arXiv:2009.08542v3 [math.DG] 1 May 2021 nioxc om,adapiain to applications physics and forms, codifferential, for anticoexact lemma Poincare The elkonfruaino h ona´ em tts[0 28 [20, states Poincar´e lemma the of formulation known Well Introduction 1 58Z05; 58A12, Classification: Subject Mathematical ope theory; Dirac Rham equations; de Maxwell equations; differ bundle; Clifford anticoexact operator; codifferential; topy lemma; Poincare Keywords: ooi tigter n t euto oteDrcequati Dirac the to reduction its and theory string occ bosonic equations Dirac-K¨aha of vacuum system Maxwell-Kalb-Ramond including coupled applied physics, tion, is fundamental method form of The differential ne tions a systems. a gives differential of decomposition exterior This decomposition solving is parts. sum It anticoexat direct and manifold. a a coexact of is differ subset of there open module that star-shaped the a whi from on e forms operator, defined anticoexact for cohomotopy forms of idea the module similar a is out a object gles Rie to main on analogy The in operator derivative. developed codifferential is for manifolds theory nian homotopy linear The ola r ka272 1 7Bn,TeCehRepublic Czech The Kotl´aˇrsk´a Brno, 37 611 267/2, aut fMtrasEgneigadPhysics and Engineering Materials of Faculty eateto ahmtc n Statistics and Mathematics of Department H asasa2,Ka´w 115 Poland Krak´ow, 31-155, 24, Warszawska k ( 2 R rcwUiest fTechnology of University Cracow a n [email protected] = ) ao a noiKycia Antoni Rados law 1 H aay University Masaryk k ( point Abstract 1 = ) ( 0 R , 1 > k , ( 2 k ,a 0) = 0) , ao;Kalb-Ramond rator; that ] nilfrs homo- forms; ential e equa- ler a of way w rigin urring on. oequa- to xterior hsin- ch shown ential man- into (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 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 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 ωac := hδω = ω − ω|x=x0 . (20) Note that the anticoexact part of the form is of the type ωac = hα and so these parts are in the kernel of the operator h by its nilpotency. We can define the anticoexact vector space as Y(U) := {ω ∈ Λ(U)|ω = hδω}, (21) that is the vector space of spanned by all anticoexact parts. Note that Y0 = 0. Anticoexact space can be alternatively defined by the vector K. To this end we have to use the following lemma Lemma 1. (Equation (1.4.7) of [1]) iα♯ ⋆ φ = ⋆(φ ∧ α), (22) for α ∈ Λ1, φ an arbitrary form, and where ♯ is a musical isomorphism such that g(α♯, X)= α(X) for an arbitrary vector field X. 5 Using this Lemma, we have Proposition 4. (K♭∧) ◦ h = 0, h ◦ K♭∧ = 0. (23) −1 Proof. Since we have (5), i.e., iK ◦ H = 0, therefore iK ⋆ ⋆ ◦ H = 0. Using (22), we have (⋆K♭∧) ◦ ⋆−1H = 0. From the definition (12) of h and the fact that ⋆ is an isomorphism, we have the result. For the second identity, using (5), i.e., H ◦ iK = 0, we get H ◦ iK⋆ = 0, so using (22) we get H ⋆ ◦K♭ ∧ ◦η = 0. Therefore, h ◦ K♭∧ = 0, as required. Using this we can characterize the vector space of anticoexact forms in an alternative way, Proposition 5. On a star-shaped region U, ♭ Y(U)= {ω ∈ Λ(U)|K ∧ ω = 0, ω|x=x0 = 0}. (24) ♭ Proof. If ω ∈Y, i.e., ω = hδω then from (23) we get K ∧ ω = 0 and ω|x=x0 = 0. ♭ In the opposite direction, let K ∧ α = 0 and take ω = α + δβ. Since K|F (t,x) = ♭ tKx and (iK ⋆ α)|F (t,x) = t ⋆ K |x ∧ ηα|F (t,x) = 0, so hα = 0. Then hω = hδβ, so ωc = δhω = δhδβ = δβ. Therefore the remaining part ω − ωc = ωac = α. Since α is the anticoexact part of ω so α = hδω and therefore αx=x0 = 0. Contrary to C being only a vector space, we have Proposition 6. Y is a C∞-module. ♭ Proof. The conditions (24) defining Y: K ∧ ω = 0 and ω|x=x0 = 0 is preserved under wedge multiplicatin of two elements from Y and under C∞ multiplication. Antiexact forms can be written as iKα for some α. Likewise, we have Proposition 7. If ω ∈Yr(U) then there exists α ∈ Λr−1(U) such that ω = K♭ ∧ α. (25) Proof. Since hδ is the projector onto Y, so for ω there is β such that ω = hβ. Since h is linear, so we can focus on a simple form which has local expression β = f(x)dxI for some multiindex I. Then 1 −1 −1 I |I|−1 ♭ hβ = η ⋆ H ⋆ β = η ⋆ iK ⋆ dx dtf(F (t,x))t = K ∧ α, (26) Z0 where 1 α = dtf(F (t,x))t|I|−1 dxI . (27) Z0 6 The final point of this section is the following Theorem 2. For a star-shaped U there is the direct sum decomposition Λk(U)= Ck(U) ⊕Yk(U). (28) Proof. For 0 For k = n we have hω = 0, so hδ = I − Sx0 . Therefore the decomposition is R n ωac = hδω = (I − Sx0 )ω and ωc = Sx0 ω ∈ ⋆ 1= C (U). Finally, if ω ∈Y∩C then using projectors hδ and δh we get ωac =0= ωc and so ω = 0. As a result, the decomposition is indeed a direct sum decomposition. 3 (Anti)coexact vs. (anti)exact forms The theory developed in the previous section can be related to the theory of antiexact and exact forms, namely, Proposition 8. In a star-shaped region • Ek = ⋆Cn−k. • Ak = ⋆Yn−k Proof. If ω ∈ E, then dω = 0. Therefore, δ⋆ω = 0, and so ⋆ω ∈ C. In a result ⋆E ⊂C. Similarly one can prove the opposite inclusion. Since ⋆ operator is an isomorphism we get the first point. For the second point, we note that from (22) we can relate in a unique way A with Y. The proposition shows that the theory of exact and antiexact forms is dual, with respect to the ⋆ isomorphism, to the theory of coexact and anticoexact forms developed above. However, all the work done above is not futile. The notion of (anti)coexact forms and operator h is helpful in applications, as we will see below. Summing up the above results and the results of [19], we have Theorem 3. On a star-shaped open region U of a manifold M, we have two ways of decompose Λ(U) into the direct sums: • Λ(U)= E(U) ⊕A(U); • Λ(U)= C(U) ⊕Y(U); 7 Since the projection operators for these different decompositions do not com- mute, we cannot simultaneously decompose an element from Λ(U) into both ways. The order of the decomposition is essential. In what follows, we need to recall the standard definitions. Definition 3. We will call H(U) := E(U) ∩C(U) = {ω ∈ Λ(U)|dω =0= δω} Hodge harmonic forms [12, 29]. By analogy, we will call elements of H¯ := A∩Y Hodge antiharmonic forms. In contrast, the typical (Kodaira) harmonic forms [12, 29] are elements of the kernel of the Laplace-Beltrami operator △ = −(δd + dδ) = (d − δ)2. However, on non-compact U, these two definitions of harmonic forms are not in general equivalent [29, 12, 8]. Due to the lack of compactness of U, the link with de Rham’s theory is obscured, as will be explained in the Appendix. In the next section the connection of above results with Clifford algebras is presented. 4 Relation to Clifford algebras For a Riemannian manifold (M, g) the Clifford bundle is isomorphic to Λ(T M) pointwise by defining the Clifford multiplication of a vector v ∈ TxM by ψ ∈ Λ(TxM) as vψ := v ∧ ψ + ivψ, (29) see e.g., [13, 2, 3, 1, 6, 24, 27]. The conversion between elements of a tangent and cotangent bundle, if needed, is made using the metric g. Then for the unique metric-compatible torsion-free connection ▽ we can define a n for an orthonormal co-frame {e }a=1 a a d := e ∧▽ea , δ := −ie ▽ea . (30) In these terms, the Dirac(-K¨ahler) operator [1] on a Clifford bundle is defined as a D := e ▽ea = d − δ. (31) Note that for the form ω Clifford multiplied by K is ♭ Kω = K ∧ ω + iKω, (32) which is the decomposition of ω into anticoexact and antiexact parts, see Propo- sition 7. In order to understand the structure of the Dirac operator on a Clifford bundle and its relation to the Poincar´elemma, we have to split it into grading of the base 8 of Fig. 1. To this end, introduce the base of grading Λ = Λ0 ⊕ . . . ⊕ Λn, where a form ω is written as the vector ω0 . ω = . , (33) ωn where ωi ∈ Λi. In this base the exterior derivative d has the simpler form 0 0 . . . 0 0 d0 0 . . . 0 0 0 d . . . 0 0 d = 1 . (34) . 0 0 .. 0 0 0 0 ... d 0 n−1 2 Since dkdk−1 = 0 therefore d = 0. The operator is nilpotent due to combination of the matrix multiplication and its (operator) elements. Likewise, we have 0 δ1 0 . . . 0 0 0 δ2 . . . 0 . δ = 0 0 . . . .. 0 , (35) 0 0 . . . 0 δn 0 0 . . . 0 0 2 where δ =0 by δk−1δk = 0. For a star-shaped region U we can define analogously the homotopy operators from Fig. 1, 0 H1 0 . . . 0 0 0 . . . 0 0 0 0 H2 . . . 0 h0 0 . . . 0 0 . 0 h . . . 0 0 H = 0 0 . . . .. 0 , h = 1 . (36) .. 0 0 . . . 0 Hn 0 0 . 0 0 0 0 . . . 0 0 0 0 ... h 0 n−1 Then the Dirac operator is 0 −δ1 0 . . . 0 d0 0 −δ2 . . . 0 . . . D = d − δ = 0 ...... 0 , (37) .. 0 . . . . 0 −δn 0 ...... dn 0 −1 9 with the Laplace-Beltrami operator δ1d0 0 . . . 0 0 0 d0δ1 + δ2d1 0 . . . 0 2 2 . D = (d − δ) = − 0 0 .. 0 0 . 0 . . . 0 dn δn + δndn 0 −2 −1 −1 0 0 . . . 0 d δ n−1 n (38) One then see that the Dirac operator mixes different grades and involves at most three neighbour grades. Similarly, on U we can define the anti-Dirac operator by D := h − H and the anti-Laplace-Beltrami operator D 2 = (h − H)2 = −(hH + Hh). A word on spinors realized by the Clifford bundle based on exterior bundle n using (29) is in order. When M is parallelizable, i.e., there is a frame {ea}a=1 such that ▽X ea = 0 for all X ∈ Γ(T M) then from this basis one can construct global idempotents and use them to project Clifford algebra to minimal left ideals obtaining spinor subbundle of a Clifford bundle [1, 13]. However, the existence of this ideal subbundle is more restrictive (M must be parallelizable) than for the existence of the spinor bundle [6] in which case the vanishing of the second Stiefel-Whitney class is needed. The realization of spinors by (exterior) Clifford bundle gives, so-called, amorphous spinor fields/Dirac-Kh¨aler spinors that do not have tensorial transformation law due to mixture of different grades, in contrast to covariant Dirac spinors used in physics. Therefore, they are not directly applicable to describe fermions (see however [18] for possible applications to lattice QCD), yet Dirac-like versions of Maxwell equations can be managed by methods presented in this paper. We will therefore focus only on sections of Clifford bundles – Clifford fields. Detailed discussion of this subject is provided in [27] and [13, 2]. In the next section, we examine local solutions of the Dirac and other physics equations using the machinery of decomposition of arbitrary form into (anti)(co)exact components. 5 Application to equations of physics In this section, the application of the above theory to some equations of physics will be presented. All considerations will be given in a star-shaped region U of a manifold M since then it is possible to use the machinery presented above. The restriction to star-shaped open regions of a manifold sacrifice generality, however, for many applications in physics is usually sufficient. Global solutions that restrict to the local ones are usually connected with topological conditions on the mani- fold and require proper ’sheafication’ procedure [26, 12]. Therefore we will limit ourselves to the local considerations only. 10 5.1 The vacuum Dirac equation First, we start from the solutions of the vacuum Dirac-K¨ahler equation [18, 6, 1] on a Clifford bundle Dψ = 0 (39) in a star-shaped region U. The simplest solution is an arbitrary Hodge harmonic form ψ ∈H(U), that is ψ ∈ ker(d)∩ker(δ). Such forms are also the solution of the Laplace equation D2ψ = 0. On a compact manifold, ψ is also a harmonic form. These solutions can be seen as a ’gauge modes’ that allows to shift other solutions, since they nullify both terms d and δ independently. More complicated solutions involve three neighbour spaces Λk−1, Λk and Λk+1 with 0 α ∈Ek−1 ∩Ck−1, β ∈Ek+1 ∩Ck+1, (41) or (when α 6∈ Ek−1 and β 6∈ Ck+1 - non-gauge case) is for dα ∈Ek ∩Ck, δβ ∈Ek ∩Ck. (42) Proof. From the first and the last equations of (40) we get α ∈Ek−1 and β ∈Ck+1. Using decomposition dH + Hd = I for 0 d(α − Hδβ) = 0 (43) Hdδβ = 0. If δβ = 0, i.e., β ∈C then dα = 0 so α ∈E and we have β ∈Ek+1 ∩Ck+1 ⇒ α ∈Ek+1 ∩Ck+1. Assume therefore that β 6∈ C, i.e., δβ 6= 0. Then the second equation of (43) gives that δβ ∈ ker(Hd)= E, so δβ ∈Ek ∩Ck. From the first equation of (43) and previous considerations dα ∈Ek ∩Ck. 11 Likewise, using decomposition δh + hδ = I for 0 5.2 Massive Dirac equation The massive Dirac equation is of the form Dψ + ψ = 0. (44) For ψ ∈ Λk belonging only to the one grade we have ψ = 0. Therefore ψ must be a sum of more grades. We consider a case of two neighbour grades ψ = α + β for α ∈ Λk and β ∈ Λk+1 for 0 12 where v ∈ Λk−1 and w ∈ Λk+2 for k + 2 < n + 1 and zero otherwise, are two arbitrary forms. Demanding that ψ consists of components of all grades give a system that decouples into independent Klein-Gordon wave equations for all grades. 5.3 Massless Dirac equation with source In the next step we will consider the massless Dirac equation Dψ = B, (48) where B ∈ Λk, 0 Theorem 5. In order to solve (49) we can either 1st approach: 1. Solve △β = dB for β; 2. Calculate α = H(δβ + B)+ dv, where v ∈ Λk−2 if k − 2 ≥ 0 and is arbitrary (k − 2)-form, or v = 0 for k − 2 < 0; 3. The solution is ψ = α + β; 2nd approach: 1. Solve △α = −δB for α; 2. Calculate β = h(dα − B)+ δw, where w ∈ Λk+2 if k + 2 ≤ n and is arbitrary (k − 2)-form, or w = 0 for k + 2 >n; 3. The solution is ψ = α + β; Proof. We will focus on the middle equation of (49) and use the decompositions from Theorem 3. First we use dH + Hd = I for 1 dα − (dH + Hd)δβ = (Hd + dH)B, 13 that is d(α − Hδβ − HB) − Hd(δβ + B) = 0, which is nontivial E⊕A-decomposition of 0, that gives α = Hδβ + HB + dφ (50) δβ + B ∈E⇔ dδβ = −dB, where ψ is arbitrary. Since dβ = 0 we get the first case. Likewise, we can use the decomposition hδ + δh = I for 1 (hδ + δh)dα − δβ = (hδ + δh)B. This gives nontivial C⊕Y-decomposition of 0, so β = hdα − hB + δψ (51) dα − B ∈C⇔ δdα − B = 0, where ψ is arbitrary. By δα = 0 we get the second case. 5.4 Maxwell equations As a preparation for describing the Kalb-Ramond equations in the next subsection, we provide application of the above theory to the solutions of the Maxwell system on Minkowski space M [1] dF = 0, δF = j, (52) where F ∈ Λ2 and the external current is j ∈ Λ1. This current is conserved since δj = 0. The typical approach on a star-shaped region U is to take a potential A ∈ A1 ⊂ Λ1 such that dA = F , since F ∈E2. Then the gauge transform A → A + χ, where χ ∈ E1, that is χ = df for some f ∈ Λ0, does not change F . The second equation is −δdA = (△ + dδ)A = −j. Since A ∈A1, so we can further decompose it into C1 and Y1. We can remove anticoexact part of A by imposing the Lorentz gauge: δA = 0, and we obtain the wave equation (in Lorentzian case) △A = −j (53) that can be solved by standard propagator methods. Then there is still a gauge freedom A → A + φ, where △φ = 0. Another approach is to use the above developed theory. First, use dF = 0, i.e., F ∈E2, to select, as before, A ∈A1 such that dA = F . Then the second equation is δdA = j. Since the current is conserved, so j ∈C1, and therefore, j = δhj. We have, δ(dA − hj) = 0, or δ(F − hj) = 0. In the end, the solution is F = δα + hj, (54) 14 where α ∈Y3 is defined up to an element of C3. Since (54) represents the decom- position of F into coexact and anticoexact parts, so by Theorem 3 the element δα is unique, and so α is unique up to a coexact form. From this solution one sees that j can be changed by an element of Y1 without affecting F . The additional constraint is dF =0= dδα + dhj, i.e., δα + hj ∈E2. (55) The constraint is equivalent to δα + hj ∈ ker(Hd). The equation (54) with the constraint (55) is simply a decomposition of an element of E2 into C2 and Y2 according to Theorem 3. We can further elaborate the condition (55). Since dF =0= dδα + dhj, we get, △α + δdα − dhj = 0. Removing from α exact part by imposing dα = 0, we get △α = dhj, (56) analogously to (53) with a potential A replaced with the three-form α. Note that, in this approach the existence of a specific A was not needed - only the fact that F ∈ E2 is sufficient. In this approach a co-potential α is more important and it has also gauge freedom. Moreover, the current also can be mod- ified by an element from Y1 without affecting F . We can also recover A. Since δα + hj ∈E2, so dA = F = dH(δα + hj), and therefore, A = df + H(δα + hj), (57) where f ∈ Λ0. This approach is more straightforward than that presented in [10] (Chapter 9) since we have the complete theory of (anti)exact and (anti)coexact forms at our disposal. The full picture is well visible when we rewrite the system (52) using the Dirac operator [1] DF = −j. (58) Then we can split F = ψ + γ, where ψ = α + β such that Dψ = 0 is the solution of the vacuum Dirac equation with α ∈ C1, β ∈ E3, and γ ∈ E2 is the solution of the nonhomogenous Dirac equation Dγ = −j, that is dγ = 0 and δγ = j. Note that if it would be that j ∈ Λ3 (hypothetical magnetic monople current) and the equations would be dF = j, δF = 0, then the procedure is similar as above with the restriction F ∈C2. Since now dj = 0, so j = dHj and the first equation is d(F −Hj) = 0, which gives F = dα+Hj for α ∈ Λ3 with the additional constraint δ(dα + Hj) = 0, i.e., dα + Hj ∈C2. Since δF = 0 so by the Poincar´elemma there exists A ∈ Λ3 such that F = δA. Then the solution for A is A = δβ + h(dα + Hj) for some β ∈ Λ4. This is dual to the classical electrodynamics presented above and by Theorem 3 it is a decomposition of an element from C2 into a sum of elements from E2 and A2. 15 5.5 Kalb-Ramond equations The Kalb-Ramond equations [17, 31] were postulated for describing charged bosonic string and, unlike Electrodynamics is the theory of a two-form F , they are equa- tions for a three-form. The literature on the subject is vast, including both phys- ical variations, e.g., [21, 9], and generalizations of an idea of using p-forms, e.g., [4, 15, 16, 30, 14]. The equations have the following form dK = 0, δK = J, (59) where K ∈ Λ3 and J ∈ Λ2. This can be further generalized to p-form electrody- namics [16], but we restrict ourselves to this simple example, since extension to different cases is straightforward. From the first equation we have that K ∈ E3 and therefore there is a Kalb- Ramond field B ∈ Λ2 such that dB = K. B is defined up to E2, therefore we can chose it as B ∈A2. From the second equation J ∈C2 and therefore J = δhJ and so δ(K − hJ) = 0. We get, analogously to the electrodynamic, that K = δβ + hJ, where β ∈ Λ4 is defined up to gauge C4. The constraint is δβ + hJ ∈E3. Since in Minkowski space dβ = 0, so the constraint is the wave equation △β = dhJ. The equations (59) can be written in the Dirac form DK = −J, (60) with the solution K = ψ + γ, where Dψ = 0 and δγ = J, as in the case of Maxwell equations. Finally, we can couple Kalb-Ramond field with the Maxwell equation. It is possible since B ∈ A2 up to closed forms, and F ∈ E2. Therefore F is a gauge field for B. We define [31] R = B + F, (61) which is possible since there is unique decomposition Λ2 = E2 ⊕A2. Then we have δR = δF + δB = j and therefore δB = 0 by Maxwell equations (52). So B ∈A2 ∩C2. Taking exterior derivative dR = dB = K, and then δK = δdB = J. Using the Dirac operator, the Kalb-Ramond-Maxwell system can be written in the compact form DR = K − j, DK = −J, (62) and can be analyzed similarly to the Maxwell equations. 5.6 Cohomotopic fermionic harmonic oscillator In [19] there was presented a homotopy analogy of a fermionic quantum harmonic oscillator and its relation to Bittner’s calculus of abstract derivative and integral [5]. Since operators δ, h are analogous to d and H, therefore, we can define the co-version of this equation. The hamiltonian operator is H¯ := hδ − δh, (63) 16 and the anticommutator relation is played by nilpotency of δ, h and the Homotopy Invariance Formula (14) for Λr, where 0 6 Conclusions The theory of anticoexact forms in analogy to antiexact forms of [10] was devel- oped in full detail. The most useful is the relation to Clifford algebra that allows us to solve the vacuum Dirac equation using (anti)(co)exact decomposition. Finally, the application of this decomposition in solving Maxwell equations of classical electrodynamics and Kalb-Ramond equations of bosonic string theory was pre- sented. Moreover, (anti)(co)exact decomposition allows tracing all ingredients of the solutions of these and other similar equations. Acknowledgments RK would like to thank Ioannis Chrysikos for the explanation of the rudiments of Clifford Algebras and Josef Silhan for support. This research was supported by the GACR grant GA19-06357S and Masaryk University grant MUNI/A/0885/2019. RK also thank the SyMat COST Action (CA18223) for partial support. A Relation to de Rham theory Here we provide some hints on how to relate the above ideas to the de Rham/Hodge theory [29, 12, 23, 8]. The Poincar´elemma is a local theory that requires a star- shaped open region, and the de Rham theory is a global one which requires in a basic formulation a compact Riemannian manifold. Therefore, the intersection of these two will be visible on a manifold compactifying a star-shaped submanifold U. Since U is diffeomorphic to an open disc D, so compactification requires two steps: making the closure of the disc D¯ and then quotient out by the relation that glues boundary giving in the end a compact manifold. This is the case for sphere or torus. The differential forms that survive such procedure must behave well under the closure and then gluing. These two topological operations impose requirements on the elements from the spaces of (co)exact, anti(co)exact forms on U. 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