Gauge Transformations Research Articles

Some disquisitions on cosmological 2-form dualities

In this work we study different aspect of self-interacting 2-form fields with special emphasis in their cosmological applications. We provide the explicit construction of how massless 2-forms are compatible with the cosmological principle without resorting to the dual scalar field formulation. In terms of the 2-form, the residual Euclidean group is non-trivially realised by means of a combination of external spatial translations and internal gauge transformations. After presenting the general discussion of the dualities in cosmological scenarios, we analyse particular examples for some singular models and discuss in some detail the dual descriptions of the DBI, the cuscuton and the ghost condensate as well as the role of the duality in the effective field theories of cosmological perturbations. We then proceed to analysing scenarios with several self-interacting massless 2-forms and we show that they naturally provide the dual description of a solid. We then show how the perfect fluid and superfluids can be obtained by taking the appropriate limits in the dual formulations. We finally consider the case of massive 2-forms and their duals and briefly discuss their potential signatures in gravitational waves astronomy.

Deforming soft algebras for gauge theory

Symmetry algebras deriving from towers of soft theorems can be deformed by a short list of higher-dimension Wilsonian corrections to the effective action. We study the simplest of these deformations in gauge theory arising from a massless complex scalar coupled to F2. The soft gauge symmetry ‘s-algebra’, compactly realized as a higher-spin current algebra acting on the celestial sphere, is deformed and enlarged to an associative algebra containing soft scalar generators. This deformed soft algebra is found to be non-abelian even in abelian gauge theory. A two-parameter family of central extensions of the s-subalgebra are generated by shifting and decoupling the scalar generators. It is shown that these central extensions can also be generated by expanding around a certain non-trivial but Lorentz invariant shockwave type background for the scalar field.

Noninvertible Time-Reversal Symmetry.

In gauge theory, it is commonly stated that time-reversal symmetry only exists at θ=0 or π for a 2π-periodic θ angle. In this Letter, we point out that in both the free Maxwell theory and massive QED, there is a noninvertible time-reversal symmetry at every rational θ angle, i.e., θ=πp/N. The noninvertible time-reversal symmetry is implemented by a conserved, antilinear operator without an inverse. It is a composition of the naive time-reversal transformation and a fractional quantum Hall state. We also find similar noninvertible time-reversal symmetries in non-Abelian gauge theories, including the N=4 SU(2) super Yang-Mills theory along the locus |τ|=1 on the conformal manifold.

A note on the asymptotic symmetries of electromagnetism

We extend the asymptotic symmetries of electromagnetism in order to consistently include angle-dependent u(1) gauge transformations ϵ that involve terms growing at spatial infinity linearly and logarithmically in r, ϵ ~ a(θ, φ)r + b(θ, φ) ln r + c(θ, φ). The charges of the logarithmic u(1) transformations are found to be conjugate to those of the mathcal{O} (1) transformations (abelian algebra with invertible central term) while those of the mathcal{O} (r) transformations are conjugate to those of the subleading mathcal{O} (r−1) transformations. Because of this structure, one can decouple the angle-dependent u(1) asymptotic symmetry from the Poincaré algebra, just as in the case of gravity: the generators of these internal transformations are Lorentz scalars in the redefined algebra. This implies in particular that one can give a definition of the angular momentum which is free from u(1) gauge ambiguities. The change of generators that brings the asymptotic symmetry algebra to a direct sum form involves non linear redefinitions of the charges. Our analysis is Hamiltonian throughout and carried at spatial infinity.

Soft gluon theorems in curved spacetime

In this paper, we derive a soft gluon theorem in the near horizon region of the Schwarzschild black hole from the Ward identity of the near horizon large gauge transformation. The flat spacetime soft gluon theorem can be recovered as a limiting case of the curved spacetime.

On the breakdown of space–time in general relativity

Based on the canonical quantization of d > 2-dimensional general relativity (GR) via the Dirac constraint formalism (also termed as ‘constraint quantization’), we propose the loss of covariance as a fundamental property of the theory. This breakdown occurs for the first-order Einstein–Hilbert action, whereby the loss of diffeomorphism invariance, besides first-class constraints, second-class constraints also exist leading to non-standard ghost fields that render the path integral non-covariant. We also attempt, for the first time, the canonical quantization via calculation of the path integral for the equivalent Hamiltonian formulation of GR, for which only first-class constraints exist. However, loss of covariance still occurs in this action due to loss of diffeomorphism invariance and structures arising from non-covariant constraints in the path integral. In contrast, we find that covariance as a symmetry is restored and quantization with perturbative calculations is possible in the weak limit of the gravitational field of these actions. Hence, we first establish, for the first time, that the breakdown in space–time is a property of GR itself as its limitation, indicating it to be an effective field theory (EFT). We further propose that the breakdown of space–time occurs as a non-perturbative feature of GR in the strong field limit of the theory. Besides GR, we also note that covariance is preserved when constraint quantization is conducted for non-Abelian gauge theories, such as the Yang–Mills theory. These findings are novel from a canonical gravity formalism and EFT approach, and are consistent with GR singularity theorems, yet are general and extend them, as the singularity theorems indicate breakdown at a strong field limit of GR in black holes. Our findings are in contrast to the asymptotic safety program. They support emergent theories of space–time and gravity, though are unique, as they do not require thermodynamics such as the entropic gravity program. From an EFT view, these indicate that new degrees of freedom and(or) principles in the non-perturbative sector of the full theory are a requirement, whereby covariance as a symmetry is broken in the high-energy (strong field) sector of GR.

Self-learning Monte Carlo for non-Abelian gauge theory with dynamical fermions

In this paper, we develop the self-learning Monte-Carlo (SLMC) algorithm for non-Abelian gauge theory with dynamical fermions in four dimensions to resolve the autocorrelation problem in lattice QCD. We perform simulations with the dynamical staggered fermions and plaquette gauge action by both in the hybrid Monte-Carlo (HMC) and SLMC for zero and finite temperature to examine the validity of SLMC. We confirm that SLMC can reduce autocorrelation time in non-Abelian gauge theory and reproduce results from HMC. For finite temperature runs, we confirm that SLMC reproduces correct results with HMC, including higher-order moments of the Polyakov loop and the chiral condensate. Besides, our finite temperature calculations indicate that four flavor ${\mathrm{QC}}_{2}\mathrm{D}$ with $\stackrel{^}{m}=0.5$ is likely in the crossover regime in the Colombia plot.

Nonthermal dynamics in a spin- 12 lattice Schwinger model

Local gauge symmetry is intriguing for the study of quantum thermalization breaking. For example, in the high-spin lattice Schwinger model (LSM), the local U(1) gauge symmetry underlies the disorder-free many-body localization (MBL) dynamics of matter fields. This mechanism, however, would not work in a spin-$\frac{1}{2}$ LSM due to the absence of electric energy in the Hamiltonian. In this paper, we show that the spin-$\frac{1}{2}$ LSM can also exhibit disorder-free MBL dynamics, as well as entropy prethermalization, by introducing a four-fermion interaction into the system. The interplay between the fermion interaction and U(1) gauge symmetry endows the gauge fields with an effectively disordered potential which is responsible for the thermalization breaking. It induces anomalous (i.e., nonthermal) behaviors in the long-time evolution of such quantities as local observables, entanglement entropy, and correlation functions. Our work offers a different platform to explore emergent nonthermal dynamics in state-of-the-art quantum simulators with gauge symmetries.

Convergence of a discretization of the Maxwell–Klein–Gordon equation based on finite element methods and lattice gauge theory

AbstractThe Maxwell–Klein–Gordon equations are a set of coupled nonlinear time‐dependent wave equations, used to model the interaction of an electromagnetic field with a particle. The solutions, expressed with a magnetic vector potential, are invariant under gauge transformations. This characteristic implies a constraint on the solution fields that might be broken at the discrete level. In this article, we propose and study a constraint preserving numerical scheme for this set of equations in dimension 2. At the semidiscrete level, we combine conforming Finite Element discretizations with the so‐called Lattice Gauge Theory to design a compatible gauge invariant discretization, leading to preservation of a discrete constraint. Relying on energy techniques and compactness arguments, we establish the convergence of this semidiscrete scheme, without a priori knowledge of the solution. Finally, at the fully discrete level, we propose a compatible explicit time‐integration strategy of leapfrog type. We implement the resulting fully discrete scheme and provide assessment on academic scenarios.

Optimal regularity and Uhlenbeck compactness for general relativity and Yang–Mills theory

We announce the extension of optimal regularity and Uhlenbeck compactness to the general setting of connections on vector bundles with non-compact gauge groups over non-Riemannian manifolds, including the Lorentzian metric connections of general relativity (GR). Compactness is the essential tool of mathematical analysis for establishing the validity of approximation schemes. Our proofs are based on the theory of the RT-equations for connections with L p curvature. Solutions of the RT-equations furnish coordinate and gauge transformations which give a non-optimal connection a gain of one derivative over its Riemann curvature (i.e. to optimal regularity). The RT-equations are elliptic regardless of metric signature, and regularize singularities in solutions of the hyperbolic Einstein equations. As an application, singularities at GR shock waves are removable, implying geodesic curves, locally inertial coordinates and the Newtonian limit all exist. By the extra derivative, we extend Uhlenbeck compactness from Uhlenbeck’s setting of vector bundles with compact gauge groups over Riemannian manifolds, to the case of compact and non-compact gauge groups over non-Riemannian manifolds. Our version of Uhlenbeck compactness can also be viewed as a ‘geometric’ improvement of the Div-Curl Lemma, improving weak continuity of wedge products to strong convergence.

Constrained discrete KP hierarchy: The constraint on the tau functions and gauge transformations

Constrained discrete KP hierarchy: The constraint on the tau functions and gauge transformations

Introduction to Renormalization Theory and Chiral Gauge Theories in Dimensional Regularization with Non-Anticommuting γ5

This review provides a detailed introduction to chiral gauge theories, renormalization theory, and the application of dimensional regularization with the non-anticommuting BMHV scheme for γ5. One goal was to show how chiral gauge theories can be renormalized despite the spurious breaking of gauge invariance and how to obtain the required symmetry-restoring counterterms. A second goal was to familiarize the reader with the theoretical basis of the renormalization of chiral gauge theories, the theorems that guarantee the existence of renormalized chiral gauge theories at all orders as consistent quantum theories. Relevant topics include BPHZ renormalization, Slavnov–Taylor identities, the BRST formalism, and algebraic renormalization, as well as the theorems guaranteeing that dimensional regularization is a consistent regularization/renormalization scheme. All of these, including their proofs and interconnections, are explained and discussed in detail. Further, these theoretical concepts are illustrated in practical applications with the example of an Abelian and a non-Abelian chiral gauge theory. Not only the renormalization procedure for such chiral gauge theories is explained step by step, but also the results of all counterterms, including the symmetry-restoring ones, necessary for the consistent renormalization, are explicitly provided.

Privacy-Preserved Distributed Optimization for Multi-Agent Systems With Antagonistic Interactions

This paper is concerned with the privacy-preserving distributed optimization problem for a class of cooperative-competitive multi-agent systems. Each agent only knows its own local objective function and interacts the state information with neighbors through a communication network. By means of the signed graph theory, the antagonistic interactions among agents are considered to characterize both the cooperative and the competitive relationships. With the help of the gauge transformation technique, a structurally balanced undirected signed graph is firstly transformed into a standard undirected graph. Then, the distributed optimization problem subject to signed network is converted into the traditional distributed optimization problem. Subsequently, a novel privacy-preserving distributed optimization algorithm is put forward to 1) minimize the sum of local objective functions; 2) achieve the bipartite consensus for all agents; and 3) avoid the information leakage caused by message exchange among agents, simultaneously. Finally, a simulation example is given to verify the effectiveness of the proposed optimization algorithm.

Deciphering colour building blocks of massive multiparton amplitudes at 4-loops and beyond

The soft function in non-abelian gauge theories exponentiate, and their logarithms can be organised in terms of the collections of Feynman diagrams called Cwebs. The colour factors that appear in the logarithm are controlled by the web mixing matrices. Direct construction of the diagonal blocks of Cwebs using the new concepts of Normal ordering, basis Cweb and Fused-Web was recently carried out in [1]. In this article we establish correspondence between the boomerang webs introduced in [2] and non-boomerang Cwebs. We use this correspondence together with Uniqueness theorem and Fused web formalism introduced in [1] to obtain the diagonal blocks of four general classes of Cwebs to all orders in perturbation theory which also cover all the four loop Boomerang Cwebs connecting four Wilson lines. We also fully construct the mixing matrix of a special Cweb to all orders in perturbation theory.

Crystalline phases at finite winding densities in a quantum link ladder

Condensed matter physics of gauge theories coupled to fermions can exhibit a rich phase structure, but are nevertheless very difficult to study in Monte Carlo simulations when they are afflicted by a sign problem. As an alternate approach, we use tensor network methods to explore the finite density physics of Abelian gauge theories without dynamical matter. As a concrete example, we consider the U(1) gauge invariant quantum link ladder with spin-1/2 gauge fields in an external electric field which cause the winding electric fluxes to condense in the ground state. We demonstrate how the electric flux tubes arrange themselves in the bulk giving rise to crystalline patterns, whose period can be controlled by tuning the external field. We propose observables to detect the transitions in ground state properties not only in numerical experiments, but also in future cold-atom realizations. A systematic procedure for reaching the thermodynamic limit, as well as extending the studies from ladders to extended geometries is outlined.

Generalized gauge transformation and the corresponding Hermitian counterparts of SU(1, 1), SU(2) pseudo-Hermitian Hamiltonians

We study in this paper both the stationary and time-dependent pseudo-Hermitian Hamiltonians consisting respectively of SU(1, 1), SU(2) generators. The pseudo-Hermitian Hamiltonians can be generated from kernel Hermitian-Hamiltonians by a generalized gauge transformation with a non-unitary but Hermitian operator. The metric operator of the biorthogonal sets of eigenstates is simply the square of the transformation operator, which is formulated explicitly. The exact solutions of pseudo-Hermitian Hamiltonians are obtained in terms of the eigenststates of the Hermitian counterparts. We observe a critical point G c of coupling constant, where all eigenstates of the stationary Hamiltonians are degenerate with a vanishing eigenvalue. This critical point is modified as G c (ω) in the time-dependent case including the frequency of external field. Returning to the original gauge we obtain analytically the wave functions and associated non-adiabatic Berry phase, which diverges at the critical point for the SU(2)Hamiltonian. Beyond the critical point Berry phase becomes a complex domain.

Residual gauge symmetry in light-cone electromagnetism

We analyze the residual gauge freedom in light-cone electromagnetism in four dimensions. The standard boundary conditions involved in the so-called lc2formalism, which contains only the two physical degrees of freedom, allow for a subset of residual gauge transformations. We relax the boundary conditions imposed on the fields in order to obtain all the residual gauge transformations. We compute the canonical generators for Poincaré and gauge transformations with these relaxed boundary conditions. This enables us to distinguish between the trivial (proper) and large (improper) gauge transformations in light-cone electromagnetism. We then employ the Newman-Penrose formalism to identify the incoming and outgoing radiation fields. We comment on the quadratic form structure of light-cone Hamiltonians, often encountered in lc2 gauge theories.

Non-Hermitian Floquet-free analytically solvable time-dependent systems [Invited

The non-Hermitian models, which are symmetric under parity (P) and time-reversal (T) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually demands precise control of nonlinear processes, limiting their application. In this paper, to bypass this obstacle, we introduce a class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet) that can describe a two-level system with temporally modulated on-site potential and couplings. We show that implementing an appropriate non-Unitary gauge transformation converts the original system to an effective one with a balanced gain and loss. This will allow us to derive the evolution of states analytically. Our proposed class of Hamiltonians can be employed in different platforms such as electronic circuits, acoustics, and photonics to design structures with hidden PT-symmetry potentially without imaginary onsite amplification and absorption mechanism to obtain an exceptional point.

Radiative phase space extensions at all orders in r for self-dual Yang-Mills and gravity

Working in the self-dual sector for Yang-Mills and gravity, we show how to construct an extended phase space at null infinity, to all orders in the radial expansion. This formalises the symmetry origin of the infrared behaviour of these theories to all subleading orders. As a corollary, we also derive a double copy mapping from a subset of YM gauge transformations to a subset of diffeomorphisms to all orders in the transformation parameters, which to our knowledge has not been presented before in the literature.

Confinement in non-Abelian lattice gauge theory via persistent homology

We investigate the structure of confining and deconfining phases in SU(2) lattice gauge theory via persistent homology, which gives us access to the topology of a hierarchy of combinatorial objects constructed from given data. Specifically, we use filtrations by traced Polyakov loops, topological densities, holonomy Lie algebra fields, as well as electric and magnetic fields. This allows for a comprehensive picture of confinement. In particular, topological densities form spatial lumps which show signatures of the classical probability distribution of instanton-dyons. Signatures of well-separated dyons located at random positions are encoded in holonomy Lie algebra fields, following the semiclassical temperature dependence of the instanton appearance probability. Debye screening discriminating between electric and magnetic fields is visible in persistent homology and pronounced at large gauge coupling. All employed constructions are gauge-invariant without a priori assumptions on the configurations under study. This work showcases the versatility of persistent homology for statistical and quantum physics studies, barely explored to date.