 5, 6]). The necessary generalities concerning integral calculus over graded commutative algebras are collected below by following the approach of [7, 8]. Let G = (G,µ) be a grading group, k a field of zero characteristic and A a G graded commutative algebra. Λ(A) = s Λs (A) stands for the k algebra of differential forms over A, which is naturally G Z graded. So, the de Rham complex over A reads 0 A d Λ 1 (A) d Λ s (A) d The A-module of graded, skew symmetric, s derivations of A with values in an A module P is denoted by D s (A,P). Put D (A,P) = s 0 D s(a,p) assuming that D 0 (A,P) = P. A modules Diff A (P,Λ s (A)), s 0, of graded linear differential operators that send P to Λ s (A) form the complex 0 Diff A (P,A) w P Diff A (P,Λ 1 (A)) w P Diff A (P,Λ s (A)) w P (1) with w P ( ) = d, Diff A (P,Λ(A)). The cohomology carries a natural graded A module structure given by P def = H(w P ) of complex (1) a[ ] = ( 1) a [ a], a A, [ ] P, Diff A (P,Λ(A)),w P ( ) = 0. A module P iscalledadjoint to P and(1)thedefining complex of P. A graded linear differential operator : P Q, Q being an A module, induces the co-chain map : Diff A (Q,Λ(A)) ( 1) Diff A (P,Λ(A)), i.e., w Q = w P. The corresponding map in cohomology : Q P, which is a differential operator, is called the adjoint to operator. Elements ofa moduleσ s (A) def = Λ s (A)arecalledintegral s forms overaandb(a) def = Â = Σ 0 (A) is called the berezinian of A. The module Σ(A) = s 0 Σ s(a) is naturally supplied with a G Z graded right Λ(A) module structure given by Σ l+m (A) Λ l (A) (Z,ω) Z,ω def = [ ω] Σ m (A) (2) where Z = [ ] Σ l+m (A), Diff A (Λ l+m (A),Λ(A)), d = 0. The complex of integral forms is the adjoint to the de Rham one: 0 B(A) d Σ 1 (A) 2 d Σ s (A)

3 The following right Leibnitz rule holds dz,ω = Z,dω +( 1) l d Z,ω, (3) with Z Σ(A), ω Λ l (A). For an A module P denote by Diff > A (P,Λ(A)) the right A module structure on the vector space of linear differential operators acting on P and with values in Λ(A), i.e., a > def = ( 1) a a, a A, Diff > A (P,Λ(A)). There takes place a natural isomorphism (see, e.g., [7]) and a natural pairing defined by Diff > A (Λs (A),Λ(A)) X D s (A,Diff > A (A,Λ(A))) D l+m (A,B(A)) Λ l (A) (X,ω) X,ω D m (A,B(A)) (4) X,ω (a 1,...,a m ) def = i X (ω da 1 da m ). a 1,...,a m A. The pairing (4) supplies D (A,B(A)) with a graded right Λ(A) module structure. A natural 0 degree homomorphism of right Λ(A) modules χ A : Σ(A) D (A,B(A)) (5) is defined as follows. If Z = [ ] Σ s (A), Diff A (Λ s (A),Λ(A)), then χ A (Z)(a 1,...,a s ) def = [X (a 1,...,a s )] B(A), a 1,...,a s A. Namely, χ A ( Z,ω ) = χ A (X),ω, Z Σ(A), ω Λ(A). Proposition 1 If Λ 1 (A) is a projective and finitely generated A module, then χ A is an isomorphism. In other words, in a smooth situation the isomorphism χ A gives an exact description of integral forms exclusively in terms of the berezinian B(A). 3

4 2 Defining Complex of Integral Forms over Iterated Differential Form Algebras In this section the complex of integral forms over the algebra of geometric iterated differential forms on a smooth manifold is described. This description and the results presented in the rest of this note are generalized almost automatically to any smooth situation, for instance, to super-manifolds. However, for simplicity s sake we do not report them here. In what follows M stands for a smooth n dimensional manifold, (x 1,...,x n ) for a local chart in it and all functors of differential calculus over commutative algebras and representing them objects are specialized to the category of geometric modules over the algebra C (M) in order to be in conformity with the standard differential geometry (see [6]). Accordingly, below Λ k = Λ k (M) stands for k times iterated geometric differential forms over C (M) (see [1]). Put also Λ s k+1 = Λs (Λ k ) Λ k+1 and note that Λ s k+1 is a projective Λ k module and Λ 1 k+1 is locally freely generated by elements d k+1 d L x µ, µ = 1,...,n, L {1,...,k}. Elements of the dual basis are denoted by d L D(Λ x µ k,λ k ) Hom Λk (Λ 1 k+1,λ k), µ = 1,...,n, L {1,...,k}, i.e., d L x µ (d J x ν ) = { 1, if ν = µ and J = L 0, otherwise. Integral geometric p forms over Λ k are cohomology classes of the complex 0 Diff Λk (Λ p k+1,λ k) w k,p Diff Λk (Λ p k+1,λ1 k+1 ) w k,p Diff Λk (Λ p k+1,λs k+1 ) w k,p where w k,p ( ) = d k+1, Diff Λk (Λ p k+1,λ k+1). The Λ k module Diff Λk (Λ k,λ s k+1 ) is locally freely generated by elements d k+1 d L1 x µ 1 d k+1 d Ls x µs d J1 x ν 1 d Jr x νr, r 0 with µ 1,...,µ s,ν 1,...,ν r = 1,...,n and L 1,...,L s,j 1,...,J r {1,...,k}. 3 Berezinian of the Algebra of Iterated Differential Forms and Adjoint de Rham Differentials Put ν(k) def = 2 k 1 n. Theorem 2 (i) H s (w k,0 ) = 0, if s ν(k), 4

5 (ii) There exists a natural isomorphism of Λ k modules β k : Λ k B(Λ k ) = H ν(k) (w k,0 ). It isworthstressing thatthe definitionof theberezinian we use inthisnoteisdifferent from the standard one in the theory of super-manifolds (see, for instance, [9, 10]). Even though these definitions are, as it can be shown, equivalent (see [8]) the conceptuality of the former makes it much more preferable for our goals. According to theorem 2 the element ζ k = β k (1 Λk ) freely generates Λ k module B(Λ k ) and its local description is as follows. Let U = {(x 1,...,x n )} be a local chart and I 1,...,I 2 k 1 (resp., J 1,...,J 2 k 1) be all subsets of {1,...,k} composed of an even (resp., odd) number of elements. It is assumed that each of these two families of subsets is ordered once for all according to the subscripts. Put Ω U def = d k+1 d I1 x 1 d k+1 d I2 k 1 x1 d k+1 d I1 x n d k+1 d I2 k 1 xn Λ ν(k) k+1 (U), U def = d J1 x 1 Then ζ k U = [ ] B(Λ k (U)) with d J2 k 1 x1 d J1 x n def = Ω U U Diff Λk (U)(Λ k (U),Λ ν(k) k+1 (U)). d J2 Diff k 1 xn Λk (U)(Λ k (U),Λ k (U)). Now define the homomorphism χ k,s : Σ s (Λ k ) D s (Λ k,λ k ) by putting (see section 1). (χ k,s (Z)(ω 1,...,ω s ))ζ k = χ Λk (Z)(ω 1,...,ω s ), Z Σ s (Λ k ), Theorem 3 χ k,s is an isomorphism of Λ k modules. So, Σ s (Λ k ) is identified with D s (Λ k,λ k ) via χ k,s and, in view of this identification, the complex of integral forms over Λ k reads dk+1 Λ k D(Λk,Λ k ) d k+1 Ds (Λ k,λ k ). A description of the differential d k+1 in these terms is as follows. Let Z D s (Λ k,λ k ). Note that the multi-derivation d k+1 Z D s 1 (Λ k,λ k ) is completely determined by its values d k+1 Z,Ω on forms Ω Λ s 1 k+1. But in view of (3) computation of these values is reduced to computation of the action of d k+1 on integral 1 forms, i.e., on D(Λ k,λ k ). The latter is described in the following proposition. Proposition 4 If Z Σ 1 (Λ k ) D(Λ k,λ k ), then d k+1 Z = Ẑ(ζ k) with Ẑ : B(Λ k) Λ k B(Λ k ) Λ k being the adjoint to Z operator. 5

6 If, locally, Z = µ,l Z µ Ld L, Z µ x µ L Λ k, with the summation running over µ = 1,...,n and all subsets L of {1,...,k}, then d k+1 Z = µ,l( 1) d L ( d L + Z ) d L x µ (Z µ L ). Theorem 5 H i ( d k+1 ) H ν(k) i (Λ 1,d 1 ). As an example we give an explicit description of the adjoint operator dl : D s (Λ k,λ k ) D s (Λ k,λ k ) to the l th iterated de Rham differential d l : Λ s k+1 Λs k+1 for l k. Proposition 6 Let Z D s (Λ k,λ k ). Then ( d l Z)(ω 1,...,ω s ) = l ( 1) d l ( Z + ω ω i 1 ) Z(ω 1,...,ω i 1,d l ω i,ω i+1,...,ω s ) d l (Z(ω 1,...,ω s )). 4 The Trace of an Endomorphism Now we shall illustrate the developed theory by a simple example showing that the notion of trace is a natural part of integral calculus. Consider with this purpose an endomorphism of the tangent bundle of M, or, equivalently, a(1,1)-tensor field on M. It is naturally interpreted as a Λ 1 (M) valued derivation of the algebra C (M) and hence as an integral form over the algebra Λ = Λ(M). Namely, let X D(C (M),Λ 1 (M)). According to the standard definition, trx, the trace of X, is the smooth function on M uniquely determined by the identity i X (ω) = trx ω, ω Λ n (M), that can be rewritten as p n i X = trx p n, where p n : Λ(M) Λ n (M) is a natural projection. Now, according to the previous section ζ 1 = [ ] B(Λ), where Diff Λ (Λ,Λ 2 ) is locally given by = d 2 x 1 d 2 x n d 1 x 1 d 1. x n The coordinate-free description of is = ( 1) n(n 1)/2 κ p n, 6

7 with κ : Λ 2 Λ 2 being the involution that interchanges differentials d 1 and d 2 (see [1]). Now, remembering that i X D(Λ,Λ) is an integral 1 form over Λ, we have: d 2 (i X ) î X (ζ 1 ) = [ i X ] = ( 1) n(n 1)/2 [κ p n i X ] = trx ζ 1. Since ζ 1 is a canonical integral form, this result allows to identify trx with the integral form d 2 (i X ) B(Λ) = Λ. 5 Final Remarks Our interest in integral calculus with iterated differential forms is motivated by the problem of unification of various natural integration procedures that one meets in differential geometry, theoretical physics, etc. From what was said above one can see that the developed calculus puts in common frames usual integrals, interpreted as the de Rham cohomology, and traces of endomorphisms. These facts give some simple arguments in favor of iterated forms in this connection. Further results, which this approach allows to get on in this direction, will be given in separate publications. References [1] A. M. Vinogradov, L. Vitagliano, Dokl. Math. 73, n 2 (2006) 169. [2] A. M. Vinogradov, L. Vitagliano, Dokl. Math. 73, n 2 (2006) 182. [3] A. M. Vinogradov, Soviet Math. Dokl. 13 (1972) [4] A. M. Vinogradov, J. Soviet Math. 17 (1981) [5] M. M. Vinogradov, Russian Math. Surveys 44 (1989) no. 3, 220. [6] J. Nestruev, Smooth Manifolds and Observables, Springer Verlag(New York) [7] I. S. Krasil shchik, A. M. Verbovetsky, Homological Methods in Equation of Mathematical Physics Open Education (Opava) See also Diffiety Inst. Preprint Series, DIPS 7/98, 98abs.htm. [8] A. M. Verbovetsky, J. Geom. Phys. 18 (1996) 195. [9] D. A. Leites, Uspekhi Mat. Nauk 35 n 1 (1980) 3; Russian Math. Surveys 35 n 1 (1980) 1. [10] F. A. Berezin, Introduction to Superanalysis, D. Reidel, Dordrecht, [11] A. M. Vinogradov, L. Vitagliano, in preparation. 7