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A Topological Model of Composite Preons from the Minimal Ideals of Two Clifford Algebras

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A Topological Model of Composite Preons from the Minimal Ideals of Two Clifford Algebras Physics Letters B 808 (2020) 135687 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb A topological model of composite preons from the minimal ideals of two Clifford algebras Niels G. Gresnigt a r t i c l e i n f o a b s t r a c t Article history: This paper demonstrates a direct correspondence between a recent algebraic characterization of leptons Received 29 April 2020 and quarks as basis elements of the minimal one-sided ideals of the complex Clifford algebras C(6) Received in revised form 2 August 2020 and C(4), shown earlier to transform as a single generation of leptons and quarks under the Standard Accepted 4 August 2020 Model’s unbroken SU(3)c × U (1)em and SU(2)L gauge symmetries respectively, and a topological Available online 11 August 2020 formulation of the Harari-Shupe preon model in which leptons and quarks are represented in terms Editor: A. Ringwald of braids. It was previously shown that mapping a Witt basis of C(6) to particular braids in the circular Artin c braid group B3 makes it possible to replicate the topological structure describing electrocolor symmetries in this preon model. This paper extends this curious correspondence, which involves only the minimal ideals of C(6) under SU(3)c × U (1)em, to include the SU(2)L chiral weak symmetry. This is achieved c by mapping a Witt basis of an additional C(4) algebra to braids in B3, taken to be a subgroup of B3. The braids corresponding to the charged vector bosons are determined, and it is demonstrated that chiral weak interactions can be described via the composition of braids. © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1 1. Introduction complex Clifford algebras. These minimal ideals are algebraic spinors, corresponding to irreducible representations of spin group. Equipping a complex Clifford algebra with a Witt basis consisting Preon models attempt to derive the quantum numbers and in- of nilpotent creation and annihilation operators, decomposes the teractions of leptons and quarks from a smaller set of constituent algebra into two minimal ideals. Each ideal consists of an exte- particles. The famous Harari-Shupe model is based on just two rior algebra generated from the creation or annihilation operators fundamental preons [1,2]. These preons must be confined, and the acting on a primitive idempotent. The associated gauge symme- appealing simplicity of the model is unfortunately sacrificed when tries correspond to the unitary symmetries that preserve these some QCD-like confinement mechanism is introduced. one-sided ideals. Using this construction, the minimal left ideals A topological realization of the Harari-Shupe model, which rep- of C(6) were previously shown to transform like one generation resents leptons and quarks in terms of simple braids composed of leptons and quarks under the unbroken SU(3)c × U (1)em gauge of three twisted ribbons was proposed in [3]. The binding of pre- symmetry [8]. Similarly, the minimal right ideals of C(4) can be ons in this case is purely topological. Color and electric charge are shown to transform as one generation of chiral weak states under represented via the twisting of ribbons, whereas the braiding of 2 SU(2)L [14]. The C(6) and C(4) ideals may be combined into these ribbons encodes the chiral weak symmetry. The weak inter- a unified picture based on C(10) [15,16]. actions are represented topologically via braid composition. One The purpose of this paper is to establish a structural correspon- exciting development has been the encoding of this braid model dence between the algebraic characterization of leptons and quarks into certain background independent theories of quantum gravity. This is possible by identifying the braids with emergent topologi- cal excitations of spin networks, and provides a novel approach to 1 At a deeper level, these Clifford algebras can often be generated from the four a unified theory of both matter and spacetime [4]. normed division algebras via their adjoint actions. For some approaches that look Another model that has become popular recently represents at the division algebra as a basis for SM physics, the reader is directed to [5–13]. 2 leptons and quarks in terms of the one-sided minimal ideals of The choice of a left or right minimal ideal is inconsequential. Here we choose to construct minimal left ideals for C(6) and minimal right ideals for C(4). This choice keeps the distinction between C(6) and C(4) apparent and later gives us a topological representation where the twist structure is written first followed by E-mail address: [email protected]. the braid structure, as in the top half of Fig. 1. https://doi.org/10.1016/j.physletb.2020.135687 0370-2693/© 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. 2 N.G. Gresnigt / Physics Letters B 808 (2020) 135687 Fig. 2. One can go from the framed braid on the left (with braiding but no twisting) Fig. 1. In the braid model, leptons and quarks are represented as braids of to the one on the right (with twisting but no braiding) by turning over the disk at three (possibly twisted) ribbons. Charged fermions come in two handedness states the top by π around the axis that passes through its center and between the first whereas the neutrino and antineutrino come in only one handedness state. Source, and second ribbons. Source, [4]. [3]. Crucial to our construction is the observation that the twist and as minimal one-sided ideals of Clifford algebras, and their topolog- braid structure are not conserved, but are interchangeable [19,21]. ical representation as braids. We set up this correspondence via a Writing the twist structure of the ribbons in terms of a vector of map from the Witt bases of Clifford algebras to elements of braid three half integers [a, b, c], an example for the braid generator σ1 groups, and find that the topological architecture assumed in [3], is given in Fig. 2. As a 3-belt, σ1 is isotopic to [1/2, 1/2, −1/2]. For follows (with some minor deviations) from the structure of the c the generators of B we write: minimal one-sided ideals of C(6) and C(4). Each basis state in 3 the exterior algebras composing the minimal ideals is mapped to σ1 ≈ [1/2, 1/2, −1/2] , σ2 ≈ [−1/2, 1/2, 1/2] , a unique braid, that coincides with the braids in [3]. The resulting correspondence provides a unification of these two models which σ3 ≈ [1/2, −1/2, 1/2] . (1) are based on very different principles. −1 The twist vectors of σ correspond to the negatives of σi , so that, In [17](see also [18]) it was shown that mapping the Witt basis i −1 c 3 for example, σ ≈[−1/2, −1/2, 1/2]. The exchange of braiding of C(6) to particular braids in the circular Artin braid group B3, 1 and subsequently exchanging the braiding of ribbons for twisting for twisting is only valid for 3-belts and not for general braided of these ribbons in an isotopic manner, that the twist structure belts. of ribbons that encodes the electrocolor symmetries in the braid 3. Standard Model particle states from the minimal ideals of model [3]can be replicated in a one-to-one manner. This paper Clifford algebras extends this curious correspondence, to include the SU(2)L chiral weak symmetry. This is achieved by mapping a Witt basis of an { } = additional C(4) algebra to braids in B3, taken to be a subgroup The algebra C(6) spanned by ei , i 1, .., 6endowed with a c Witt basis of B3. The braids corresponding to the charged vector bosons are determined, and it is demonstrated that chiral weak interactions ≡ 1 − + ≡ 1 − + ≡ 1 − + can be described via the composition of braids. α1 ( e5 ie4), α2 ( e3 ie1), α3 ( e6 ie2), 2 2 2 † 1 † 1 † 1 2. A topological model of composite preons α ≡ (e5 + ie4), α ≡ (e3 + ie1), α ≡ (e6 + ie2), (2) 1 2 2 2 3 2 → − We provide a brief overview of the building blocks and rules decomposes the algebra into minimal ideals. Here, †takes i i, of the braid model [3]. The fundamental preonic object is a ribbon e j → −e j , and reverses the order of multiplications. This Witt basis which may be twisted. Ribbons are combined into triplets with satisfies non-trivial braiding, and connected together at the top and bot- † † = = † = tom to a parallel disk. It is assumed that ribbons with twists in αi , α j αi, α j 0, αi , α j δij. (3) opposite directions are not permitted in the same triplet. The re- † sulting topological objects, corresponding to framed braids in the { } { } Both αi and αi are bases for maximally isotropic subspaces circular Artin braid group Bc , are called 3-belts [19,20]. The first † † 3 χ and χ respectively, and generate the exterior algebras χ generation SM fermions in terms of braids is shown in Fig. 1. and χ via the Clifford product. The first minimal left ideal u † † † † In this representation, the twist structure of the ribbons accounts is then given by S ≡ C(6)ωω = χ ωω , where ωω = for the electrocolor symmetries, with charges of ±e/3represented † † † α α α α α α is a primitive idempotent.
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