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The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications

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The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications The Poincare lemma for codifferential, anticoexact forms, and applications to physics Rados law Antoni Kycia1,2,a 1Masaryk University Department of Mathematics and Statistics Kotl´aˇrsk´a267/2, 611 37 Brno, The Czech Republic 2Cracow University of Technology Faculty of Materials Engineering and Physics Warszawska 24, Krak´ow, 31-155, Poland [email protected] Abstract The linear homotopy theory for codifferential operator on Rieman- nian manifolds is developed in analogy to a similar idea for exterior derivative. The main object is the cohomotopy operator, which sin- gles out a module of anticoexact forms from the module of differential forms defined on a star-shaped open subset of a manifold. It is shown that there is a direct sum decomposition of a differential form into coexact and anticoexat parts. This decomposition gives a new way of solving exterior differential systems. The method is applied to equa- tions of fundamental physics, including vacuum Dirac-K¨ahler equa- tion, coupled Maxwell-Kalb-Ramond system of equations occurring in a bosonic string theory and its reduction to the Dirac equation. Keywords: Poincare lemma; codifferential; anticoexact differential forms; homo- topy operator; Clifford bundle; Maxwell equations; Dirac operator; Kalb-Ramond equations; de Rham theory; arXiv:2009.08542v3 [math.DG] 1 May 2021 Mathematical Subject Classification: 58A12, 58Z05; 1 Introduction Well known formulation of the Poincar´elemma states [20, 28] that R, (k = 0) Hk(Rn)= Hk(point)= , (1) 0 (k > 0) 1 for n > 0. This means that in Rn each differential form which is closed, i.e., it is in the kernel of exterior derivative operator d, is also exact, i.e., is also in the image of d. This can be also extended to a star-shaped open region of a smooth manifold M, which by definition is an open set U of M that is diffeomorphic to an open ball in Rn, where n = dim(M). In full generality, the Poincar´elemma is valid in any contractible submanifold. In practical calculations, especially in physics, there is need for finding, if possible, potential from dα = 0 for some differential form α, i.e., finding the exact formula for a differential form β such that dβ = α. This can be done locally in a star-shaped region U using (linear) homotopy operator [10, 11, 28, 20, 19, 7], i.e., 1 k−1 Hω := iKω|F (t,x)t dt, (2) Z0 where a k-form ω ∈ Λk(U), k = deg(ω), i K := (x − x0) ∂i, (3) and F (t,x) = x0 + t(x − x0), x0 ∈ U, is a linear homotopy between the constant map sx0 : x → x0 and the identity map I : x → x. The form ω under the integral is evaluated at the point F (t,x). Here iK = Ky is the insertion antiderivative. The assumption about star-shapedness of U is imposed to ensure the correct definition of homotopy F and the H operator. Below we will be exclusively dealing with U being star-shaped. There is general construction of homotopy operator for various (non-linear) homotopies [19, 28, 20, 7], however the linear homotopy operator has additional advantage as it was pointed out by D.G.B. Edelen [10, 11, 19]. The homotopy operator H has many properties useful in calculations [11, 10], e.g., H2 = 0, HdH = H, dHd = d, (4) iK ◦ H = 0, H ◦ iK = 0. (5) This operator fulfils Homotopy Invariance Formula [10, 11, 28, 20, 19] ∗ dH + Hd = I − sx0 , (6) ∗ where sx0 is the pullback along the constant map sx0 (x)= x0, and I is the identity map. It is well-known that the kernel of d defines the closed vector space E(U) = {ω ∈ Λ(U)|dω = 0} that is a vector subspace of Λ(U). Since we focus on open star- shaped regions U of a smooth manifold M with or without boundary, by Poincar´e lemma, elements of E(U) are also exact. Therefore, we will be using ’closed forms’ and ’exact forms’ interchangeably in what follows. Similarly, the kernel of H on U defines a module over Λ0(U) of antiexact forms A = {ω ∈ Λ(U)|Hω = 0}, which was described in [11, 10]. It was also proved [11, 10] that A = {ω ∈ Λ(U)|iKω = 0, ω|x=x0 = 0}, (7) 2 and that there is a direct sum decomposition [11, 10, 19] Λk(U)= Ek(U) ⊕Ak(U), (8) for 0 ≤ k ≤ n. On Riemannian manifolds with non-degenerate metric tensor g one can define the Hodge star operator [25, 1], ⋆ : Λr → Λn−r, that fulfils ⋆ ⋆ω = (−1)r(n−r)ω = (−1)r(n−r)sig(g)ω, ω ∈ Λr(U), (9) with the inverse ⋆−1 = (−1)r(n−r)sig(g)⋆ = sig(g)ηn−1⋆ = sig(g) ⋆ ηn−1, (10) where η is an involutive automorophis: ηω = (−1)pω for ω ∈ Λp, and where det(g) sig(g)= |det(g)| is the signature of the metric g. For clarity of presentation we will focus on the Riemannian case (sig(g) = 1) only, and the other signatures, e.g., Lorentzian one, can be analyzed similarly. Then the codifferential is defined as δ = ⋆−1d⋆η. (11) The Poincar´elemma for codifferential, an easy corollary of the Poincar´elemma, is as follows: Theorem 1. (The Poincar´elemma for codifferential) For a star-shaped region U, if δω = 0 for ω ∈ Λk(U), then there exists α ∈ Λk+1(U) for k<n = dim(U), such that ω = δα. This paper aims to build a theory analogous to antiexact forms, in which cod- ifferential δ is in the central place - anticoexact forms, and then apply it to various equations and systems of equations containing d and δ operators. Anti(co)exact forms are local objects valid in a star-shaped open region of a manifold; however, local problems are essential to physics applications. Therefore, we also provide some examples of physics equations that can be solved locally by the methods presented here. The paper is organized as follows: In the next section, we develop the theory of anticoexact forms that allows us to decompose arbitrary differential form into coexact and anticoexact parts. Then we relate this decomposition with exact- antiexact direct sum decomposition of [10, 11]. Next, the connection with Clifford algebras will be presented. Finally, application of (anti)(co)exact decomposition to various cases of Dirac(-K¨ahler) equations [1], Maxwell equations of classical electrodynamics and their coupling with the Kalb-Ramond equations of bosonic string theory [17, 31] will be presented. In the Appendix the relation to the de Rham theory is discussed. 3 H H H H δ Ñ q δ Ñ δ Ó p δ ...Õ t r−1 U q r U r r+1 U p ... 0 Λ ( ) 2 Λ ( ) 1 Λ ( ) 3 D h C h A h A h d d d d Figure 1: The action of d, H, δ and h on Λ. Here 1 <r<n − 1. 2 Anticoexact forms This section defines an analog of the theory for antiexact forms, which we call anticoexact forms. The presentation will be along with Chapter 5 of [10] with marking differences between antiexact and defined below anticoexact forms. We start from the homotopy operator for δ: Definition 1. We define the cohomotopy operator for δ for a star-shaped region U as h : Λ(U) → Λ(U), h = η ⋆−1 H ⋆ . (12) In particular, r r−1 r+1 −1 hr : Λ (U) → Λ (U), hr = (−1) ⋆ H⋆, r> 0. (13) Fig. 1 presents interplay between all the operators. Since (Hω)|x=x0 = 0, so (hω)|x=x0 = 0. Such a definition makes the Homotopy Invariance Formula for δ and h similar to (6), namely, Proposition 1. δh + hδ = I − Sx0 , (14) −1 ∗ n where Sx0 = ⋆ sx0 ⋆. The operator Sx0 is nonzero for Λ (U) and it evaluates top n forms at x = x0, i.e., Sx0 ω = ω|x=x0 for ω ∈ Λ (U). Proof. Using the Homotopy Invariance Formula 6 restricted to Λr, we have ⋆−1(dH + Hd)⋆ = ((−1)r+1 ⋆−1 d⋆)((−1)r+1 ⋆−1 H⋆)+ r −1 r −1 (15) ((−1) ⋆ H⋆)((−1) ⋆ d⋆)= δh + hδ = I − Sx0 , r since η|Λr = (−1) I. As a simple extension of the properties of d and H [10, 11], we have Proposition 2. h2 = 0, δhδ = δ, hδh = h. (16) 4 Proof. Since H2 = 0 so, by (12), h2 = 0. For the second property, we have δhδ = ⋆−1d⋆ηη⋆−1 h ⋆ ⋆−1d ∗ η = ⋆−1d ⋆ η = δ, since η2 = 1 and dHd = d. Similarly, using HdH = H, we get the third property. Define now the coclosed (that is also coexact in a star-shaped U) vector space C := {ω ∈ Λ(U)|δω = 0}. (17) Note that Cn = R ⋆ 1. Since E0 consists of constant functions [19], coexact top forms are dual exact ones: ⋆E0 = Cn. We have that Proposition 3. The operator δh is the projector δh : Λ →C. Proof. For any form ω, the form δhω is coexat, so δh : Λ →C. It is idempotent since δ(hδh) = δh. Finally, for ω ∈ C we have from the Poincar´elemmat that there exists α such that δα = ω, and hω = hδα. Then δhω = δhδα = δα = ω. Therefore on C the operator δh is the identity. We can therefore define the coexact part of the form by the projection ωc := δhω. (18) By treating Homotopy Invariance Formula (14) as decomposition of the identity operator into projections, we can define Definition 2. (Anticoexact part of a form) We define anticoexact part of a form ω ∈ Λk(U),k<n, on a star-shaped region U as ωac := hδω = ω − δhω.
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