Closed Algebras for Higher Rank, non Abelian Tensor Gauge Fields

Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 963 (2021) 115285 www.elsevier.com/locate/nuclphysb Closed algebras for higher rank, non-Abelian tensor gauge fields ✩ Spyros Konitopoulos Nuclear and Particle Physics, NCSR Demokritos, Greece Received 22 August 2020; received in revised form 20 November 2020; accepted 15 December 2020 Available online 21 December 2020 Editor: Hubert Saleur Abstract A systematic method is presented for the construction and classification of algebras of gauge transformations for arbitrary high rank tensor gauge fields. For any tensor gauge field of a given rank, the gauge transformation will be stated, in a generic way, via an ansatz that contains all the possible terms, with arbitrary coefficients and the maximum number of tensor gauge functions. Although the requirement for the closure of the algebra will prove to be restrictive, a variety of legitimate choices remains. Adjusting properly the values of the initial coefficients and imposing restrictions on the gauge functions, one can, on the one hand, recover the so far analysed algebras and on the other, construct new ones. The presentation of a new algebra for tensor gauge transformations is the central result of this article.  2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 . 1. Introduction The investigation of theories describing fields of arbitrary high spin dates back to 1936, when P. Dirac formulated the Relativistic Wave Equations for massless and massive fermions of spin higher than 1/2 [1]. His work was extended by M. Fierz and W. Pauli, who followed a field theoretical approach to include bosons of higher spin [2], and tuned a few months later [4]. All the attempts gave elegant results for the free particle case, but did not succeed to include ✩ This work is based on the model of Tensor Gauge Field Theory built by G. Savvidy. We provide an extension, a further development and an analytical comparison of this model in the area of Higher Spin Theory. E-mail address: [email protected]. https://doi.org/10.1016/j.nuclphysb.2020.115285 0550-3213/ 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 . S. Konitopoulos Nuclear Physics B 963 (2021) 115285 the electromagnetic interactions in a satisfactory and free of ambiguities unified framework. So far, the physical requirements for constructing higher spin theories were the invariance under Poincaré transformations and the positivity of energy after quantisation [9]. The revolutionary, group theoretical approach carried out in [3,5], together with the establishment of the Principle of Invariance under Gauge Transformations [6,7], opened the way for the introduction of new models for the description of higher spin fields with less restrictive principal requirements [8–11]. Nevertheless, a satisfactory and self contained, interacting higher spin field theory was yet to be formulated both for the abelian (electromagnetic) and the non-abelian interactions (weak, strong). In 2005, it was proposed that the description of higher spin particles can be carried out following a natural extension of the Yang-Mills (Y-M) principle [14]. The central idea was motivated from the study of Fock space of tensionless string theory with perimeter action, where it has been predicted that the ground state is infinitely degenerate and contains massless gauge fields of arbitrary high integer spin [13]. The theory for higher spin particles was developed quite extensively under the name Tensor Gauge Field Theory. It has been theoretically tested with regards to self consistency issues [16,19], properly analysed and extrapolated to provide both phenomenological and theoretical predictions [17,21–23,27–29] and served as a basis for further developments [31–33]. In particular, along with the elucidation the theory possibly offers to the aforementioned string theory grounds from which it emerged, it suggests natural extensions of the Standard Model [17] and the Poincaré algebra [25], NLO corrections to the QCD beta function [28], provides a framework for a possible resolution of the proton spin crisis [12,29] and can be implemented in the discovery of new topological invariants and potential gauge anomalies [26,32,33]. As far as one departs from the Y-M case, where the mediators of the forces are described by vector gauge fields, one faces the challenging problem of defining new algebras for tensor gauge transformations. Under the light and guidance of the Y-M principle, the starting point for the formulation of a consistent higher spin theory is the extension of gauge transformations to higher rank tensor fields. Fortunately, this investigation is facilitated by the requirement that new proposed transformations are legitimate candidates as long as they form a closed algebraic structure and therefore belong to the class of Lie algebras. Quantitatively, this is translated to the imperative that the commutator of two infinitesimal successive transformations belongs to the same set of transformations. In the following section, we give a brief review of all closed algebras that have been proposed under the framework of Tensor Gauge Field Theory and make some remarks concerning their internal structure. In section 3, we present a general method that will, on the one hand, allow the embedding of the previously mentioned algebras in a generalised framework and on the other, facilitate the investigation of new ones. Finally, we conclude with a review of the new algebra constructed by the introduced method and comment on the potentials of its usage. 2. Known closed algebras In recent papers, [14,15,18,20,24], a number of gauge transformations for bosonic tensor fields of arbitrary high rank have been constructed and proven to form a closed algebraic structure. Each of these algebras were constructed to serve a particular purpose. The first that appeared historically, under the framework of Tensor Gauge Field Theory, served as the building block for the construction of field strength tensors, by the aid of which gauge invariant Langrangians for higher rank fields were presented. We will call these transformations the standard ones, to dis2 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 criminate from the others that followed. The dual gauge transformations appeared in two versions and elucidated the fact that two seemingly different algebras can be equivalent and therefore be related by a similarity transformation. The implementation of symmetric gauge transformations was part of an attempt to formulate a theory solely for the irreducible representations of the Poincaré group, but ended to be technically impossible. Finally, the algebra of the gauge transformations used for the construction of new topological invariants in higher dimensions is our last example of extended algebras. For classification purposes, they will be called conjugate transformations. 2.1. Standard extended gauge transformations The algebra of standard extended gauge transformations appeared for the first time in [14], as the fundamental building block of non-Abelian Tensor Gauge Field Theory [15,16]. It deals with higher rank bosonic fields, Aμλ1 λ2 ...λs , which are by construction symmetric under the permutation of their lambda indices, but bear no symmetry with respect to the index μ. Besides, the tensor gauge functions, ξλ1 λ2 ...λs , introduced to define the gauge transformations of the fields Aμλ1 λ2 ...λs , are totally symmetric.1 For successively higher rank fields, the gauge transformations are given below, δξ Aμ = ∂μ ξ − ig[Aμ , ξ ]   δξ Aμλ = ∂μ ξλ − ig [Aμ , ξλ ] + [Aμλ , ξ ]   δξ Aμλ1 λ2 = ∂μ ξλ1 λ2 − ig [Aμ , ξλ1 λ2 ] + [Aμλ1 , ξλ2 ] + [Aμλ2 , ξλ1 ] + [Aμλ1 λ2 , ξ ] ...... δξ Aμλ1 ...λs = ∂μ ξλ1 ...λs − ig s   [Aμλ1 ...λi , ξλi+1 ...λs ]. (2.1) i=0 P s The first is the well known Y-M gauge transformation for the vector field. The rest defines the way the higher rank tensors transform. In particular, the non-homogeneous part of the transformations involves differentiation with respect to the first index, μ. As one can show, they form an infinite dimensional gauge group, G, with a closed algebraic structure [15], [δξ , δη ]Aμλ1 λ2 ...λs = −igδζ Aμλ1 λ2 ...λs (2.2) where, ζ ζλ ζλ1 λ2 = [ξ, η] = [ξ, ηλ ] + [ξλ , η] = [ξ, ηλ1 λ2 ] + [ξλ1 , ηλ2 ] + [ξλ2 , ηλ1 ] + [ξλ1 λ2 , η] ...... ζλ1 ...λs = [ξ, ηλ1 ...λs ] + s  [ξλi , ηλ1 ...λi−1 λi+1 ...λs ] + · · · + [ξλ1 ...λs , η] (2.3) i=1 1 The tensor gauge fields and the tensor gauge functions carry a colour index which enumerates the independent generators of the underlying SU (N ) Lie algebra. Hence, both the quantities are summation shortcuts with respect to the fundamental representation matrices of the generators of the Lie algebra, Aμλ1 λ2 ...λs = Aaμλ λ ...λ La and ξλ1 λ2 ...λs = 1 2 ξλa λ ...λ La , where [La , Lb ] = if abc Lc . s 1 2 3 s S. Konitopoulos Nuclear Physics B 963 (2021) 115285 The above algebra allowed for the definition of consistent field strength tensors which gauge transform homogeneously, for the construction of two classes of gauge invariant Lagrangians for each rank of tensor gauge fields, and for a fundamental extension of the Poincaré group [25]. 2.2. Dual extended gauge transformations Parallelly to the standard extended gauge transformations, one can define complementary ones, which end up to be equivalent to them. Dual extended gauge transformations have been formulated in two different versions [18,20], which will be reviewed below. 2.2.1. 1st Version In the first version [18], the roles of the μ and the λ indices are properly interchanged so that the inhomogeneous terms entail derivatives with respect to each of the latter indices. Up to the third rank tensor, the transformations are, δ̃ξ Aμ = ∂μ ξ − ig[Aμ , ξ ]   δ̃ξ Aμλ = ∂λ ξμ − ig [Aλ , ξμ ] + [Aμλ , ξ ]  δ̃ξ Aμλ1 λ2 = ∂λ1 ξμλ2 + ∂λ2 ξμλ1 − ig [Aλ1 , ξμλ2 ] + [Aλ2 , ξμλ1 ] + [Aμλ1 , ξλ2 ] +  +[Aμλ2 , ξλ1 ] + [Aλ1 λ2 , ξμ ] + [Aλ2 λ1 , ξμ ] + [Aμλ1 λ2 , ξ ] ...... (2.4) It has been proven [18], that the above transformations form a closed algebraic structure. Indeed, the commutator of two successive transformations is a third dual gauge transformation, [δ̃ξ , δ̃η ]Aμλ1 ...λs = −ig δ̃ζ Aμλ1 ...λs , (2.5) with = [ξ, η] ζ = [ξ, ηλ ] + [ξλ , η] ζλ = [ξ, ηλ1 λ2 ] + [ξλ1 , ηλ2 ] + [ξλ2 , ηλ1 ] + [ξλ1 λ2 , η] ζλ1 λ2 ...... ζλ1 ...λs = [ξ, ηλ1 ...λs ] + s  [ξλi , ηλ1 ...λi−1 λi+1 ...λs ] + · · · + [ξλ1 ...λs , η] (2.6) i=1 Besides this, in the same article, it has been shown that the introduction of the transpose tensor gauge fields, Ãμλ Ãμλ1 λ2 Ãμλ1 ...λs = Aλμ  1 1 Aλ1 μλ2 + Aλ2 μλ1 − Aμλ1 λ2 = 2 2 ...... s  s−1 1  Aμλ1 ...λs , Aλi μλ1 ...λi−1 λi+1 ...λs − = s s (2.7) i=1 illuminates the fact that the duality map serves as a similarity transformation between two representations of the same algebra [20]. Nevertheless, a bug in the definition of the transposition 4 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 operator which transforms the fields Aμλ1 ...λs into the fields Ãμλ1 ...λs and in particular the fact that the former is not an idempotent, resulted in the necessity of a second version of the dual gauge transformations. 2.2.2. 2nd Version In the article [20] it has been shown that if, instead of (2.7), we use the following definition for the transposed fields, Ãμλ Ãμλ1 λ2 Ãμλ1 ...λs = Aλμ  1 2 Aλ1 μλ2 + Aλ2 μλ1 − Aμλ1 λ2 = 3 3 ...... s  s −1 2  Aμλ1 ...λs , Aλi μλ1 ...λi−1 λi+1 ...λs − = s +1 s +1 (2.8) i=1 we see that the transposition operator becomes an idempotent, as it should be. However, the change in the definition of the transposition operator reflects itself upon the definition of the gauge transformations, which now acquire the form [20], δ̃ξ Aμ δ̃ξ Aμλ δ̃ξ Aμλ1 λ2 = ∂μ ξ − ig[Aμ , ξ ] = ∂λ ξμ − ig[Aλ , ξμ ] − ig[Aμλ , ξ ]  2 ∂λ1 ξμλ2 − ig[Aλ1 , ξμλ2 ] + ∂λ2 ξμλ1 − ig[Aλ2 , ξμλ1 ] − = 3  1 2 − ∂μ ξλ1 λ2 − ig[Aμ , ξλ1 λ2 ] − ig [Aμλ1 , ξλ2 ] − 3 3 2 1 2 −ig [Aλ1 λ2 , ξμ ] + ig [Aλ1 μ , ξλ2 ] − ig [Aμλ2 , ξλ1 ] − 3 3 3 2 1 −ig [Aλ2 λ1 , ξμ ] + ig [Aλ2 μ , ξλ1 ] − ig[Aμλ1 λ2 , ξ ] 3 3 ...... (2.9) which been proven to form a closed algebraic structure. We observe that the price we paid for casting the transposition operator an idempotent, was to introduce derivatives with respect to the index μ. 2.3. Symmetrized extended gauge transformations According to E. Wigner, one particle states fall into irreducible representations of the Poincaré group [3,5]. Since the irreducible component of a tensor, that describe the physical propagating modes, is its symmetric component, it sounds reasonable to specialise the gauge transformations for symmetric, over all their indices, tensor gauge fields [15]. To simplify the notation, we replace the index μ with the index λ1 , so that the symmetry properties of the tensor become more apparent. The totally symmetric version of the transformations (2.1) is given below,   δ̄ξ ASλ1 λ2 = ∂λ1 ξλ2 + ∂λ2 ξλ1 − ig [Aλ1 , ξλ2 ] + [Aλ2 , ξλ1 ] + [ASλ1 λ2 , ξ ]  δ̄ξ ASλ1 λ2 λ3 = ∂λ1 ξλ2 λ3 + ∂λ2 ξλ3 λ1 + ∂λ3 ξλ1 λ2 − ig [Aλ1 , ξλ2 λ3 ] + [Aλ2 , ξλ3 λ1 ] +  +[Aλ3 , ξλ1 λ2 ] + [ASλ2 λ3 , ξλ1 ] + [ASλ3 λ1 , ξλ2 ] + [ASλ1 λ2 , ξλ3 ] + [ASλ1 λ2 λ3 , ξ ] 5 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 ...... δ̄ξ ASλ1 ...λs s  = ∂λi ξλ1 ...λi−1 λi+1 ...λs − ig s  [Aλi , ξλ1 ...λi−1 λi+1 ...λs ] + i=1 i=1 s   + [ASλi λj , ξλ1 ...λi−1 λi+1 ...λj −1 λj +1 ...λs ] + · · · + [ASλ1 ...λs , ξ ] (2.10) i<j The above transformations have been proven to form another set of a closed algebra [15], [δ̄ξ , δ̄η ]ASλ1 λ2 ...λs = −ig δ̄ζ ASλ1 λ2 ...λs , (2.11) where ζ ζλ ζλ1 λ2 = [ξ, η] = [ξ, ηλ ] + [ξλ , η] = [ξ, ηλ1 λ2 ] + [ξλ1 , ηλ2 ] + [ξλ2 , ηλ1 ] + [ξλ1 λ2 , η] ...... ζλ1 ...λs = [ξ, ηλ1 ...λs ] + s  [ξλi , ηλ1 ...λi−1 λi+1 ...λs ] + · · · + [ξλ1 ...λs , η] (2.12) i=1 Defining, ASμλ1 ...λs ≡ s  (2.13) Aλi λ1 ...λi−1 μλi+1 ...λs , i=1 we observe2 that [18], 1 Aμλ1 ...λs + Ãμλ1 ...λs = ASμλ1 ...λs (2.14) s Nevertheless, from the above transformations, a homogeneously gauge transformed field strength tensor cannot be defined. Therefore, a gauge invariant Lagrangian, consisting solely of symmetric tensor gauge fields bilinears, cannot be constructed. This is probably a hint that the construction of a higher spin theory dictates for a further departure of the common practices which focus primarily on irreducible representations of the Poincaré group and hence, on one particle states. This is, however, the case with the Dirac equation which, inevitably describes both the particle and its antiparticle in the same 4-spinor, reducible representation. The above observation underlines the central role played by the index μ in the standard transformations, without which the construction of a gauge invariant Lagrangian would be impossible. The further relaxation of the symmetry of the indices and in particular the formulation of the theory without any symmetry properties between them was the primary motivation for the current study. 2.4. Conjugate extended gauge transformations Here, we review the algebra found in [24] and served as the basis for the construction of new topological invariants and Chern-Simons forms in higher dimensions [26,27]. The higher rank 2 Here we follow the definition of the transpose tensor gauge fields as given in the 1st version. 6 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 tensors fields and tensor gauge functions are antisymmetric under permutations of indices of the same letter, e.g. (σ1 ↔ σ2 ) and symmetric under permutations of pairs of indices of different letters (e.g. (σ1 σ2 ) ↔ (ρ1 ρ2 )). There is no symmetry with respect to the index μ. δAμ δAμσ1 σ2 δAμσ1 σ2 ρ1 ρ2 = ∂μ ξ − ig[Aμ , ξ ]   = ∂μ ξσ1 σ2 − ig [Aμ , ξσ1 σ2 ] + [Aμσ1 σ2 , ξ ]  = ∂μ ξσ1 σ2 ρ1 ρ2 − ig [Aμ , ξσ1 σ2 ρ1 ρ2 ] + [Aμσ1 σ2 , ξρ1 ρ2 ] + [Aμρ1 ρ2 , ξσ1 σ2 ] +  +[Aμσ1 σ2 ρ1 ρ2 , ξ ] (2.15) ...... The above transformations follow the spirit of the standard ones. Indeed, if we treat the additional pairs of indices, as one index, σ1 σ2 = σ̂ , the two algebras coincide. By construction, only tensor fields of odd number of indices participate in the algebra. This, as it might seem restrictive, has been proven sufficient for the construction of new topological invariants [26,27]. 3. A systematic treatment of extended gauge algebras In this section we present a systematic method for constructing algebras of gauge transformations for arbitrary high rank tensor gauge fields. For any tensor gauge field of a given rank, the gauge transformation will be stated, in a generic way, via an ansatz that contains all the possible terms, with arbitrary coefficients and the maximum number of tensor gauge functions. Although the requirement for the closure of the algebra will prove to be restrictive, a variety of legitimate choices remains. Adjusting properly the values of the initial coefficients and imposing restrictions on the gauge functions, one can, on the one hand, recover the so far analysed algebras and on the other, construct new ones. 3.1. Vector gauge fields We begin with the fundamental building block of the Standard Model, the Y-M vector gauge transformations. Since we know that the group of gauge transformations is uniquely defined, it will suffice to present a proof for the closure of the algebra, just to clarify the details of the formalism that will be adopted throughout the article. The infinitesimal, non-abelian gauge transformations are known to be, δξ Aμ = ∂μ ξ − ig[Aμ , ξ ] (3.1) In order to check that the algebra of these transformations has a closed structure, let us calculate the commutator of two successive operations on the vector field Aμ . During the process, we take into account that the operators δξ and δη , act only on Aμ , since gauge functions are not themselves gauge transformed. We have,    i i  [δξ , δη ]Aμ = δξ δη Aμ − δη δξ Aμ = [δξ Aμ , η] − [δη Aμ , ξ ] g g   = [∂μ ξ, η] − [∂μ η, ξ ] − ig [Aμ , ξ ], η] − [Aμ , η], ξ ]   = ∂μ [ξ, η] − ig Aμ , [ξ, η] , where in the last step we used the Jacobi Identity. We conclude that, 7 (3.2) S. Konitopoulos Nuclear Physics B 963 (2021) 115285 [δξ , δη ]Aμ = −igδζ Aμ , (3.3) where ζ = [ξ, η]. This shows that the commutator of two successive Y-M gauge transformations results in a third one, thus proving that they form a closed algebraic structure. 3.2. Second rank tensor gauge fields We wish to follow an analogous procedure to examine the structure of the extended nonabelian gauge transformations. In order to begin with the most general transformations for 2nd rank tensor gauge fields, we introduce two vector gauge functions, ξμ1 , ξμ2 , four coefficients, ci , and begin with the ansatz,   δξ Aμν = ∂μ ξν1 + ∂ν ξμ2 − ig [Aμ , c1 ξν1 + c2 ξν2 ] + [Aν , c3 ξμ1 + c4 ξμ2 ] + [Aμν , ξ ] . (3.4) We shall attempt a determination of the coefficients under the restriction of the closure of the algebra. The commutator of two successive operations on the tensor field Aμν is,   i  i δξ , δη Aμν = δξ (δη Aμν ) − δη (δξ Aμν ) = g g   1 2 + δξ Aμν , η − + c4 ημ = δξ Aμ , c1 ην1 + c2 ην2 + δξ Aν , c3 ημ   − δη Aμ , c1 ξν1 + c2 ξν2 − δη Aν , c3 ξμ1 + c4 ξμ2 − δη Aμν , ξ (3.5) To determine the coefficients let us first examine the structure of the zeroth order terms over the gauge coupling g. Substituting (3.4), we get,  i  1 2 2 δξ , δη Aμν = [∂μ ξ, c1 ην1 + c2 ην2 ] + [∂ν ξ, c3 ημ + c4 ημ ] + [ξ, ∂μ ην1 + ∂ν ημ ]− g − [∂μ η, c1 ξν1 + c2 ξν2 ] − [∂ν η, c3 ξμ1 + c4 ξμ2 ] − [η, ∂μ ξν1 + ∂ν ξμ2 ] + O(g) =     2 ] − [η, ξμ2 ] + = ∂μ [ξ, ην1 ] − [η, ξν1 ] + ∂ν [ξ, ημ + [∂μ ξ, c2 ην2 + (c1 − 1)ην1 ] − [∂μ η, c2 ξν2 + (c1 − 1)ξν1 ] + 1 2 + [∂ν ξ, c3 ημ + (c4 − 1)ημ ] − [∂ν η, c3 ξμ1 + (c4 − 1)ξμ2 ] + O(g), (3.6) where, O(g) = −ig 1 2 − [Aμ , ξ ], c1 ην1 + c2 ην2 + [Aν , ξ ], c3 ημ + c 4 ημ − [Aμ , η], c1 ξν1 + c2 ξν2 − [Aν , η], c3 ξμ1 + c4 ξμ2 +   + [Aμ , c1 ξν1 + c2 ξν2 ], η + [Aν , c3 ξμ1 + c4 ξμ2 ], η + [Aμν , ξ ], η −   1 2 − [Aμ , c1 ην1 + c2 ην2 ], ξ − [Aν , c3 ημ + c4 ημ ], ξ − [Aμν , η], ξ (3.7) What is obvious from (3.4) is that if the commutator of two successive transformations is to represent another transformation, the zeroth order terms in g must be written solely as partial derivatives over the two indices of the 2nd rank tensor. Hence, the only choices for the terms of 8 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 the last two lines of (3.6) are either to eliminate them by properly choosing the coefficients ci , or to absorb them in the terms of the first two lines, by imposing suitable symmetry properties and restricting the vector gauge functions ξμi . Following the first option, we keep the two vector gauge functions ξμi , i = 1, 2, independent and eliminate the last four terms of (3.6) by setting c1 = c4 = 1 , c2 = c3 = 0 Then we get,  i  δξ , δη Aμν = ∂μ ζν1 + ∂ν ζμ2 + O(g) g (3.8) where, i ζμi = [ξ, ημ ] + [ξμi , η] , i = 1, 2 (3.9) In order to guarantee that the algebra closes for this specific choice of the ci s and the new gauge functions ζμ1 , ζμ2 , and furthermore to determine ζ , we need to examine the O(g) terms. We have, i 2 2 O(g) = [[Aμ , ξ ], ην1 ] + [[ην1 , Aμ ], ξ ] + [[Aν , ξ ], ημ ] + [[ημ , Aν ], ξ ] + g +[[Aμ , ξν1 ], η] + [[η, Aμ ], ξν1 ] + [[Aν , ξμ2 ], η] + [[η, Aν ], ξμ2 ] + +[[Aμν , ξ ], η] + [[η, Aμν ], ξ ] = 2 = [Aμ , [ξ, ην1 ] − [η, ξν1 ]] + [Aν , [ξ, ημ ] − [η, ξμ2 ]] + [Aμν , [ξ, η]], (3.10) where in the last step we employed Jacobi Identity, once for each pair of adjacent terms. We conclude that, O(g) = −ig [Aμ , ζν1 ] + [Aν , ζμ2 ] + [Aμν , ζ ] , with ζ = [ξ, η]. Hence, the gauge transformations,   δξ Aμν = ∂μ ξν1 + ∂ν ξμ2 − ig [Aμ , ξν1 ] + [Aν , ξμ2 ] + [Aμν , ξ ] , (3.11) (3.12) form a closed group algebra with [δξ , δη ]Aμν = −igδζ Aμν (3.13) and ζ = [ξ, η] 1 ζμ1 = [ξ, ημ ] + [ξμ1 , η] 2 ζμ2 = [ξ, ημ ] + [ξμ2 , η] (3.14) What we observe from (3.12) is that the vector field, Aμ , participates in the gauge transformation of Aμν only as part of the covariant derivative of the two gauge functions.3 Thus far, we have managed to merge the two independent, standard and dual, extended gauge transformations into a more general one, indicating a higher symmetry for the system of 2nd rank 3 The standard geometrical interpretation of A , as a principal bundle connection is not directly extrapolated for μ analogous interpretations of the higher rank gauge fields. This will be examined in subsequent studies. 9 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 tensor gauge fields. Before we continue the same procedure for the 3rd rank tensor gauge fields, let us recover the algebras of the standard (2.1) and dual (2.4) transformations for the 2nd rank tensor gauge fields. Let us return to (3.6) and check if there is an alternative way to nullify the terms that cannot be written as partial derivatives with respect to the indices μ and ν. If we set the second vector gauge function equal to zero, we see that the algebra closes if and only if, c1 = 1, c3 = 0. This, obviously, recovers the standard transformations for the 2nd rank field (2.1). If, on the other hand, we set the first vector gauge function to equal to zero, the algebra closes provided, c2 = 0, c4 = 1. This recovers the dual transformations for the 2nd rank field (2.4). Lastly, setting the two vector gauge functions equal and symmetrising the tensor gauge field over its two indices, we recover the first equation of the symmetrized transformations (2.10). 3.3. Third rank tensor gauge fields In an analogous way followed in the case of the 2nd rank tensor gauge fields, let us introduce i , i = 1, 2, 3, and begin with the ansatz,4 three 2nd rank gauge functions ξμν  1 2 3 1 2 3 δξ Aμνλ = ∂μ ξνλ + ∂ν ξλμ + ∂λ ξμν − ig [Aμ , ξνλ ] + [Aν , ξλμ ] + [Aλ , ξμν ]+  + [Aμν , c1 ξλ1 + c2 ξλ2 ] + [Aνλ , c3 ξμ1 + c4 ξμ2 ] + [Aλμ , c5 ξν1 + c6 ξν2 ] + [Aμνλ , ξ ] (3.15) We will try to determine the 6 coefficients ci so that the algebra of the gauge transformations forms a closed structure. We need to underline that the 3rd rank tensor gauge field, Aμνλ , contrary to its standard definition [14], bares no symmetry under the permutation of its last two indices. i which are no longer symmetric. This is The same holds for the three tensor gauge functions, ξνλ a significant departure which will hopefully lead to an enhancing of the symmetry of the theory of non-abelian tensor gauge fields. The commutator of two successive gauge transformations on the field Aμνλ gives,  i i [δξ , δη ]Aμνλ = δξ (δη Aμνλ ) − δη (δξ Aμνλ ) = g g 2 3 1 2 1 ]− ] − [δη Aν , ξλμ ] + [δξ Aλ , ημν ] − [δη Aμ , ξνλ ] + [δξ Aν , ηλμ = [δξ Aμ , ηνλ 3 1 2 −[δη Aλ , ξμν ] + [δξ Aμν , c1 ηλ1 + c2 ηλ2 ] + [δξ Aνλ , c3 ημ + c4 ημ ]+ +[δξ Aλμ , c5 ην1 + c6 ην2 ] − [δη Aμν , c1 ξλ1 + c2 ξλ2 ] − [δη Aνλ , c3 ξμ1 + c4 ξμ2 ]− −[δη Aλμ , c5 ξν1 + c6 ξν2 ] + [δξ Aμνλ , η] − [δη Aμνλ , ξ ] (3.16) To determine the coefficients for the gauge transformation (3.15), we will first examine the structure of the zeroth order terms over the gauge coupling g. As is indicated in (3.15) a necessary condition for the closure of the algebra is that the zeroth order terms must be written as partial derivatives with respect to the indices μ, ν and λ. Focusing on the zeroth order terms over the coupling g, and substituting (3.1), (3.12) and (3.15) in the above equation,5 one gets, 4 Directed from the case of the 2nd rank field, the non-homogeneous, derivative terms combine with the first O(g) terms to give the covariant derivative of the higher rank gauge functions. With this in mind, we do not add coefficients on the first three O(g) terms. 5 The substitution of (3.12) does not harm generality since it is the most general transformation for 2nd rank fields which is compatible with the closure of the algebra. 10 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 i 1 2 3 + ∂ν ξ, ηλμ + ∂λ ξ, ημν − [δξ , δη ]Aμνλ = ∂μ ξ, ηνλ g 1 2 3 − ∂μ η, ξνλ − ∂ν η, ξλμ − ∂λ η, ξμν + 1 2 + + c4 ημ + ∂μ ξν1 + ∂ν ξμ2 , c1 ηλ1 + c2 ηλ2 + ∂ν ξλ1 + ∂λ ξν2 , c3 ημ 2 , c1 ξλ1 + c2 ξλ2 − + ∂λ ξμ1 + ∂μ ξλ2 , c5 ην1 + c6 ην2 − ∂μ ην1 + ∂ν ημ 1 + ∂μ ηλ2 , c5 ξν1 + c6 ξν2 + − ∂ν ηλ1 + ∂λ ην2 , c3 ξμ1 + c4 ξμ2 − ∂λ ημ 1 2 3 1 2 3 +[∂μ ξνλ + ∂ν ξλμ + ∂λ ξμν , η] − [∂μ ηνλ + ∂ν ηλμ + ∂λ ημν , ξ ] + O(g) =     1 2 2 1 = ∂μ [ξ, ηνλ ] − [η, ξνλ ] + ∂ν [ξ, ηλμ ] − [η, ξλμ ] +   3 3 +∂λ [ξ, ημν ] − [η, ξμν ] +   2 , ξλ1 + +c1 ∂μ ξν1 + ∂ν ξμ2 , ηλ1 − ∂μ ην1 + ∂ν ημ   2 +c2 ∂μ ξν1 + ∂ν ξμ2 , ηλ2 − ∂μ ην1 + ∂ν ημ , ξλ2 +   1 − ∂ν ηλ1 + ∂λ ην2 , ξμ1 + +c3 ∂ν ξλ1 + ∂λ ξν2 , ημ   2 − ∂ν ηλ1 + ∂λ ην2 , ξμ2 + +c4 ∂ν ξλ1 + ∂λ ξν2 , ημ   1 +c5 ∂λ ξμ1 + ∂μ ξλ2 , ην1 − ∂λ ημ + ∂μ ηλ2 , ξν1 +   1 (3.17) +c6 ∂λ ξμ1 + ∂μ ξλ2 , ην2 − ∂λ ημ + ∂μ ηλ2 , ξν2 + O(g) Up to the zeroth order over the gauge coupling, we have nine terms in total. The first three have exactly the desired property for the closure of the algebra. We can isolate the parts of the last six terms, that contain commutators of both the vector gauge functions and can be written as partial differentials with respect to the three indices, and insert them in the first three terms.6    i 1 1 [δξ , δη ]Aμνλ = ∂μ [ξ, ηνλ ] + c2 [ξν1 , ηλ2 ] − [ην1 , ξλ2 ] + ] − [η, ξνλ g    2 2 2 +∂ν [ξ, ηλμ ] − [ηλ1 , ξμ2 ] + ] + c1 [ξλ1 , ημ ] − [η, ξλμ    3 3 1 ] − [η, ξμν ] + c3 [ξμ1 , ην2 ] − [ημ , ξν2 ] + +∂λ [ξ, ημν     2 +c1 [∂μ ξν1 , ηλ1 ] − [∂μ ην1 , ξλ1 ] + c4 [∂λ ξν2 , ημ ] − [∂λ ην2 , ξμ2 ] +     2 1 +c2 [∂ν ξμ2 , ηλ2 ] − [∂ν ημ , ξλ2 ] + c5 [∂λ ξμ1 , ην1 ] − [∂λ ημ , ξν1 ] +     1 +c3 [∂ν ξλ1 , ημ ] − [∂ν ηλ1 , ξμ1 ] + c6 [∂μ ξλ2 , ην2 ] − [∂μ ηλ2 , ξν2 ] +   2 +(c4 − c1 ) [ξμ2 , ∂ν ηλ1 ] − [ημ , ∂ν ξλ1 ] +   +(c5 − c2 ) [ξν1 , ∂μ ηλ2 ] − [ην1 , ∂μ ξλ2 ] +   1 +(c6 − c3 ) [ξν2 , ∂λ ημ ] − [ην2 , ∂λ ξμ1 ] + O(g) (3.18) 6 This will simplify the subsequent calculations and especially the cases where one of the vector gauge functions is neglected. 11 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 Up to the zeroth order over the gauge coupling, we end up with twelve terms. Further, one can see that the symmetric parts of the fourth up to the ninth term can also be isolated and written as partial derivatives with respect to the indices μ, ν λ. For example, one can easily show that the symmetric part, over the indices ν and λ, of the fourth term can be written as a partial derivative with respect to the index μ,  1  1 1 1 1 (3.19) ] = ∂μ [ξν1 , ηλ1 ] + [ξλ1 , ην1 ] , ξλ) ] − [∂μ η(ν , ηλ) [∂μ ξ(ν 2 Such is the case for the remaining terms, symmetrized over the proper indices. With these in mind we get,    i 1 1 [δξ , δη ]Aμνλ = ∂μ [ξ, ηνλ ] − [η, ξνλ ] + c2 [ξν1 , ηλ2 ] − [ην1 , ξλ2 ] + g  c6  2 2  c1  1 1 [ξν , ηλ ] + [ξλ1 , ην1 ] + [ξλ , ην ] + [ξν2 , ηλ2 ] + + 2 2   1 2  2 2 +∂ν [ξ, ηλμ ] − [η, ξλμ ] + c1 [ξλ , ημ ] − [ηλ1 , ξμ2 ] +  c3  1 1  c2  2 2 2 [ξμ , ηλ ] + [ξλ2 , ημ [ξλ , ημ ] + [ξμ1 , ηλ1 ] + + ] + 2 2    3 3 1 +∂λ [ξ, ημν ] − [η, ξμν ] + c3 [ξμ1 , ην2 ] − [ημ , ξν2 ] +  c5  1 1  c4  2 2 1 + [ξν , ημ ] + [ξμ2 , ην2 ] + [ξμ , ην ] + [ξν1 , ημ ] + 2 2     2 2 2 2 1 1 1 1 ] + , ξμ] ] − [∂λ η[ν , ημ] +c1 [∂μ ξ[ν , ηλ] ] − [∂μ η[ν , ξλ] ] + c4 [∂λ ξ[ν     1 1 1 1 2 2 2 2 ] + , ξν] ] − [∂λ η[μ , ην] ] + c5 [∂λ ξ[μ , ξλ] ] − [∂ν η[μ , ηλ] +c2 [∂ν ξ[μ     2 2 2 1 2 1 1 1 ] + , ξν] ] − [∂μ η[λ ] + c6 [∂μ ξ[λ , ην] , ξμ] ] − [∂ν η[λ , ημ] +c3 [∂ν ξ[λ   2 +(c4 − c1 ) [ξμ2 , ∂ν ηλ1 ] − [ημ , ∂ν ξλ1 ] +   +(c5 − c2 ) [ξν1 , ∂μ ηλ2 ] − [ην1 , ∂μ ξλ2 ] +   1 ] − [ην2 , ∂λ ξμ1 ] + O(g) (3.20) +(c6 − c3 ) [ξν2 , ∂λ ημ Now we have to nullify the last 9 terms and there are many ways to do this. The forth till ninth terms can be omitted, either by nullifying the respective coefficients, or by imposing suitable symmetric properties over the indices of the 3rd rank tensor gauge field or by restricting one vector gauge function (neglecting it, equating it with the other etc.). The last three terms vanish either by equating the suitable pairs of ci s, or by equating the two vector gauge functions and imposing proper symmetries on their indices, or by neglecting one of them. Let us examine the different cases. If we keep all the tensor gauge functions independent, and do not assume any symmetry properties on the indices of the tensor gauge fields, then, for the closure of the algebra, it is necessary to set, c1 = c2 = c3 = c4 = c5 = c6 = 0 (3.21) Then we get,     i 2 2 1 1 [δξ , δη ]Aμνλ = ∂μ [ξ, ηνλ ] + ] − [η, ξλμ ] + ∂ν [ξ, ηλμ ] − [η, ξνλ g   3 3 ] − [η, ξμν ] +∂λ [ξ, ημν 12 (3.22) S. Konitopoulos Nuclear Physics B 963 (2021) 115285 Let us now examine the closure of the full algebra of the transformations,   1 2 3 1 2 3 δAμνλ = ∂μ ξνλ + ∂ν ξλμ + ∂λ ξμν − ig [Aμ , ξνλ ] + [Aν , ξλμ ] + [Aλ , ξμν ] + [Aμνλ , ξ ] (3.23) taking into consideration all the higher order terms in g. We get, i 1 2 3 [δξ , δη ]Aμνλ = [δξ Aμ , ηνλ ] + [δξ Aν , ηλμ ] + [δξ Aλ , ημν ] + [δξ Aμνλ , η] − g 1 2 3 ] − [δη Aν , ξλμ ] − [δη Aλ , ξμν ] − [δη Aμνλ , ξ ] = −[δη Aμ , ξνλ 2 1 ]+ ] + [∂ν ξ − ig[Aν , ξ ], ηλμ = [∂μ ξ − ig[Aμ , ξ ], ηνλ 3 1 2 3 ] + ∂μ ξνλ + ∂ν ξλμ + ∂λ ξμν − +[∂μ ξ − ig[Aλ , ξ ], ημν   3 2 1 −ig [Aμ , ξνλ ] + [Aλ , ξμν ] + [Aμνλ , ξ ] , η − ] + [Aν , ξλμ 1 2 ] − [∂ν η − ig[Aν , η], ξλμ ]− − [∂μ η − ig[Aμ , η], ξνλ 3 2 3 1 −[∂μ η − ig[Aλ , η], ξμν + ∂λ ημν − + ∂ν ηλμ ] − ∂μ ηνλ   1 2 3 −ig [Aμ , ηνλ ] + [Aν , ηλμ ] + [Aλ , ημν ] + [Aμνλ , η] , ξ =     2 2 1 1 ] + ] − [η, ξλμ ] + ∂ν [ξ, ηλμ ] − [η, ξνλ = ∂μ [ξ, ηνλ   3 3 +∂λ [ξ, ημν ] − [η, ξμν ] −  1 1 2 2 −ig Aμ , [ξ, ηνλ ] − [η, ξνλ ] + Aν , [ξ, ηλμ ] − [η, ξλμ ] +    3 3 + Aλ , [ξ, ημν ] − [η, ξμν ] + Aμνλ , [ξ, η] , (3.24) where in the second step we substituted (3.1), (3.23) and in last we employed the Jacobi Identity. We conclude that, [δξ , δη ]Aμνλ = −igδζ Aμνλ , (3.25) with, i i i ζμν = [ξ, ηνλ ] + [ξνλ , η] , i = 1, 2, 3 ζ = [ξ, η], (3.26) which proves that the transformations (3.23) form a closed algebraic structure. As in the case of the 2nd rank tensor field, the vector field, Aμ , participates in the gauge transformation of Aμνλ only as part of the covariant derivative with respect to the three gauge functions. Now, let us explore the case where the second vector gauge function and the second and third 2nd rank tensor gauge functions are set to zero, 2 3 ξμ2 = ξμν = ξμν =0 (3.27) Also, let us symmetrize the 3rd rank tensor gauge field over its last two indices, ν and λ. The last three, together with the fifth, sixth and ninth terms of (3.20), vanish identically because all of 2 ). The fourth term vanishes because of them contain the second vector gauge function (ξμ2 or ημ the symmetrization over the indices ν and λ. In order to get rid of the seventh and eighth terms, it is sufficient to set, 13 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 c3 = c5 (3.28) Then, taking the only surviving 2nd rank tensor gauge function ξμν symmetric,7 the gauge transformation (3.15) simplifies to,    δAμλ1 λ2 = ∂μ ξλ1 λ2 − ig [Aμ , ξλ1 λ2 ] + c1 [Aμλ1 , ξλ2 ] + [Aμλ2 , ξλ1 ] +    + c3 [Aλ1 μ , ξλ2 ] + [Aλ2 μ , ξλ2 ] + [Aλ1 λ2 , ξμ ] + [Aλ2 λ1 , ξμ ] + [Aμλ1 λ2 , ξ ] (3.29) which for c1 = 1 and c3 = 0, coincides with the standard gauge transformation (2.1). In an analogous way, the 1st version of the dual transformations for the 3rd rank field can be 1 = ξ 2 = ξ 1 = 0, symmetrising the third tensor gauge function, ξ 3 , and recovered by setting ξμν μν μ μν 3rd rank tensor gauge field over its last two indices. Then, the closure of the algebra forces, c2 = c4 , (3.30) so that the gauge transformation becomes,  δξ Aμλ1 λ2 = ∂λ1 ξμλ2 + ∂λ2 ξμλ1 − ig [Aλ1 , ξμλ2 ] + [Aλ2 , ξμλ1 ] +   +c2 [Aμλ1 , ξλ2 ] + [Aμλ2 , ξλ1 ] + [Aλ1 λ2 , ξμ ] + [Aλ2 λ1 , ξμ ] +    +c6 [Aλ1 μ ξλ2 ] + [Aλ2 μ ξλ1 ] + [Aμλ1 λ2 , ξ ] , (3.31) which for the particular choice c2 = 1, c6 = 0, coincides with (2.4). It is not hard to see that the 2nd version of the dual transformation is recovered if we set, 3 2 1 ξμν = 0 , ξμν = −4ξμν , ξμ1 = 0 , c2 = c4 = 1 , c6 = −1/2, (3.32) and symmetrise and the 3rd rank tensor gauge field over its last two indices together with the surviving 2nd rank tensor gauge functions. Then the exact form of (2.9) is recovered if we normalise 4 1 1 (3.33) ξμν = − ξμν , ξμ2 = ξμ 3 3 Finally, let us examine the case where the 3rd rank tensor is antisymmetric over its last two indices. If we also set, 2 3 ξμ2 = ξμν = ξμν =0 (3.34) it is easy to see from (3.18) that the nullification of the terms that cannot be written as partial differentials is achieved only if we set, c1 = 0 , c5 = −c3 (3.35) Then (3.15) reduces to,  c3  [Aσ1 σ2 , ξμ ] − [Aσ2 σ1 , ξμ ] + [Aσ1 μ , ξσ2 ] − δξ Aμσ1 σ2 = ∂μ ξσ1 σ2 − ig [Aμ , ξσ1 σ2 ] + 2  (3.36) −[Aσ2 μ , ξσ1 ] + [Aμσ1 σ2 , ξ ] , 7 We renamed the indices (ν, λ) to (λ , λ ) respectively, so that the their permuting symmetry is more apparent. 1 2 14 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 where the tensor gauge function, ξμν , is antisymmetric under the permutation of its two indices. Now the commutator of two successive transformations (3.18) reduces to,   i [δξ , δη ]Aμσ1 σ2 = ∂μ [ξ, ησ1 σ2 ] + [ξσ1 σ2 , η] + g      +c3 ∂σ1 [ξσ2 , ημ ] + [ξμ , ησ2 ] − ∂σ2 [ξσ1 , ημ ] + [ξμ , ησ1 ] + O(g) (3.37) Since the closure of the algebra requires that partial derivatives in terms of the indices σ1 and 2 , ξ 3 , the only legitimate possibility σ2 should be absent when we ignore the gauge functions ξμν μν is to set c3 = 0. Thus, we recover the algebra (2.15) which has been proven to form a closed structure [24]. 3.4. The general case The method we developed seems to be sufficiently generic to accomplish two aims. On the one hand to recover all the existing closed algebras of higher rank tensor gauge fields and thus to embed them in a more general framework, on the other to provide a tool for the investigation of new ones. Implementing it as a tool, the existence of a new closed algebra for higher rank tensor gauge fields became apparent. Hence, we have proved that up to the tensor of the third tank, the following transformations provide a closed algebraic structure, δξ Aμ = ∂μ ξ − ig[Aμ , ξ ]   δξ Aμν = ∂μ ξν1 + ∂ν ξμ2 − ig [Aμ , ξν1 ] + [Aν , ξμ2 ] + [Aμν , ξ ]   1 2 3 1 2 3 + ∂ν ξλμ + ∂λ ξμν ] + [Aν , ξλμ ] + [Aλ , ξμν − ig [Aμ , ξνλ ] + [Aμνλ , ξ ] δξ Aμνλ = ∂μ ξνλ (3.38) It seems natural to postulate that the gauge transformations for the general case of a r-th rank gauge field are given by, δξ Aμ1 ...μr = r  ∂μi ξμi i+1 ...μr μ1 ...μi−1 − ig i=1 r   [Aμi , ξμi i+1 ...μr μ1 ...μi−1 ] + [Aμ1 ...μr , ξ ] . i=1 (3.39) Let us prove that the postulated generalisation forms a closed algebra. The commutator of two successive transformations gives,  i i [δξ , δη ]Aμ1 ...μr = δξ (δη Aμ1 ...μr ) − δη (δξ Aμ1 ...μr ) = g g r r   i = [δξ Aμi , ημ ] − [δη Aμi , ξμi i+1 ...μr μ1 ...μi−1 ] + ...μ μ ...μ r 1 i+1 i−1 i=1 i=1 +[δξ Aμ1 ...μr , η] − [δη Aμ1 ...μr , ξ ] = = r  i ∂μi ξ − ig[Aμi , ξ ], ημ − i+1 ...μr μ1 ...μi−1 i=1 15 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 − r  ∂μi η − ig[Aμi , η], ξμi i+1 ...μr μ1 ...μi−1 + i=1 + r  ∂μi ξμi i+1 ...μr μ1 ...μi−1 − ig Aμi , ξμi i+1 ...μr μ1 ...μi−1 , η − r  i i ∂μi ημ ,ξ + − ig Aμi , ημ i+1 ...μr μ1 ...μi−1 i+1 ...μr μ1 ...μi−1 i=1    −ig Aμ1 ...μr , ξ , η − − i=1    +ig Aμ1 ...μr , η , ξ = r    i i + ξ , η − ∂μi ξ, ημ = μi+1 ...μr μ1 ...μr−1 i+1 ...μr μ1 ...μr−1 i=1 −ig r  i=1    i  i Aμi , ξ, ημ + ξμi+1 ...μi μ1 ...μr−1 , η + i+1 ...μi μ1 ...μr−1    + Aμ1 ...μr , [ξ, η] (3.40) In the third step we substituted (3.1) and (3.39) and in the last, we employed the Jacobi Identity, where needed. We conclude that, [δξ , δη ]Aμ1 ...μr = −igδζ Aμ1 ...μr , (3.41) with ζ = [ξ, η] ζμi i+1 ...μr μ1 ...μi−1 i + ξμi i+1 ...μr μ1 ...μi−1 , η , = ξ, ημ i+1 ...μr μ1 ...μi−1 (3.42) hence that the general gauge transformation (3.39) forms a closed algebraic structure. At this point it is would be instructive to examine separately the case of totally antisymmetric higher rank fields. In this case, the gauge transformations of the fields can be cast into a free index notation formalism, which will allow a close comparison to the algebra of gauge transformations as predicted in the Free Differential Algebra’s framework [30,32,33]. It is easy to see that for the case of totally antisymmetric fields, the above stated algebra, acquires the form, δξ Aμ = ∂μ ξ − ig[Aμ , ξ ]   δξ Aμν = ∂μ ξν − ∂ν ξμ − ig [Aμ , ξν ] − [Aν , ξμ ] + [Aμν,ξ ]   δξ Aμνλ = ∂μ ξνλ + ∂ν ξλμ + ∂λ ξμν − ig [Aμ , ξνλ ] + [Aν , ξλμ ] + [Aλ , ξμν ] + [Aμνλ , ξ ] .........   δξ Aμ1 ...μr = ∂[μ1 ξμ2 ...μr ] − ig [A[μ1 , ξμ2 ...μr ] ] + [Aμ1 ...μr , ξ ] (3.43) It is straightforward to verify that the above transformations form a closed algebraic structure which can be reformulated in a free index notation, 16 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 δA = dξ + [A, ξ ] = Dξ δA2 = dξ + [A, ξ1 ] + [A2 , ξ ] = Dξ1 + [A2 , ξ ] δA3 = dξ2 + [A, ξ2 ] + [A3 , ξ ] = Dξ2 + [A3 , ξ ] ......... δAn = dξn−1 + [A, ξn−1 ] + [An , ξ ] = Dξn−1 + [An , ξ ], (3.44) where Ap = Aaμ1 ...μn dx μ1 ∧ · · · ∧ dx μp , ξp = ξμa 1 ...μp La dx μ1 ∧ · · · ∧ dx μp , the gauge potential and gauge function p-forms respectively and Dξp = dξp + [A, ξp ], the (p+1)-form of the exterior covariant derivative of a p-form gauge function. On the other hand, the components of Y-M gauge fields, Aaμ , can be identified with the soft, left-invariant, cotangent, 1-form fields, μA , which satisfy the softly deformed Maurer-Cartan equations, 1 dμA + C ABC μB ∧ μC = R A , 2 (3.45) along with the integrability condition d 2 μA = 0 (3.46) The extension of the above dual formulation to a multi-form basis, { i(p) }, leads to the so called Free Differential Algebra (FDA) diffeomorphisms which allow the identification of the basis pforms of the FDA with the totally antisymmetric p-rank fields, {Aaμ1...μp }, of tensor gauge field theory. Hence, FDA’s diffeomorphisms imply the extended gauge transformations [33], δA(p+1) = dξ (p) + N  n=1 A(p) CB1(p B ...Bn(pn ) 1 ) 2(p2 ) AB1 (p1 ) ∧ ξ B2 (p2 ) ∧ · · · ∧ ξ Bn (pn ) , (3.47) where in the case of a minimal FDA [30] and for n = 2, the structure constants C ABC are identified with the structure constants of the underlying, SU (N), Lie algebra of extended Y-M fields, f abc [14], δA = Dξ δA2 = Dξ1 + [A2 , ξ ] δA3 = Dξ2 + [A3 , ξ ] δA5 = Dξ4 + [A3 , ξ2 ] + [A5 , ξ ] (3.48) ......... The above diffeomorphisms have been shown [33] to coincide, up to coefficients factors, with the algebra of Conjugate Extended Gauge Transformations (2.15). As regards the new algebra, it is obvious that the results of the minimal FDAs, coincide with (3.44), up to the diffeomorphisms of the 3-form tensor. Following the spirit of this comparison, it can be stated that the new algebra (3.44) constitutes a subset of the minimal FDA where the intermediate fields are truncated over so that the only ones that participate in the expressions of exterior derivatives and diffeomorphisms of the p-form basis co-vectors, are the fields themselves and the 1-form connection. 17 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 4. Conclusions We presented a general method for constructing extended gauge transformations that include bosonic fields of arbitrary high spin under the requirement that they should form a closed algebraic structure. After having recovered the known closed algebras of extended gauge transformations (2.1), (2.4), (2.9), (2.10) and (2.15), by properly adjusting the initial coefficients and restricting the tensor gauge functions of the transformations, we advocated for the existence of the following new algebra of extended gauge transformations, δξ Aμ = ∂μ ξ − ig[Aμ , ξ ]   δξ Aμν = ∂μ ξν1 + ∂ν ξμ2 − ig [Aμ , ξν1 ] + [Aν , ξμ2 ] + [Aμν , ξ ]   3 2 3 1 2 1 ] + [Aλ , ξμν ] + [Aμνλ , ξ ] ] + [Aν , ξλμ + ∂λ ξμν − ig [Aμ , ξνλ + ∂ν ξλμ δξ Aμνλ = ∂μ ξνλ ......... δξ Aμ1 ...μr = r  ∂μi ξμi i+1 ...μr μ1 ...μi−1 − ig r   [Aμi , ξμi i+1 ...μr μ1 ...μi−1 ] + [Aμ1 ...μr , ξ ] , i=1 i=1 (4.1) which was proven to be closed under the commutator, [δξ , δη ]Aμ1 ...μr = −igδζ Aμ1 ...μr , (4.2) with, ζ = [ξ, η] ζμi i+1 ...μr μ1 ...μi−1 i + ξμi i+1 ...μr μ1 ...μi−1 , η = ξ, ημ i+1 ...μr μ1 ...μi−1 (4.3) The fact that the new algebra introduces the same number of gauge functions as the rank of the tensor gauge field indicates a higher symmetry for the system and sounds promising as regards the cancellation of non-propagating degrees of freedom. It is worth mentioning that at each step of the transformations, apart from the field transformed, no lower rank tensor fields participate other than the Y-M field. The latter participate exactly in the way to define the covariant derivative on each of the tensor gauge functions, ∇μi ξμi i+1 ...μr μ1 ...μi−1 ≡ ∂μi ξμi i+1 ...μr μ1 ...μi−1 − ig[Aμi , ξμi i+1 ...μr μ1 ...μi−1 ], hence playing its custom, geometrical role as a principal bundle connection. Furthermore, it has been seen that implementing the new algebra gauge transformations for the case of totally antisymmetric higher rank fields, results in a closed structure that coincides with a subset of minimal FDA diffeomorphisms where fields of intermediate forms do not participate. The potential usage of the new algebra is still under investigation. It remains an open question whether an extended gauge invariant Lagrangian can be constructed or whether the extended symmetry manifests itself only in critical phenomena. Besides, the extended algebra for tensor gauge fields can be put in conjunction with the vierbein and spin connection field transformations on the grounds of Poincaré’s gauge theory [34–36]. The latter is the main objective for subsequent research. 18 S. Konitopoulos Nuclear Physics B 963 (2021) 115285 CRediT authorship contribution statement Spyros Konitopoulos: Conceptualization, Methodology, Presentation, Editing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The author would like to thank prof. G. Savvidy for his useful remarks and creative discussions we had together. This article is dedicated to my students in the University of Patras, during the years 2016-2019. Their direct expression of support was invaluable. References [1] P. Dirac, Relativistic wave equations, Proc. R. Soc. A 155 (1936) 447. [2] M. 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