Gauge field interactions in spaces with arbitrary torsion

The modern formulation of non-Abelian lattice gauge theories is due to Wilson [Wil74]. Independently, gauge theories were discussed on a lattice by Wegner [Weg71] as a gauge-invariant extension of the Ising model and in an unpublished work by A. Polyakov in 1974 which deals mostly with Abelian theories. Placing gauge fields on a lattice provides, first, a nonperturbative regularization of ultraviolet divergences. Secondly, the lattice formulation of QCD possesses some nonperturbative terms in addition to perturbation theory. A result of this is that one has a nontrivial definition of QCD beyond perturbation theory which guarantees confinement of quarks. The lattice formulation of gauge theories deals with phase-factor-like quantities, which are elements of the gauge group, and are natural variables for quantum gauge theories. The gauge group on the lattice is therefore compact, offering the possibility of nonperturbative quantization of gauge theories without fixing the gauge. The lattice quantization of gauge theories is performed in such a way as to preserve the compactness of the gauge group. The continuum limit of lattice gauge theories is reproduced when the lattice spacing is many times smaller than the characteristic scale. This is achieved when the non-Abelian coupling constant tends to zero as it follows from the renormalization-group equation. In this chapter we consider the Euclidean formulation of lattice gauge theories. First, we introduce the lattice terminology and discuss the action of lattice gauge theory at the classical level.

The phase shift due to gravitational field and all gauge fields in the interference of two coherent beams is obtained. In the case of gravitation, it is shown, for a particle with arbitrary spin, that there exists a phase shift due to the coupling of spin to space-time curvature. This and the corresponding phase shift for gauge fields are analogous to the Aharonov-Bohm effect. The classical limit for particles moving in gravitational and gauge fields is obtained from the phase shift. For gravitation, in the absence of torsion, this is the Mathisson-Papapetrou equation, which is thereby shown to be the classical limit of the Dirac and Bargmann-Wigner wave equations, generalized to curved space-time. In the presence of torsion, a modification of this equation, given by Hehl, is obtained. It is pointed out that gravity is not a pure gauge field and that it must be placed in the more general category of an «interference field» which contains both gravity and gauge fields as special cases. The field equations for gauge fields and gravity are obtained from the heuristic assumption that a particle acts on a field in a manner which depends on how it responds to the field via the phase shift. For gauge fields, they contain the Yang-Mills equations as a special case. For gravity, a modification of Einstein’s field equations is obtained, which corresponds to the Lagrangian (1/16πK) ·, · (2Λ +R) + (1/32πf)RμνϱσRμνσϱ, where the Riemann tensor contains torsion andK, f, Λ are constants (Λ may be zero). The relevance of the phase shift, due to rotation, to the quantization of vortices in superfluid helium is pointed out. This suggests that the curl of the superfluid velocity may obey a system of equations analogous to Maxwell’s equations and the analogue of the magnetic monopole for superfluid helium is also introduced.

We present new constraints on spectator axion-U(1) gauge field interactions during inflation using the latest Planck (PR4) and BICEP/Keck 2018 data releases. This model can source tensor perturbations from amplified gauge field fluctuations, driven by an axion rolling for a few e-folds during inflation. The gravitational waves sourced in this way have a strongly scale-dependent (and chiral) spectrum, with potentially visible contributions to large/intermediate scale B-modes of the CMB. We first derive theoretical bounds on the model imposing validity of the perturbative regime and negligible backreaction of the gauge field on the background dynamics. Then, we determine bounds from current CMB observations, adopting a frequentist profile likelihood approach. We study the behaviour of constraints for typical choices of the model's parameters, analyzing the impact of different dataset combinations. We find that observational bounds are competitive with theoretical ones and together they exclude a significant portion of the model's parameter space. We argue that the parameter space still remains large and interesting for future CMB experiments targeting large/intermediate scales B-modes.

We present a gauge and Lorentz invariant effective field theory model for the interaction of a charged scalar matter field with a magnetic monopole source, described by an external magnetic current. The quantum fluctuations of the monopole field are described effectively by a strongly coupled “dual” Ud(1) gauge field, which is independent of the electromagnetic Uem(1) gauge field. The effective interactions of the charged matter with the monopole source are described by a gauge invariant mixed Chern-Simons-like (Pontryagin-density) term between the two U(1) gauge fields. The latter interaction coupling is left free, and a lattice study of the system is performed with the aim of determining the phase structure of this effective theory. Our study shows that, in the spontaneously broken-symmetry phase, the monopole source triggers, via the mixed Chern-Simons term, which is nontrivial in its presence, the generation of a dynamical singular configuration (magnetic-monopolelike) for the respective gauge fields. The scalar field also behaves in the broken phase in a way similar to that of the scalar sector of the ’t Hooft-Polyakov monopole. Moreover, we show that the modest size of the lattices involved does not have significant effects on the main conclusions of our analysis. Published by the American Physical Society 2024

We consider a generalization of non-relativistic Schrödinger–Higgs Lagrangian by introducing a nonstandard kinetic term. We show that this model is Galilean invariant, we construct the conserved charges associated to the symmetries and realize the algebra of the Galilean group. In addition, we study the model in the presence of a gauge field. We also show that the gauged model is Galilean invariant. Finally, we explore relations between the twin models and their solutions.

A basic problem of classical field theory, which has attracted growing attention over the past decade, is to find and classify all nonlinear deformations of linear abelian gauge theories. The physical interest in studying deformations is to address uniqueness of known nonlinear interactions of gauge fields and to look systematically for theoretical possibilities for new interactions. Mathematically, the study of deformations aims to understand the rigidity of the nonlinear structure of gauge field theories and to uncover new types of nonlinear geometrical structures. The first part of this paper summarizes and significantly elaborates a field-theoretic deformation method developed in earlier work. Some key contributions presented here are, firstly, that the determining equations for deformation terms are shown to have an elegant formulation using Lie derivatives in the jet space associated with the gauge field variables. Secondly, the obstructions (integrability conditions) that must be satisfied by lowest-order deformations terms for existence of a deformation to higher orders are explicitly identified. Most importantly, a universal geometrical structure common to a large class of nonlinear gauge theory examples is uncovered. This structure is derived geometrically from the deformed gauge symmetry and is characterized by a covariant derivative operator plus a nonlinear field strength, related through the curvature of the covariant derivative. The scope of these results encompasses Yang–Mills theory, Freedman–Townsend theory, and Einstein gravity theory, in addition to their many interesting types of novel generalizations that have been found in the past several years. The second part of the paper presents a new geometrical type of Yang–Mills generalization in three dimensions motivated from considering torsion in the context of nonlinear sigma models with Lie group targets (chiral theories). The generalization is derived by a deformation analysis of linear abelian Yang–Mills Chern–Simons gauge theory. Torsion is introduced geometrically through a duality with chiral models obtained from the chiral field form of self-dual (2+2) dimensional Yang–Mills theory under reduction to (2+1) dimensions. Field-theoretic and geometric features of the resulting nonlinear gauge theories with torsion are discussed.

We consider SU(2) gauge potentials over a space with a compactified\ndimension. A non-Abelian Fourier transform of the gauge potential in the\ncompactified dimension is defined in such a way that the Fourier coefficients\nare (almost) gauge invariant. The functional measure and the gauge field\nstrengths are expressed in terms of these Fourier coefficients. The emerging\nformulation of the non-Abelian gauge theory turns out to be an Abelian gauge\ntheory of a set of fields defined over the initial space with the compactified\ndimension excluded. The Abelian theory contains an Abelian gauge field, a\nscalar field, and an infinite tower of vector matter fields, some of which\ncarry Abelian charges. Possible applications of this formalism are discussed\nbriefly.\n

Local quartic interaction of higher-spin gauge field with a scalar field is considered. In this special case, the nontrivial task of construction of interacting Lagrangian for the higher spin field in physical gauge was solved using the full power of Noether's procedure. As a result, the linear on-field gauge transformation is obtained and the corresponding commutator of transformation is analyzed. To understand the closure of this algebra the right-hand side of this commutator is classified in respect to gauge transformations coming from cubic interactions with different higher spin symmetric tensor fields and with mixed symmetry tensor fields transformations.

We construct a unified (quantum) description, by the gauge principle, of gravity and Standard Model (SM), that generalizes the Dirac-Born-Infeld action to the SM and Weyl geometry, hereafter called Weyl-Dirac-Born-Infeld action (WDBI). The theory is formulated in d = 4 − 2 ε dimensions. The WDBI action is a general gauge theory of SM and Weyl group (of dilatations and Poincaré symmetry), in the Weyl gauge covariant (metric) formulation of Weyl geometry. The theory is SM and Weyl gauge invariant in d = 4 − 2 ε dimensions and there is no Weyl anomaly. The WDBI action has the unique elegant feature, not present in other gauge theories or even in string theory, that it is mathematically well defined in d = 4 − 2 ε dimensions with no need to introduce in the action a UV regulator scale or field. This action actually that gravity, through (Weyl covariant) space-time curvature R ^ , acts as UV regulator of both SM and gravity in d = 4 . A series expansion of the WDBI action (in dimensionless couplings) recovers in the leading order a Weyl gauge invariant version of SM and the Weyl (gauge theory of) quadratic gravity. The SM and Einstein-Hilbert gravity are recovered in the Stueckelberg broken phase of Weyl gauge symmetry, which restores Riemannian geometry below Planck scale. Subleading orders are suppressed by powers of (dimensionless) gravitational coupling ( ξ ) of Weyl quadratic gravity.

The Wess-Zumino-Witten (WZW) model and the two-dimensional chiral gravitational theory of Polyakov are understood to encode anomalous symmetry breaking at the level of the path integral. In each case, we can rediscover the corresponding action by constructing an action on coadjoint orbits of the Kac-Moody and Virasoro Lie algebras, respectively. This coadjoint action construction reveals additional couplings to both the WZW model and Polyakov’s model. In the Kac-Moody case, we find a coupling of a gauge field A to the WZW model, and, in the Virasoro case, we see a coupling of a field, dubbed the diffeomorphism field, D to Polyakov’s model. The Yang-Mills action is well-known to give dynamics to the gauge field A and, since they appear in an analogous fashion, we are motivated to seek a dynamical action for the diffeomorphism field D. Through the projective connections of Thomas and Whitehead, we recover an object corresponding to the diffeomorphism field D. Then, using an action constructed from projective curvature invariants, we can give appropriate dynamics to D in much the same way that the Yang-Mills action gives dynamics to A. This action mimics the form of the Gauss-Bonnet curvature term so that no higher-order derivatives of the metric are introduced. Overall, the action giving rise to dynamics for D is what we call TW Gravity and encodes the dynamics for the metric Ɡₐ{b}, the diffeomorphism field Dₐ{b}, and an object known as the fundamental projective invariant IIª{bc}. We show that the TW Gravity model can be applied to the setting of cosmology, where it offers a path toward solving long-standing problems. We show that a projective angular momentum constant J₀ arises in a way that can be understood as producing a realistic bare cosmological constant. Here, J₀ ≈ 10⁸⁶ J· s and can be thought of as a new cosmological angular momentum scale. We also show that the trace degree of freedom corresponding to Dₐ{b} gives rise to a non-minimally coupled inflaton action. This model is shown to yield parameters n{s}, A{s}, and r of the nearly scale-invariant perturbations to the Cosmic Microwave Background radiation, which are within current constraints. Finally, we cast TW Gravity in a general cosmological setting via the use of the symmetries corresponding to a Friedmann-Robertson-Walker-Lemaitre background.

A class of new non-Abelian gauge theories for vector fields on three manifolds is presented. The theories describe a generalization of three-dimensional Yang–Mills theory featuring a novel nonlinear gauge symmetry and field equations for Lie-algebra-valued vector potential fields. The nonlinear form of the gauge symmetry and field equations relies on the vector cross-product and vector curl operator available only in three dimensions, and makes use of an auxiliary Lie bracket together with the Lie bracket used in Yang–Mills theory. A gauge covariant formulation of the new theories is given which utilizes the covariant derivative and curvature from the geometrical formulation of Yang–Mills theory. Further features of the new theories are discussed.

We discuss the Dirac-K\"ahler and staggered formulations of lattice gauge theory with fermions and show that, although the noninteracting parts of the Lagrangians are identical, the Lagrangians differ in the interactions between the gauge field and the fermion field. However, we argue that, in general, the continuum limit of a two-dimensional gauge theory with fermions is independent of the precise details of the gauging of the fermion terms in the Lagrangian, provided that the resulting Lagrangian is invariant under an arbitrary infinitesimal gauge transformation. Hence, we conclude that, in two dimensions, the Dirac-K\"ahler and staggered formulations are equivalent in the continuum limit. This equivalence is in agreement with explicit calculations of the axial-vector anomaly in the two formulations.

We present variational formulations of gauge theories and Einstein–Yang–Mills equations in the spirit of Kaluza–Klein theories. For gauge theories, only a topological fibration is assumed. For gravitation coupled with gauge fields, no fibration is assumed: Fields are defined on a ‘space-time’ \({\mathcal {Y}}\) of dimension \(4 + r\) without any structure a priori, where r is the dimension of the structure group. If the latter is compact and simply connected, classical solutions allow to construct a manifold \({\mathcal {X}}\) of dimension 4 to be the physical space-time, in such a way that \({\mathcal {Y}}\) acquires the structure of a principal bundle over \({\mathcal {X}}\) and leads to solutions of the Einstein–Yang–Mills systems. The special case of the Einstein–Maxwell system is also discussed: It suffices that at least one fiber closes in on a circle to deduce that the five-dimensional space-time has a fiber bundle structure.

We present a dual formulation of the Cosserat theory of elasticity. In this theory a local element of an elastic body is described in terms of local displacement and local orientation. Upon the duality transformation these degrees of freedom map onto a coupled theory of a U(1) vector-valued one-form gauge field and an ordinary U(1) gauge field. We discuss the degrees of freedom in the corresponding gauge theories, relation to symmetric tensor gauge theories, the defect matter and coupling to the curved space.

We develop a new theoretical framework for describing and analyzing exotic phases of strongly correlated electrons which support excitations with fractional quantum numbers. Starting with a class of microscopic models believed to capture much of the essential physics of the cuprate superconductors, we derive a new gauge theory - based upon a {\it discrete} Ising or Z_2 symmetry - which interpolates naturally between an antiferromagnetic Mott insulator and a conventional d-wave superconductor. We explore the intervening regime, and demonstrate the possible existence of an exotic fractionalized insulator - the nodal liquid - as well as various more conventional insulating phases exhibiting broken lattice symmetries. A crucial role is played by vortex configurations in the Z_2 gauge field. Fractionalization is obtained if they are uncondensed. Within the insulating phases, the dynamics of these Z_2 vortices in two dimensions (2d) is described, after a duality transformation, by an Ising model in a transverse field - the Ising spins representing the Z_2 vortices. The presence of an unusual Berry's phase term in the gauge theory, leads to a doping-dependent "frustration" in the dual Ising model, being fully frustrated at half-filling. The Z_2 gauge theory is readily generalized to a variety of different situations - in particular, it can also describe 3d insulators with fractional quantum numbers. We point out that the mechanism of fractionalization for d>1 is distinct from the well-known 1d spin-charge separation. Other interesting results include a description of an exotic fractionalized superconductor in two or higher dimensions.