Gauge Orbit Research Articles

Simplicial graviton from selfdual Ashtekar variables

In perturbative gravity, it is straight-forward to characterize the two local degrees of freedom of the gravitational field in terms of a mode expansion of the linearized perturbation. In the non-perturbative regime, we are in a more difficult position. It is not at all obvious how to construct Dirac observables that can separate the gauge orbits. Standard procedures rely on asymptotic boundary conditions or formal Taylor expansions of relational observables. In this paper, we lay out a new non-perturbative lattice approach to tackle the problem in terms of Ashtekar’s self-dual formulation. Starting from a simplicial decomposition of space, we introduce a local kinematical phase space at the lattice sites. At each lattice site, we introduce a set of constraints that replace the generators of the hypersurface deformation algebra in the continuum. We show that the discretized constraints close under the Poisson bracket. The resulting reduced phase space describes two complex physical degrees of freedom representing the two radiative modes at the discretized level. The paper concludes with a discussion of the key open problems ahead and the implications for quantum gravity.

Vacuum stability conditions of the general two-Higgs-doublet potential

In this paper, we present the novel analytical expressions for the bounded-from-below or the vacuum stability conditions of scalar potential for a general two-Higgs-doublet model by using the concepts of co-positivity and the gauge orbit spaces. More precisely, several sufficient conditions and necessary conditions are established for the vacuum stability of the general 2HDM potential, respectively. We also give an equivalent condition of the vacuum stability of the general 2HDM potential in theory, and then, apply it to derive the analytical necessary conditions of the general 2HDM potential. Meanwhile, the semi-positive definiteness is proved for a class of 4th-order two-dimensional complex tensor.

Switching quantum reference frames in the N-body problem and the absence of global relational perspectives

Given the importance of quantum reference frames (QRFs) to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of QRFs, which is valid in both fields. Here we continue with such a unifying approach, begun in [Quantum 4, 225 (2020)], whose key ingredient is a symmetry principle, which enforces physics to be relational. Thanks to gauge related redundancies, this leads to a perspective-neutral structure which contains all frame choices at once and via which frame perspectives can be consistently switched. Formulated in the language of constrained systems, the perspective-neutral structure is the constraint surface classically and the gauge invariant Hilbert space in the Dirac quantized theory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. QRF changes thus amount to a gauge transformation. We show that they take the form of `quantum coordinate changes'. We illustrate this in a general mechanical model, namely the relational N-body problem in 3D space with rotational and translational symmetry. This model is especially interesting because it features the Gribov problem so that globally valid gauge fixing conditions, and hence relational frame perspectives, are absent. The constraint surface is topologically non-trivial and foliated by 3-, 5- and 6-dimensional gauge orbits, where the lower dimensional orbits are a set of measure zero. The N-body problem also does not admit globally valid canonically conjugate pairs of Dirac observables. These challenges notwithstanding, we exhibit how one can construct the QRF transformations for the 3-body problem. Our construction also sheds new light on the generic inequivalence of Dirac and reduced quantization through its interplay with QRF perspectives.

Poisson gauge models and Seiberg-Witten map

The semiclassical limit of full non-commutative gauge theory is known as Poisson gauge theory. In this work we revise the construction of Poisson gauge theory paying attention to the geometric meaning of the structures involved and advance in the direction of a further development of the proposed formalism, including the derivation of Noether identities and conservation of currents. For any linear non-commutativity, Θab(x) = {f}_c^{ab} xc, with {f}_c^{ab} being structure constants of a Lie algebra, an explicit form of the gauge Lagrangian is proposed. In particular a universal solution for the matrix ρ defining the field strength and the covariant derivative is found. The previously known examples of κ-Minkowski, λ-Minkowski and rotationally invariant non-commutativity are recovered from the general formula. The arbitrariness in the construction of Poisson gauge models is addressed in terms of Seiberg-Witten maps, i.e., invertible field redefinitions mapping gauge orbits onto gauge orbits.

Gauge and scalar fields on CP2 : A gauge-invariant analysis. II. The measure for gauge fields and a 4d WZW theory

We consider the volume of the gauge orbit space for gauge fields on four-dimensional complex projective space. The analysis uses a parametrization of gauge fields where gauge transformations act homogeneously on the fields, facilitating a manifestly gauge-invariant analysis. The volume element contains a four-dimensional Wess-Zumino-Witten (WZW) action for a hermitian matrix-valued field. There is also a mass-like term for certain components of the gauge field. We discuss how the mass term could be related to results from lattice simulations as well as Schwinger-Dyson equations. We argue for a kinematic regime where the Yang-Mills theory can be approximated by the 4d-WZW theory. The result is suggestive of the instanton liquid picture of QCD. Further it is also indicative of the mechanism for confinement being similar to what happens in two dimensions.

Simulating Yang-Mills theories with a complex coupling

We propose a novel simulation strategy for Yang-Mills theories with a complex coupling, based on the Lefschetz thimble decomposition. We envisage, that the approach developed in the present work, can also be adapted to QCD at finite density, and real time simulations. Simulations with Lefschetz thimbles offer a potential solution to sign problems in Monte Carlo calculations within many different models with complex actions. We discuss the structure of Generalized Lefschetz thimbles for pure Yang-Mills theories with a complex gauge coupling $\beta$ and show how to incorporate the gauge orbits. We propose to simulate such theories on the union of the tangential manifolds to the relevant Lefschetz thimbles attached to the critical manifolds of the Yang-Mills action. We demonstrate our algorithm on a (1+1)-dimensional U(1) model and discuss how, starting from the main thimble result, successive subleading thimbles can be taken into account via a reweighting approach. While we face a residual sign problem, our novel approach performs exponentially better than the standard reweighting approach.

Darboux Transforms for the $${\hat{B}}_n^{(1)}$$-Hierarchy

The $${\hat{B}}_n^{(1)}$$ -hierarchy is constructed from the standard splitting of the affine Kac–Moody algebra $${\hat{B}}_n^{(1)}$$ , the Drinfeld–Sokolov $${\hat{B}}_n^{(1)}$$ –KdV hierarchy is obtained by pushing down the $${\hat{B}}_n^{(1)}$$ -flows along certain gauge orbit to a cross section of the gauge action. In this paper, we

A note on Faddeev-Popov action for doubled-yet-gauged particle and graded Poisson geometry

The section condition of Double Field Theory has been argued to mean that doubled coordinates are gauged: a gauge orbit represents a single physical point. In this note, we consider a doubled and at the same time gauged particle action, and show that its BRST formulation including Faddeev-Popov ghosts matches with the graded Poisson geometry that has been recently used to describe the symmetries of Double Field Theory. Besides, by requiring target spacetime diffeomorphisms at the quantum level, we derive quantum corrections to the classical action involving dilaton, which might be comparable with the Fradkin-Tseytlin term on string worldsheet.

Vacuum stability and symmetry breaking in left-right symmetric model

We derive analytic necessary and sufficient conditions for the vacuum stability of the left-right symmetric model by using the concepts of copositivity and gauge orbit spaces. We also derive the conditions sufficient for successful symmetry breaking and the existence of a correct vacuum. We then compare results obtained from the derived conditions with those from numerical minimization of the scalar potential. Finally, we discuss the renormalization group analysis of the scalar quartic couplings through an example study that satisfies vacuum stability, perturbativity, unitarity and experimental bounds on the physical scalar masses.

Learning a local symmetry with neural networks.

We explore the capacity of neural networks to detect a symmetry with complex local and non-local patterns: the gauge symmetry Z_{2}. This symmetry is present in physical problems from topological transitions to quantum chromodynamics, and controls the computational hardness of instances of spin-glasses. Here, we show how to design a neural network, and a dataset, able to learn this symmetry and to find compressed latent representations of the gauge orbits. Our method pays special attention to system-wrapping loops, the so-called Polyakov loops, known to be particularly relevant for computational complexity.

An approach to stability analyses in general relativity via symplectic geometry

We begin with a review of the statements of non-linear, linear and mode stability of autonomous dynamical systems in classical mechanics, using symplectic geometry. We then discuss what the Arnowitt–Deser–Misner (ADM) phase space and the ADM Hamiltonian of general relativity are, what constitutes a dynamical system, and subsequently present a nascent attempt to draw a formal analogy between the notions of stability in these two theories. We wish to note that we have not discussed here the construction of the reduced phase space by forming the quotient space of the constrained phase space with the gauge orbits. Our approach here is pedagogical and geometric, and the motivation is to unify and simplify a formal understanding of the statements regarding the stability of stationary solutions of general relativity. That is, typically the governing equations of motion of a Hamiltonian dynamical system are simply the flow equations of the associated symplectic Hamiltonian vector field, defined on phase space, and the non-linear stability analysis of its critical points have simply to do with the divergence of its flow there. Further, the linear stability of a critical point is related to the properties of the tangent flow of the Hamiltonian vector field. In the second half of this work, we posit that a study of the genericity of a particular black hole or naked singularity spacetime forming as an endstate of gravitational collapse is equivalent to an inquiry of how sensitive the orbits of the symplectic Hamiltonian vector field of general relativity are to changes in initial data. We demonstrate this by conducting a restricted non-linear stability analysis of the formation of a Schwarzschild black hole, working in the usual initial value formulation of general relativity.

The Hilbert space costratification for the orbit type strata of SU(2)-lattice gauge theory

We construct the Hilbert space costratification of $G=\mathrm{SU}(2)$-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work where we have implemented the classical gauge orbit strata on quantum level within a suitable holomorphic picture. In this picture, each element $\tau$ of the classical stratification corresponds to the zero locus of a finite subset $\{p_i\}$ of the algebra $\mathcal R$ of $G$-invariant representative functions on the complexification of $G^N$. Viewing the invariants as multiplication operators $\hat p_i$ on the Hilbert space $\mathcal H$, the union of their images defines a subspace of $\mathcal H$ whose orthogonal complement $\mathcal H_\tau$ is the element of the costratification corresponding to $\tau$. To construct $\mathcal H_\tau$, one has to determine the images of the $\hat p_i$ explicitly. To accomplish that goal, we construct an orthonormal basis in $\mathcal H$ and determine the multiplication law for the basis elements, that is, we determine the structure constants of $\mathcal R$ in this basis. This part of our analysis applies to any compact Lie group $G$. For $G = \mathrm{SU}(2)$, the above procedure boils down to a problem in combinatorics of angular momentum theory. Using this theory, we obtain the union of the images of the operators $\hat p_i$ as a subspace generated by vectors whose coefficients with respect to our basis are given in terms of Wigner's $3nj$ symbols. The latter are further expressed in terms of $9j$ symbols. Using these techniques, we are also able to reduce the eigenvalue problem for the Hamiltonian of this theory to a problem in linear algebra.

On a renormalizable class of gauge fixings for the gauge invariant operator [formula omitted

The dimension two gauge invariant non-local operator Amin2, obtained through the minimization of ∫d4xA2 along the gauge orbit, allows to introduce a non-local gauge invariant configuration Aμh which can be employed to built up a class of Euclidean massive Yang–Mills models useful to investigate non-perturbative infrared effects of confining theories. A fully local setup for both Amin2 and Aμh can be achieved, resulting in a local and BRST invariant action which shares similarities with the Stueckelberg formalism. Though, unlike the case of the Stueckelberg action, the use of Amin2 gives rise to an all orders renormalizable action, a feature which will be illustrated by means of a class of covariant gauge fixings which, as much as ’t Hooft’s Rζ-gauge of spontaneously broken gauge theories, provide a mass for the Stueckelberg field.

Stringy Gravity: Solving the Dark Problems at ‘short’ distance

Dictated by Symmetry Principle, string theory predicts not General Relativity but its own gravity which assumes the entire closed string massless sector to be geometric and thus gravitational. In terms of R/(MG), i.e. the dimensionless radial variable normalized by mass, Stringy Gravity agrees with General Relativity toward infinity, but modifies it at short distance. At far short distance, gravitational force can be even repulsive. These may solve the dark matter and energy problems, as they arise essentially from small R/(MG) observations: long distance divided by much heavier mass. We address the pertinent differential geometry for Stringy Gravity, stringy Equivalence Principle, stringy geodesics and the minimal coupling to the Standard Model. We highlight the notion of ‘doubled-yet-gauged’ coordinate system, in which a gauge orbit corresponds to a single physical point and proper distance is defined between two gauge orbits by a path integral.

Nonperturbative aspects of Euclidean Yang-Mills theories in linear covariant gauges: Nielsen identities and a BRST-invariant two-point correlation function

In order to construct a gauge invariant two-point function in a Yang-Mills theory, we propose the use of the all-order gauge invariant transverse configurations A^h. Such configurations can be obtained through the minimization of the functional A^2_{min} along the gauge orbit within the BRST invariant formulation of the Gribov-Zwanziger framework recently put forward in [1,2] for the class of the linear covariant gauges. This correlator turns out to provide a characterization of non-perturbative aspects of the theory in a BRST invariant and gauge parameter independent way. In particular, it turns out that the poles of <A^h A^h> are the same as those of the transverse part of the gluon propagator, which are also formally shown to be independent of the gauge parameter entering the gauge condition through the Nielsen identities. The latter follow from the new exact BRST invariant formulation introduced before. Moreover, the correlator <A^h A^h> enables us to attach a BRST invariant meaning to the possible positivity violation of the corresponding temporal Schwinger correlator, giving thus for the first time a consistent, gauge parameter independent, setup to adopt the positivity violation of <A^h A^h> as a signature for gluon confinement. Finally, in the context of gauge theories supplemented with a fundamental Higgs field, we use <A^h A^h> to probe the pole structure of the massive gauge boson in a gauge invariant fashion.

Lifting a weak Poisson bracket to the algebra of forms

We detail the construction of a weak Poisson bracket over a submanifold Σ of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle Π T M , interpreted as a weak Koszul bracket on differential forms on M . This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian function that generates this structure. Such a construction has direct physical interpretation. For a generic gauge system, the submanifold Σ may be viewed as a stationary surface or a constraint surface, with the foliation given by the foliation of the gauge orbits. Through this interpretation, the lift of the weak Poisson structure is simply a lift of the action generating the corresponding BRST operator of the system.

On the gauge-invariant operator [formula omitted] in Euclidean Yang-Mills theories

We review our recent work on the gauge-invariant non-local dimension-two operator Amin2, whose minimization is defined along the gauge orbit. Albeit non-local, the operator Amin2 can be cast in local form through the introduction of an auxiliary Stueckelberg field. The whole procedure results into a local action which turns out to be renormalizable to all orders.

Local and renormalizable framework for the gauge-invariant operatorAmin2in Euclidean Yang-Mills theories in linear covariant gauges

We address the issue of the renormalizability of the gauge-invariant non-local dimension-two operator $A^2_{\rm min}$, whose minimization is defined along the gauge orbit. Despite its non-local character, we show that the operator $A^2_{\rm min}$ can be cast in local form through the introduction of an auxiliary Stueckelberg field. The localization procedure gives rise to an unconventional kind of Stueckelberg-type action which turns out to be renormalizable to all orders of perturbation theory. In particular, as a consequence of its gauge invariance, the anomalous dimension of the operator $A^2_{\rm min}$ turns out to be independent from the gauge parameter $\alpha$ entering the gauge-fixing condition, being thus given by the anomalous dimension of the operator $A^2$ in the Landau gauge.

Fermions, mass-gap and Landau levels: gauge invariant Hamiltonian for QCD in D = 2+1

A gauge-invariant reformulation of QCD in three spacetime dimensions is presented within a Hamiltonian formalism, extending previous work to include fermion fields in the adjoint and fundamental representations. A priori there are several ways to define the gauge-invariant versions of the fermions; a consistent prescription for choosing the fermionic variables is presented. The fermionic contribution to the volume element of the gauge orbit space and the gluonic mass-gap is computed exactly and this contribution is shown to be closely related to the mechanism for induction of Chern–Simons terms by parity-odd fermions. The consistency of the Hamiltonian scheme with known results on index theorems, Landau levels and renormalization of Chern–Simons level numbers is shown in detail. We also comment on the fermionic contribution to the volume element in relation to issues of confinement and screening.

Gauge invariance in simple mechanical systems

This article discusses and explains the Hamiltonian formulation for a class of simple gauge invariant mechanical systems consisting of point masses and idealized rods. The study of these models may be helpful to advanced undergraduate or graduate students in theoretical physics to understand, in a familiar context, some concepts relevant to the study of classical and quantum field theories. We use a geometric approach to derive the Hamiltonian formulation for the model considered in the paper: four equal masses connected by six ideal rods. We obtain and discuss the meaning of several important elements, in particular, the constraints and the Hamiltonian vector fields that define the dynamics of the system, the constraint manifold, gauge symmetries, gauge orbits, gauge fixing, and the reduced phase space.