Gauge Invariant Action Research Articles

Note on higher-group structure in 6d self-dual gauge theory

We analyze higher-group structure of a 6d model coupled with a self-dual 2-form gauge field. This model is defined from 6d axion-electrodynamics with a 1-form Chern-Weil(CW) symmetry gauged dynamically. The gauging leads to a Green-Schwarz-West-Sagnotti(GSWS) term, which gives rise to an anomaly through a GSWS transformation acting on the 2-form gauge field. We cancel this anomaly by gauging a 3-form CW symmetry in 6d axion-electrodynamics. We find out the global symmetries in the resultant model and derive the gauge invariant action in the presence of the background gauge fields. It is argued that a discrete 1-form symmetry is anomalous because turning on the associated background gauge field causes quantum inconsistency due to an operator-valued ambiguity. Higher-group structure in this model that is manifested as a Green-Schwarz-like transformation for CW background gauge fields is discussed.

Spin-(s, j) projectors and gauge-invariant spin-s actions in maximally symmetric backgrounds

Given a maximally symmetric d-dimensional background with isometry algebra g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{g} $$\\end{document}, a symmetric and traceless rank-s field ϕa(s) satisfying the massive Klein-Gordon equation furnishes a collection of massive g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{g} $$\\end{document}-representations with spins j ∈ {0, 1, · · · , s}. In this paper we construct the spin-(s, j) projectors, which are operators that isolate the part of ϕa(s) that furnishes the representation from this collection carrying spin j. In the case of an (anti-)de Sitter ((A)dSd) background, we find that the poles of the projectors encode information about (partially-)massless representations, in agreement with observations made earlier in d = 3, 4. We then use these projectors to facilitate a systematic derivation of two-derivative actions with a propagating massless spin-s mode. In addition to reproducing the massless spin-s Fronsdal action, this analysis generates new actions possessing higher-depth gauge symmetry. In (A)dSd we also derive the action for a partially-massless spin-s depth-t field with 1 ≤ t ≤ s. The latter utilises the minimum number of auxiliary fields, and corresponds to the action originally proposed by Zinoviev after gauging away all Stückelberg fields. Some higher-derivative actions are also presented, and in d = 3 are used to construct (i) generalised higher-spin Cotton tensors in (A)dS3; and (ii) topologically-massive actions with higher-depth gauge symmetry. Finally, in four-dimensional 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} = 1 Minkowski superspace, we provide closed-form expressions for the analogous superprojectors.

On the gauge invariance of the higher-derivative Yang–Mills–Chern–Simons action

In this work, we study the perturbative generation of the gauge invariant effective action for the non-Abelian gauge field in a (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional spacetime. We present a detailed analysis of the two, three and four-point functions in order to determine the non-Abelian Chern–Simons terms (parity odd) and Yang–Mills terms (parity even). Moreover, these terms are supplemented by the higher-derivative corrections which resulted in the Alekseev–Arbuzov–Baikov effective action (parity even) plus the higher-derivative (HD) corrections to the Chern–Simons terms (parity odd). In order to highlight some features about the perturbative generation of the effective action, we present a discussion based on the dimensional analysis, which allows us to establish the general structure of the permissible terms to guarantee the gauge invariance of the higher-derivative parts.

More on β-symmetry

Recent work has proposed a method for imposing T-duality on the metric, B-field, and dilaton of the classical effective action of string theory without using Kaluza-Klein reduction. Specifically, the D-dimensional effective action should be invariant under global O(D,D) transformations, provided that the partial derivatives along the β-parameters of the non-geometrical elements of the O(D,D) group become zero. In this paper, we speculate that the global β-symmetry can be utilized to identify both bulk and boundary couplings.We demonstrate that the Gibbons-Hawking term at the leading order of α′ is reproduced by this symmetry. Additionally, for closed spacetime manifolds at order α′, we show that the 20 independent geometrical couplings at this order are fixed by this symmetry up to field redefinitions. Specifically, we show that the invariance of the most general covariant and gauge invariant bulk action at order α′ under the most general covariant deformed β-transformation at order α′ fixes the action up to 13 parameters. These parameters reflect the field redefinitions freedom for the closed spacetime manifolds. For particular values of these parameters, we recover the effective action in the Metsaev-Tseytlin and in the Meissner schemes.

Symplectic groupoids and Poisson electrodynamics

We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative U(1) gauge theory. Our framework is based on an integrating symplectic groupoid for the underlying Poisson brackets, which we interpret as the classical phase space of a point particle on noncommutative spacetime. In this picture gauge fields arise as bisections of the symplectic groupoid while gauge transformations are parameterized by Lagrangian bisections. We provide a geometric construction of a gauge invariant action functional which minimally couples a dynamical charged particle to a background electromagnetic field. Our constructions are elucidated by several explicit examples, demonstrating the appearances of curved and even compact momentum spaces, the interplay between gauge transformations and spacetime diffeomorphisms, as well as emergent gravity phenomena.

On gauge-invariant deformation of reducible gauge theories

New method for construction of gauge-invariant deformed theory from an initial gauge theory proposed in our previous papers (Buchbinder and Lavrov in JHEP 06:854, 2021; Buchbinder and Lavrov in Eur Phys J C 81:856, 2021) for closed/open gauge algebras is extended to the case of reducible gauge algebras. The deformation procedure is explicitly described with the help of generating functions of anticanonical transformations depending on fields of the initial gauge action only. The deformed gauge-invariant action and the deformed gauge generators are described with the help of the generating functions in a closed and simple form. As an example of reducible gauge systems we consider the free fermionic p-form fields or, in another words, the antisymmetric tensor-spinor fields. It is proved that gauge-invariant deformation of fermionic p-form fields leads always to non-local deformed theory which does not contain a closed local sector. In its turn the model based on two fermionic 2-form fields and a real massive scalar field admits local interactions between these fields in local sector of the deformed action.

One-particle irreducible Wilson action in the gradient flow exact renormalization group formalism

Abstract We define a one-particle irreducible (1PI) Wilson action in the gradient flow exact renormalization group (GFERG) formalism as the Legendre transform of a Wilson action. We consider quantum electrodynamics in particular, and show that the GFERG flow equation preserves the invariance of the 1PI Wilson action (excluding the gauge-fixing term) under the conventionalU(1) gauge transformation. This is in contrast to the invariance of the original Wilson action under a modified U(1) gauge transformation. The global chiral transformation also takes the conventional form for the 1PI Wilson action. Despite the complexity of the GFERG flow equation, the conventional form of the gauge and global chiral transformations may allow us to introduce a non-perturbative Ansatz for gauge and chiral invariant 1PI Wilson actions.

Gauge invariance and localization of vector Kaluza\u2013Klein modes

We investigate the gauge invariance and localization of vector KK modes for a bulk U(1) gauge field under three kinds of localization mechanism on a brane with one extra dimension. By a general KK decomposition for the bulk U(1) gauge field, there are both vector and scalar KK modes on the brane, which couple with each other. We demonstrate that for a localization mechanism with a gauge invariant bulk action of the U(1) gauge field, the effective action of the KK modes on the brane can be formalized to gauge invariant form. However, only the massive vector KK modes and their accompanying scalar ones can be both localized on the brane, which depends on the solution of the brane, the gauge invariance of the massive vector field is finally preserved. For a localization mechanism with a broken gauge invariant bulk action of the field, it is impossible to rebuild the gauge invariance on the brane.

Covariant effective action for generalized Proca theories

We investigate the quantum stability of generalised Proca theories in curved spacetime treating gravity as a dynamical field. To compute the quantum gravitational corrections, we evaluate the divergent part of the effective action at one-loop level using Vilkovisky-DeWitt formalism, which gives us a gauge invariant and gauge condition independent effective action. It is shown that the quantum corrections are suppressed by a UV cutoff parametrically higher than the Proca mass, if the coupling constants are restricted to lie in a certain range. Furthermore, it is shown that the quantum corrections remain suppressed even at scales where classical non-linearities dominate over kinetic terms, allowing Vainshtein mechanism to work.

Poisson gauge theory

The Poisson gauge algebra is a semi-classical limit of complete non- commutative gauge algebra. In the present work we formulate the Poisson gauge theory which is a dynamical field theoretical model having the Poisson gauge algebra as a corresponding algebra of gauge symmetries. The proposed model is designed to investigate the semi-classical features of the full non-commutative gauge theory with coordinate dependent non-commutativity Θab(x), especially whose with a non-constant rank. We derive the expression for the covariant derivative of matter field. The commutator relation for the covariant derivatives defines the Poisson field strength which is covariant under the Poisson gauge transformations and reproduces the standard U(1) field strength in the commutative limit. We derive the corresponding Bianchi identities. The field equations for the gauge and the matter fields are obtained from the gauge invariant action. We consider different examples of linear in coordinates Poisson structures Θab(x), as well as non-linear ones, and obtain explicit expressions for all proposed constructions. Our model is unique up to invertible field redefinitions and coordinate transformations.

Gradient flow exact renormalization group: Inclusion of fermion fields

Abstract The gradient flow exact renormalization group (GFERG) is an exact renormalization group motivated by the Yang--Mills gradient flow and its salient feature is a manifest gauge invariance. We generalize this GFERG, originally formulated for the pure Yang--Mills theory, to vector-like gauge theories containing fermion fields, keeping the manifest gauge invariance. For the chiral symmetry we have two options: one possible formulation preserves the conventional form of the chiral symmetry and the other simpler formulation realizes the chiral symmetry in a modified form à la Ginsparg--Wilson. We work out a gauge-invariant local Wilson action in quantum electrodynamics to the lowest nontrivial order of perturbation theory. This Wilson action reproduces the correct axial anomaly in $D=2$.

On a gauge-invariant deformation of a classical gauge-invariant theory

We consider a general gauge theory with independent generators and study the problem of gauge-invariant deformation of initial gauge-invariant classical action. The problem is formulated in terms of BV-formalism and is reduced to describing the general solution to the classical master equation. We show that such general solution is determined by two arbitrary generating functions of the initial fields. As a result, we construct in explicit form the deformed action and the deformed gauge generators in terms of above functions. We argue that the deformed theory must in general be non-local. The developed deformation procedure is applied to Abelian vector field theory and we show that it allows to derive non-Abelain Yang-Mills theory. This procedure is also applied to free massless integer higher spin field theory and leads to local cubic interaction vertex for such fields.

Higher-spin Cotton tensors and massive gauge-invariant actions in AdS3

In a conformally flat three-dimensional spacetime, the linearised higher-spin Cotton tensor ℭα(n)(h) is the unique conserved conformal current which is a gauge-invariant descendant of the conformal gauge prepotential hα(n). The explicit form of ℭα(n)(h) is well known in Minkowski space. Here we solve the problem of extending the Minkowskian result to the case of anti-de Sitter (AdS) space and derive a closed-form expression for ℭα(n)(h) in terms of the AdS Lorentz covariant derivatives. It is shown that every conformal higher-spin action {S}_{mathrm{CS}}^{(n)}left[hright]propto int {mathrm{d}}^3{xeh}^{alpha (n)}{mathrm{mathfrak{C}}}_{alpha (n)}(h) factorises into a product of (n − 1) first-order operators that are associated with the spin-n/2 partially massless AdS values. Our findings greatly facilitate the on-shell analysis of massive higher-spin gauge-invariant actions in AdS3. The main results are extended to the case of mathcal{N} = 1 AdS supersymmetry. In particular, we derive simple expressions for the higher-spin super-Cotton tensors in AdS3.

AdS superprojectors

Within the framework of mathcal{N} = 1 anti-de Sitter (AdS) supersymmetry in four dimensions, we derive superspin projection operators (or superprojectors). For a tensor superfield {mathfrak{V}}_{alpha (m)overset{cdot }{alpha }(n)}:= {mathfrak{V}}_{left(alpha 1dots alpha mright)left({overset{cdot }{alpha}}_1dots {overset{cdot }{alpha}}_nright)} on AdS superspace, with m and n non-negative integers, the corresponding superprojector turns {mathfrak{V}}_{alpha (m)overset{cdot }{alpha }(n)} into a multiplet with the properties of a conserved conformal supercurrent. It is demonstrated that the poles of such superprojectors correspond to (partially) massless multiplets, and the associated gauge transformations are derived. We give a systematic discussion of how to realise the unitary and the partially massless representations of the mathcal{N} = 1 AdS4 superalgebra mathfrak{osp} (1|4) in terms of on-shell superfields. As an example, we present an off-shell model for the massive gravitino multiplet in AdS4. We also prove that the gauge-invariant actions for superconformal higher-spin multiplets factorise into products of minimal second-order differential operators.

Elasticity theory in general relativity

The general relativistic theory of elasticity is reviewed from a Lagrangian, as opposed to Eulerian, perspective. The equations of motion and stress–energy–momentum tensor for a hyperelastic body are derived from the gauge–invariant action principle first considered by DeWitt. This action is a natural extension of the action for a single relativistic particle. The central object in the Lagrangian treatment is the Landau–Lifshitz radar metric, which is the relativistic version of the right Cauchy–Green deformation tensor. We also introduce relativistic definitions of the deformation gradient, Green strain, and first and second Piola–Kirchhoff stress tensors. A gauge-fixed description of relativistic hyperelasticity is also presented, and the nonrelativistic theory is derived in the limit as the speed of light becomes infinite.

Generalised superconformal higher-spin multiplets

We propose generalised mathcal{N} = 1 superconformal higher-spin (SCHS) gauge multiplets of depth t, {Upsilon}_{alpha (n)overset{cdot }{alpha }(m)}^{(t)} , with n ≥ m ≥ 1. At the component level, for t > 2 they contain generalised conformal higher-spin (CHS) gauge fields with depths t − 1, t and t + 1. The supermultiplets with t = 1 and t = 2 include both ordinary and generalised CHS gauge fields. Super-Weyl and gauge invariant actions describing the dynamics of {Upsilon}_{alpha (n)overset{cdot }{alpha }(m)}^{(t)} on conformally-flat superspace backgrounds are then derived. For the case n = m = t = 1, corresponding to the maximal-depth conformal graviton supermultiplet, we extend this action to Bach-flat backgrounds. Models for superconformal non-gauge multiplets, which are expected to play an important role in the Bach-flat completions of the models for {Upsilon}_{alpha (n)overset{cdot }{alpha }(m)}^{(t)} , are also provided. Finally we show that, on Bach-flat backgrounds, requiring gauge and Weyl invariance does not always determine a model for a CHS field uniquely.

Background independent exact renormalisation

A geometric formulation of Wilson’s exact renormalisation group is presented based on a gauge invariant ultraviolet regularisation scheme without the introduction of a background field. This allows for a manifestly background independent approach to quantum gravity and gauge theories in the continuum. The regularisation is a geometric variant of Slavnov’s scheme consisting of a modified action, which suppresses high momentum modes, supplemented by Pauli–Villars determinants in the path integral measure. An exact renormalisation group flow equation for the Wilsonian effective action is derived by requiring that the path integral is invariant under a change in the cutoff scale while preserving quasi-locality. The renormalisation group flow is defined directly on the space of gauge invariant actions without the need to fix the gauge. We show that the one-loop beta function in Yang–Mills and the one-loop divergencies of General Relativity can be calculated without fixing the gauge. As a first non-perturbative application we find the form of the Yang–Mills beta function within a simple truncation of the Wilsonian effective action.

Kaluza-Klein modes of U(1) gauge vector field on brane with codimension-d

From the paper of Fu et al. (2019) [44], it is known that the effective action of a massless U(1) gauge vector field on a codimension-2 brane is gauge invariant due to the coupling between the vector Kaluza-Klein (KK) modes with two types of scalar KK modes. It is interesting to generalize this result to a brane world model with an arbitrary number of extra dimensions. In this work, we first investigate the case with three extra dimensions. After KK decomposition, there are three types of scalar KK modes. In addition to the mutual coupling between these scalar modes, there are also coupling between the scalar and the vector KK modes. The coupling constants are not all independent. The relationships between the coupling constants enable us to obtain a gauge invariant effective action, from which we can see that the masses of the vector KK modes are contributed by all the three extra dimensions. The masses of the scalar modes, however, are contributed only by two of the three extra dimensions. Then we generalize our results to branes with codimension d (d=1,2...).

Generalised Fayet–Iliopoulos Terms in Supergravity

The U(1) vector multiplet theory with the Fayet-Iliopoulos (FI) term is one of the oldest and simplest models for spontaneously broken rigid supersymmetry. Lifting the FI term to supergravity requires gauged $R$-symmetry, as was first demonstrated in 1977 by Freedman within ${\cal N}=1$ supergravity. There exists an alternative to the standard FI mechanism, which is reviewed in this conference paper. It is obtained by replacing the FI model with a manifestly gauge-invariant action such that its functional form is determined by two arbitrary real functions of a single complex variable. One of these functions generates a superconformal kinetic term for the vector multiplet, while the other yields a generalised FI term. Coupling such a vector multiplet model to supergravity does not require gauging of the $R$-symmetry. These generalised FI terms are consistently defined for any off-shell formulation for ${\cal N}=1$ supergravity, and are compatible with a supersymmetric cosmological term.

New locally (super)conformal gauge models in Bach-flat backgrounds

For every conformal gauge field {h}_{alpha (n)overset{cdot }{alpha }(m)} in four dimensions, with n ≥ m > 0, a gauge-invariant action is known to exist in arbitrary conformally flat backgrounds. If the Weyl tensor is non-vanishing, however, gauge invariance holds for a pure conformal field in the following cases: (i) n = m = 1 (Maxwell’s field) on arbitrary gravitational backgrounds; and (ii) n = m + 1 = 2 (conformal gravitino) and n = m = 2 (conformal graviton) on Bach-flat backgrounds. It is believed that in other cases certain lower-spin fields must be introduced to ensure gauge invariance in Bach-flat backgrounds, although no closed-form model has yet been constructed (except for conformal maximal depth fields with spin s = 5/2 and s = 3). In this paper we derive such a gauge-invariant model describing the dynamics of a conformal gauge field {h}_{alpha (3)overset{cdot }{alpha }} coupled to a self-dual two-form. Similar to other conformal higher-spin theories, it can be embedded in an off-shell superconformal gauge-invariant action. To this end, we introduce a new family of mathcal{N} = 1 superconformal gauge multiplets described by unconstrained prepotentials ϒα(n), with n > 0, and propose the corresponding gauge-invariant actions on conformally-flat backgrounds. We demonstrate that the n = 2 model, which contains {h}_{alpha (3)overset{cdot }{alpha }} at the component level, can be lifted to a Bach-flat background provided ϒα(2) is coupled to a chiral spinor Ωα. We also propose families of (super)conformal higher-derivative non-gauge actions and new superconformal operators in any curved space. Finally, through considerations based on supersymmetry, we argue that the conformal spin-3 field should always be accompanied by a conformal spin-2 field in order to ensure gauge invariance in a Bach-flat background.