Gauge Group Research Articles

Computation of the masses of the elementary particles

An approach to gauge theory in the context of locally conformally flat space–time is described. It is discussed how there are a number of natural principal bundles associated with any given locally conformally flat space–time X. The simplest of these principal bundles is the bundle PX(G) with structure group G = U(2, 2). An 11-dimensional bundle Q with structure group a certain seven-dimensional group K is constructed by a method involving a reduction of structure group for the bundle PX(G). It is shown how the gauge groups U(1), SU(2), and SU(3) can be derived from the geometry of locally conformally flat space–time. Fock spaces of multiparticle states for the fields of the standard model are constructed in the context of bundles with these groups as structure groups. Scattering and other particle interaction processes are defined in terms of linear maps between multiparticle state spaces. A technique for computing analytically and/or computationally the masses of the elementary particles is described. This method involves the computation of a certain quantity called the integral mass spectrum for a given family of particles and then the masses of the particles in the family are determined to be the locations of the peaks of the integral mass spectrum. The method is applied successfully in the electroweak sector to the cases of the charged leptons, μ and τ, and the Z0 particle.

Geometric constructions in the theory of analytic complexity

Two geometric constructions are considered in the context of analytic complexity. Using the first construction, on the set of analytic functions, we build a metric invariant under the action of the gauge group. With the help of the second construction, we obtain a necessary differential algebraic condition for membership of a function in the tangent space to the class of bivariate functions of analytic complexity $\le 2$ at the point $z_0=x^3 y^2 +xy$. From this result we show that the polynomial $z=x^3y^2+xy + \pi x^2 y^3$ of degree 5 has analytic complexity 3.

The discreet charm of the discrete series in dS2

We study quantum field theories placed on a two-dimensional de Sitter spacetime (dS2) with an eye on the group-theoretic organization of single and multi-particle states. We explore the distinguished role of the discrete series unitary irreducible representation (UIR) in the Hilbert space. By employing previous attempts to realize these states in free tachyonic scalar field theories, we propose how the discrete series may contribute to the Källén–Lehmann decomposition of an interacting scalar two-point function. We also study BF gauge theories with SL(N,R) gauge group in dS2 and establish a relation between the discrete series UIRs and the operator content of these theories. Although present at the level of the operators, states carrying discrete series quantum numbers are projected out of the gauge-invariant Hilbert space. This projection is reminiscent of what happens for quantum field theories coupled to semiclassical de Sitter gravity, where we must project onto the subspace of de Sitter invariant states. We discuss how to impose the diffeomorphism constraints on local field-theory operators coupled to two-dimensional gravity in de Sitter, with particular emphasis on the role of contact terms. Finally, we discuss an SYK-type model with a random two-body interaction that encodes an infinite tower of discrete series operators. We speculate on its potential microscopic connection to the SL(N,R) BF theory in the large-N limit.

Anomaly cancellation in string theory

The anomaly cancellation in superstring theory is known to hold at leading order in the curvature for the gauge groups [Formula: see text] and [Formula: see text]. The coefficients of the next-to-leading order terms may be evaluated, and a mechanism for cancellation is described, which would remain valid at higher genus when there exists a global splitting of the coordinates of supermoduli space. Since the spin-[Formula: see text] fields transform under the adjoint representation in these models, compactification of the heterotic string over [Formula: see text] or a fundamental 12-dimensional theory, from which superstrings are produced through an elliptic fibration, over [Formula: see text], provides another phenomenologically viable theory at lower energies compatible with the standard description of the elementary particle interactions. The 96 spin-[Formula: see text] fields now would transform under the fundamental representation of [Formula: see text] and the spin-one gauge fields would belong to the adjoint representation. The sum of the anomaly polynomials for the particle content of the [Formula: see text] model vanishes at [Formula: see text]. The contributions to the gravitational anomaly from the particles and antiparticles cancel by the CPT theorem and the duality transformations of polynomials of degree 6 in the curvature and the field strength. The existence of the interaction of a spin-2 charge, which is conserved only over a finite time interval, can be traced to nonlocal terms in the reduction of the string field theory to the gravitational sector. The source of the global gravitational anomaly cancellation in the modular form equations derived from an elliptic fibration of a 12-dimensional theory would restrict the compactifications and provide a method for preserving the absence of anomalies in four dimensions.

Higher category theory and n-groups as gauge symmetries for quantum gravity

Higher category theory can be employed to generalize the notion of a gauge group to the notion of a gauge n-group. This novel algebraic structure is designed to generalize notions of connection, parallel transport and holonomy from curves to manifolds of dimension higher than one. Thus it generalizes the concept of gauge symmetry, giving rise to a topological action called nBF action, living on a corresponding n-principal bundle over a spacetime manifold. Similarly as for the Plebanski action, one can deform the topological nBF action by adding appropriate simplicity constraints, in order to describe the correct dynamics of both gravity and matter fields. Specifically, one can describe the whole Standard Model coupled to gravity as a constrained 3BF or 4BF action. The split of the full action into a topological sector and simplicity constraints sector is adapted to the spinfoam quantization technique, with the aim to construct a full model of quantum gravity with matter. In addition, the properties of the gauge n-group structure open up a possibility of a nontrivial unification of all fields. An n-group naturally contains additional novel gauge groups which specify the spectrum of matter fields present in the theory, in a similar way to the ordinary gauge group that prescribes the spectrum of gauge vector bosons in the Yang-Mills theory. The presence and the properties of these new gauge groups has the potential to explain fermion families, and other structure in the matter spectrum of the theory.

On Yang–Mills fields from anti-de Sitter spaces

Motivated by some recent progress involving a non-compact gauge group, we obtain classical gauge fields using non-compact foliations of anti-de Sitter space in 4 dimensions (required dimensionality for conformal invariance of Yang–Mills theory) and transfer these to Minkowski spacetime using a series of conformal maps. This construction also builds upon some previous works involving SU(2) gauge group in that we now use its non-compact counterpart SU(1, 1) here. We note down gauge fields in both Abelian as well as non-Abelian settings and find them to be divergent at some hyperboloid, which is a hypersurface of co-dimension 1 inside the conformal boundary of AdS4. In spite of this hurdle we find a physically relevant field configuration in the Abelian case, reproducing a known result.

Holographic RG flows from four-dimensional N = 2 gauged supergravities

In quantum field theories, the examination of particle interactions is facilitated through the application of Feynman diagrams. The computation of loop Feynman diagrams often yields divergent or infinite values. To tackle this issue, a method known as renormalization is employed, which entails the removal of these infinities. The phenomenon of a physical system undergoing changes under different scales gives rise to a concept referred to as the renormalization group. Certain renormalization group trajectories, denoted as RG flows, describe transformations of a conformal field theory (CFT) into other conformal or non-conformal theories. These transformations result in deformations of a UV conformal fixed point into another fixed point or into a non-conformal phase within the infrared (IR) regime. In this study, we investigate holographic RG flows originating from N = 2 gauged supergravity with an SO(2) × SO(6) gauge group. These solutions delineate RG flows from the N = 2 CFT to a three-dimensional non-conformal field theory driven by mass deformations. This transformation is elucidated through the AdS/CFT correspondence, also known as AdS/CFT holography.

Mass Generation of the Scalars Bosons in a Left-Right Symmetric Model with two Bidoublets

The search for new Physics justified by new discoverys, motivate among others, the realization of new models called Extensions of the Standard Model (ESM) which must include at low energy the Standar Model. (SM) The electroweak symmetry group SU(2) L ⊗ U(1) Y to the gauge group with left-right symmetry SU(2) L ⊗ SU(2) R ⊗ U(1) B − L ⊗ 𝒫, represents one of the minimal extensions of the SM, in which 𝒫 is a parity discrete symmetry such that left-right coupling constants satisfied gL = gR , and according to the hierarchy in the symmetry breaking (conditions that must be met by the vacuum expectation values introduced in the model) allow to obtain the SM. The aim of this work is to identify the Higgs boson of the SM, considering a more general scalar potential that must respect all the symmetries (local gauge invariant, discrete symmetries and Lorentz invariant) established in the model. We will take into account those previous works about the hierarchy conditions and certain approximations that must satisfy the vacuum expectation values for obtain greater simplicity in the caculations of scalar masses of the model.

A sedenion algebraic representation of three colored fermion generations

Three generations of fermions with SU(3) C symmetry are represented algebraically in terms of the algebra of sedenions, 𝕊, generated from the octonions, 𝕆, via the Cayley-Dickson process. Despite significant recent progress in generating the Standard Model gauge groups and particle multiplets from the four normed division algebras, an algebraic motivation for the existence of exactly three generations has been difficult to substantiate. In the sedenion model, one generation of leptons and quarks with SU(3) C symmetry is represented in terms of two minimal left ideals of ℂℓ(6), generated from a subset of all left actions of the complex sedenions on themselves. Subsequently, the finite group S 3, which are automorphisms of 𝕊 but not of 𝕆, is used to generate two additional generations. The present paper highlight the key aspects and ideas underlying this construction.

Stefan–Boltzmann law and Casimir effect for dark photons

In this paper, the existence of a massive dark photon, associated with a new gauge group is considered. The dark photon can be kinetically mixed with the photon. To study some applications, the thermo field dynamics formalism is used. Exploring the topological structure of this approach, the influence of dark photons on the Stefan–Boltzmann law and the Casimir effect at zero and finite temperature is calculated.

Energy–momentum and angular momentum densities in the Poincaré gauge theory of gravity

Abstract In the $\overline{\rm {Poincar\acute{e}}}$ gauge theory of gravity, we examine the energy–momentum and angular momentum densities of the source of the gravitational field (gauge potentials $A^{k}{}_{\mu }, \, A^{kl}{}_{\mu }$). There are two types of these quantities, the Hilbert-type densities ${\boldsymbol \Theta}_{k}{}^{\mu },\, {\boldsymbol \Phi}_{kl}{}^{\mu }$, which appear in the right-hand sides of the gravitational field equations, and $\, ^{M}{\boldsymbol T}_{k}{}^{\mu }, \, ^{M}{\boldsymbol S}_{kl}{}^{\mu }$, which are the densities (Noether currents) of the generators of $T^{4}\underline{\otimes }SL(2,C)$ transformations. We show that these are identical to each other, respectively, i.e., ${\boldsymbol \Theta}_{k}{}^{\mu }\equiv \, ^{M}{\boldsymbol T}_{k}{}^{\mu } ,\, {\boldsymbol \Phi}_{kl}{}^{\mu }\equiv \, ^{M}{\boldsymbol S}_{kl}{}^{\mu }$, in parallel with the cases of the electromagnetic and Yang–Mills fields.

Contraction of the Standard Model Gauge Group Is Not in Conflict with the LHC Data

Contraction of the Standard Model Gauge Group Is Not in Conflict with the LHC Data

3d exceptional gauge theories and boundary confinement

We find boundary confining dualities of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric gauge theories with exceptional gauge groups. The half-indices which enumerate the boundary BPS local operators in the presence of Neumann and Dirichlet boundary conditions for gauge fields are identified with the Askey-Wilson type q-beta integrals and Macdonald type sums respectively. New conjectural identities of E6 and E7 integrals and sums are derived from the boundary confining dualities. We also consider theories with a vector multiplet and adjoint chiral, which correspond to an N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 vector multiplet, with appropriate boundary conditions. We argue for the boundary confinement of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 vector multiplet and comment on such theories also with classical gauge groups.

Soft particles and infinite-dimensional geometry

In the sigma model, soft insertions of moduli scalars enact parallel transport of S-matrix elements about the finite-dimensional moduli space of vacua, and the antisymmetric double-soft theorem calculates the curvature of the vacuum manifold. We explore the analogs of these statements in gauge theory and gravity in asymptotically flat spacetimes, where the relevant moduli spaces are infinite-dimensional. These models have spaces of vacua parameterized by (trivial) flat connections on the celestial sphere, and soft insertions of photons, gluons, and gravitons parallel transport S-matrix elements about these infinite-dimensional manifolds. We argue that the antisymmetric double-soft gluon theorem in d + 2 bulk dimensions computes the curvature of a connection on the infinite-dimensional space Map, where G is the global part of the gauge group. The analogous metrics in abelian gauge theory and gravity are flat, as indicated by the vanishing of the antisymmetric double-soft theorems in those models. In other words, Feynman diagram calculations not only know about the vacuum manifold of Yang–Mills theory, they can also be used to compute its curvature. The results have interesting implications for flat space holography.

SO(1,3) Yang-Mills solutions on Minkowski space via cosets

We present a novel family of Yang-Mills solutions, with gauge group SO(1,3), on Minkowski space that are geometrically distinguished into two classes, viz. interior and exterior of the lightcone. We achieve this by foliating the former with SO(1,3)/SO(3) cosets and the latter with SO(1,3)/SO(1,2) cosets and analytically solving the Yang-Mills equation of an SO(1,3)-invariant gauge field. The resulting fields and their stress-energy tensor, when translated to the Minkowski space, diverge at the lightcone, but we demonstrate how this stress-energy tensor could be regularised due to its unique algebraic structure.

Thermodynamics and tachyon condensation of the dressed-dynamical unstable Dp-branes

Using the boundary state formalism and thermo field dynamics approach, we study a Dp-brane at finite temperature in which the background Kalb–Ramond field, a U(1) gauge potential, and tachyon field are turned on together with a general tangential dynamics. The thermal entropy of the brane will be studied. In addition, the behavior of the entropy after the tachyon condensation process will be investigated and some thermodynamic interpretations will be extracted.

Ground states of helicoidal spin-orbit-coupled spin-1 bosons

The ground states of spin-1 Bose–Einstein condensate under helicoidal spin–orbit-coupling are investigated, by solving the corresponding Gross–Pitaevskii equations, and via the time-splitting Fourier pseudospectral method. Taking into account the ferromagnetic interaction as well as the antiferromagnetic case, we identify some novel structures on the ground states, including the spot phase, particle-like supersolid phase, hole-like supersolid phase, particle-hole-like supersolid phase, and wave-shaped stripe phase. With the variance of helicoidal gauge potential, the system would undergo abundant quantum phases.

Meson spectrum of SU(2) QCD1+1 with quarks in Large representations

We consider SU(2) quantum chromodynamics in 1 + 1 dimensions with a single quark in the spin J representation of the gauge group and study the theory in the large J limit where the gauge coupling g2 → 0 and J → ∞ with λ = g2J2 fixed. We work with a Dirac spinor field for arbitrary J, and with a Majorana spinor for integer J since the integer spin representations of SU(2) are real, and analyze the two cases separately.The theory is reformulated in terms of global colour non-singlet fermion bilocal operators which satisfy a W∞ × U(2J + 1) algebra. In the large J limit, the dynamics of the bilocal fields is captured by fluctuations along a particular coadjoint orbit of the W∞ algebra. We show that the global colour-singlet sector of the bilocal field fluctuations satisfy the same integral equation for meson wavefunctions that appears in the ’t Hooft model. For Majorana spinors in the integer spin J representation, the Majorana condition projects out half of the meson spectrum, as a result of which the linear spacing of the asymptotic meson spectrum for Majorana fermions is double that of Dirac fermions. The Majorana condition also projects out the zero mass bound state that is present for the Dirac quark at zero quark mass.We also consider the formulation of the model in terms of local charge densities and compute the quark spectral function in the large J limit: we see evidence for the absence of a pole in the quark propagator.

Mechanism for conversion between twist and writhe of DNA based on chiral gauge potential

It is well known that the linking number is the sum of the writhing number and twisting number based on white’s formula. In this work, we reveal the underlying physics of the transformation between the twist and the writhe of deoxyribonucleic acid (DNA). In the framework of the Landau model, we describe the local interaction between the base pairs by using the chiral gauge potential. With the chiral gauge potential, we develop a geometrical model where the free energy density is given by the geometrical parameters, including the normal curvature, the geodesic curvature, and the geodesic torsion. It is shown that the chiral gauge potential plays a crucial role in the conversion between the twist and the writhe of DNA. By using the invariance of the self-linking number, we find that when the total twisting number is changed by −Δ Q, the writhing number is changed by Δ Q. Our findings not only deepen the understanding of the topological nature of DNA, but also provide a physical model for the research of the conformation of DNA.

Anisotropic compactification of nonrelativistic M-theory

We study a decoupling limit of M-theory where the three-form gauge potential becomes critical. This limit leads to nonrelativistic M-theory coupled to a non-Lorentzian spacetime geometry. Nonrelativistic M-theory is U-dual to M-theory in the discrete light cone quantization, a non-perturbative approach related to the Matrix theory description of M-theory. We focus on the compactification of nonrelativistic M-theory over a two-torus that exhibits anisotropic behaviors due to the foliation structure of the spacetime geometry. We develop a frame covariant formalism of the toroidal geometry, which provides a geometrical interpretation of the recently discovered polynomial realization of SL(2 , ℤ) duality in nonrelativistic type IIB superstring theory. We will show that the nonrelativistic IIB string background fields transform as polynomials of an effective Galilean “boost velocity” on the two-torus. As an application, we construct an action principle describing a single M5-brane in nonrelativistic M-theory and study its compactification over the anisotropic two-torus. This procedure leads to a D3-brane action in nonrelativistic IIB string theory that makes the SL(2 , ℤ) invariance manifest in the polynomial realization.