Gauge-fixed Action Research Articles

Conformal symmetry in quantum gravity

We study the problem of how to derive conformal symmetry in the framework of quantum gravity. We start with a generic gravitational theory which is invariant under both the general coordinate transformation (GCT) and Weyl transformation (or equivalently, local scale transformation), and then construct its BRST formalism by fixing the gauge symmetries by the extended de Donder gauge and scalar gauge conditions. These gauge-fixing conditions are invariant under global GL(4) and global scale transformations. The gauge-fixed and BRST invariant quantum action possesses a huge Poincaré-like IOSp(10|10) global symmetry, from which we can construct an extended conformal symmetry in a flat Minkowski background in the sense that the Lorentz symmetry is replaced with the GL(4) symmetry. Moreover, we construct the conventional conformal symmetry out of this extended symmetry. With a flat Minkowski background ⟨gμν⟩=ημν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle g_{\\mu \ u } \\rangle = \\eta _{\\mu \ u }$$\\end{document} and a non-zero scalar field ⟨ϕ⟩≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\phi \\rangle \ e 0$$\\end{document}, the GL(4) and global scale symmetries are spontaneously broken to the Lorentz symmetry, thereby proving that the graviton and the dilaton are respectively the corresponding Nambu–Goldstone bosons, and therefore they must be exactly massless at nonperturbative level. One of remarkable aspects in our findings is that in quantum gravity, a derivation of conformal symmetry does not depend on a classical action, and its generators are built from only the gauge-fixing and the FP ghost actions. Finally, we address a generalized Zumino theorem in quantum gravity.

A novel background field approach to the confinement-deconfinement transition

We propose a novel approach to the confinement-deconfinement transition in Yang-Mills theories in the context of gauge-fixed calculations. The method is based on a background-field generalisation of the Landau gauge (to which it reduces at vanishing temperature) with a given, center-symmetric background. This is to be contrasted with most implementations of background field methods in gauge theories, where one uses a variable, self-consistent background. Our proposal is a bona fide gauge fixing that can easily be implemented on the lattice and in continuum approaches. The resulting gauge-fixed action explicitly exhibits the center symmetry of the nonzero temperature theory that controls the confinement-deconfinement transition. We show that, in that gauge, the electric susceptibility diverges at a second order transition [e.g., in the SU(2) theory], so that the gluon propagator is a clear probe of the transition. We implement our proposal in the perturbative Curci-Ferrari model, known for its successful description of various infrared aspects of Yang-Mills theories, including the confinement-deconfinement transition. Our one-loop calculation confirms our general expectation for the susceptibility while providing transition temperatures in excellent agreement with the SU(2) and SU(3) lattice values. Finally, the Polyakov loops above the transition show a more moderate rise, in contrast to previous implementations of the Curci-Ferrari model using a self-consistent background, and our SU(3) result agrees quite well with the lattice data in the range [0,2T_c][0,2Tc].

SL(2,R) Wess-Zumino-Novikov-Witten spin-chain σ -model

The $\textrm{SL}(2,\mathbb{R})$ Wess-Zumino-Novikov-Witten model realises bosonic-string theory in $\textrm{AdS}_{3}$ with pure Neveu-Schwarz-Neveu-Schwarz flux. We construct an effective action in the semi-classical limit of the model, which corresponds to a $\textrm{SL}(2,\mathbb{R})$ spin-chain $\sigma$-model. We adopt two complementary points of view. Firstly, we consider the classical action. We identify fast and slow target-space coordinates. We impose a gauge-fixing condition to the former. By expanding the gauge-fixed action in an effective coupling, we obtain the effective action for the slow coordinates. Secondly, we consider the spin chain of the model. We postulate a set of coherent states to express a transition amplitude in the spin chain as a path integral. We observe that the temporal interval is discretised in terms of the step length of the spatial interval. This relationship implies that the Landau-Lifshitz limit of the spin chain involves both intervals. The limit yields a semi-classical path integral over coherent states, wherein we identify the effective action again.

Anomaly inflow and holography

We systematically study the perturbative anomaly inflow by the bulk Chern-Simons (CS) theory in a slice of five-dimensional anti-de Sitter spacetime (AdS5). The introduction of UV and IR 3-branes makes the anomaly story remarkably rich and many interesting aspects can be obtained, including weakly gauging and spontaneous symmetry breaking of the global symmetries of the dual 4D CFT. Our main contribution is to provide a unified and comprehensive discussion of the subject, together with a detailed description of the dual CFT picture for each case. To this end, we employ a gauge-fixed effective action suitable for a holographic study, which allows us to incorporate general UV and IR boundary conditions (BCs). As part of the process, we reproduce many known results in the literature, such as ’t Hooft anomaly matching for unbroken symmetry (Neumann IR-BC) and (gauged) Wess-Zumino-Witten (WZW) action for broken symmetry (IR-BC breaks the bulk group G → H). In addition, we show that anomaly matching occurs for ABJ anomalies as well as ’t Hooft anomalies, which suggests anomalies inflowed from the bulk CS theory are necessarily free of mixed anomalies with the confining gauge force of the 4D dual CFT. In the case of broken symmetry, we prove that the “would-be” Goldstone bosons associated with the weakly gauged symmetry are completely removed by a proper field redefinition, provided the anomaly from the bulk is exactly cancelled by the boundary contribution, hence confirming the standard expectation. Moreover, we present a holographic formulation of Witten’s argument for the quantization condition for the WZW action, and argue in favor of an alternative way to obtain the same condition using a “deformed” theory (different BCs). We work out several examples, including a product group with mixed anomaly, and identify the corresponding dual CFT picture. We consider a fully general case typically arising in the context of dynamical electroweak symmetry breaking.

Soft-collinear effective theory: BRST formulation

We provide a BRST formalism for the soft-collinear effective theory describing interactions of soft and collinear degrees of freedom in the presence of a hard interaction. In particular, we develop a BRST symmetry transformation for SCET theory. We further generalize the BRST formulation by making the transformation parameter field dependent. This establishes a mapping between several SCET actions consistently when defined in different gauge conditions. In fact, a definite structure of gauge-fixed actions corresponding to any particular gauge condition can be generated for SCET theory using our formulation.

Single extra dimension from \u03ba-Poincar\xe9 and gauge invariance

We show that κ-Poincaré invariant gauge theories on κ-Minkowski space with physically acceptable commutative (low energy) limit must be 5-d. The gauge invariance requirement of the action fixes the dimension of the κ-Minkowski space to d = 5 and selects the unique twisted differential calculus with which the construction can be achieved. We characterize a BRST symmetry related to the 5-d noncommutative gauge invariance though the definition of a nilpotent operation, which is used to construct a gauge-fixed action. We also consider standard scenarios assuming (compactification of) flat extra dimension, for which the 5-d deformation parameter κ can be viewed as the bulk 5-d Planck mass. We study physical properties of the resulting 4-d effective theories. Recent data from collider experiments require κ ≳ mathcal{O} (1013) GeV. The use of standard test of in-vacuo dispersion relations of Gamma Ray Burst photons increases this lower bound by 4 orders of magnitude. The robustness of this bound is discussed in the light of possible new features of noncommutative causal structures.

Physical Hamiltonian for mimetic gravity

Starting from a local action for mimetic gravity that includes higher derivatives of a scalar field $\phi$, we derive a gauge-fixed canonical action of the theory in the ADM canonical formalism in the time gauge $\phi=t$. This reduced action reveals (i) a non-vanishing conserved physical Hamiltonian that is a sum of two terms, the expression for the Hamiltonian constraint of general relativity and a function of the expansion scalar, and (ii) a reduced symplectic structure that geometrically provides the Dirac brackets. As applications of our general analysis, we compute the physical Hamiltonians and canonical equations for perturbations around Minkowski spacetime, homogeneous cosmologies, and spherically symmetric spacetimes.

A SO(1,3) gauge theory of quantum gravity: quantization and renormalizability proof

A new SO(1,3) gauge field theory classically equivalent to general relativity in a limiting case is quantized and the gauge-fixed path integral representation of the quantum effective action (QEA) is derived. Both the gauge-fixed classical action and the QEA are shown to be invariant under nilpotent BRST variations of the gauge, matter, ghost, antighost and Nakanishi–Lautrup fields defining the theory and a Zinn-Justin equation constraining the QEA is derived. Dimensional analysis and the various linear constraints put on the QEA plus the ones from the non-linear Zinn-Justin equation are deployed to demonstrate full renormalizability such that all infinities appearing in a perturbative expansion of the QEA can be absorbed into the gauge-fixed classical action solely by field renormalizations and coupling redefinitions—providing the third step in consistently quantizing the SO(1,3) gauge field theory at hands, and with it potentially gravity.

Hamiltonian BRST formalism for gauge fields on black hole spacetimes

We investigate the Becchi-Rouet-Stora-Tyutin (BRST) formalism for gauge theories on spherically symmetric black hole spacetimes, with or without a cosmological constant ($\Lambda\geq0$). This is illustrated through the example of scalar electrodynamics. We first demonstrate that the horizons contribute additional surface terms to the Gauss law constraint of the theory when gauge transformations are not required to vanish on the horizons. We then consider the BRST invariant path integral including these surface terms following the Hamiltonian BRST formalism. We fix a radiation-like gauge which involves null components of the electromagnetic field at the horizons. We find that the presence of the surface terms in the constraint forces the ghost and gauge fixing actions to include additional surface integrals at the horizons. The null combination of the gauge fields at the horizons is shown to modify the ghost number charge of the theory through additional terms at the horizons. We also demonstrate how one can construct a gauge-fixing fermion which generates its own nilpotent symmetry transformations, called co-BRST transformations, that leave the theory invariant. The BRST and co-BRST transformations are further used to identify dressed (gauge invariant) fields of theory, whose dressings are affected by the presence of Killing horizons. We conclude with a discussion of potential applications of our results in soft limits and thermal field theories on black hole backgrounds.

Chiral strings, topological branes, and a generalised Weyl-invariance

We introduce a sigma model Lagrangian generalising a number of new and old models which can be thought of as chiral, including the Schild string, ambitwistor strings, and the recently introduced tensionless AdS twistor strings. This “chiral sigma model” describes maps from a [Formula: see text]-brane worldvolume into a symplectic space and is manifestly invariant under diffeomorphisms as well as under a “generalised Weyl invariance” acting on space–time coordinates and worldvolume fields simultaneously. Construction of the Batalin–Vilkovisky master action leads to a BRST operator under which the gauge-fixed action is BRST-exact; we discuss whether this implies that the chiral brane sigma model defines a topological field theory.

Perturbative study of the QCD phase diagram for heavy quarks at nonzero chemical potential: Two-loop corrections

We extend a previous investigation of the QCD phase diagram with heavy quarks in the context of background field methods by including the two-loop corrections to the background field effective potential. The nonperturbative dynamics in the pure-gauge sector is modeled by a phenomenological gluon mass term in the Landau-DeWitt gauge-fixed action, which results in an improved perturbative expansion. We investigate the phase diagram at nonzero temperature and (real or imaginary) chemical potential. Two-loop corrections yield an improved agreement with lattice data as compared to the leading-order results. We also compare with the results of nonperturbative approaches. We further study the equation of state as well as the thermodynamic stability of the system at two-loop order. Finally, we show, using simple thermodynamic arguments, that the behavior of the Polyakov loops as functions of the chemical potential complies with their interpretation in terms of quark and anti-quark free energies.

Fundamental theorem on gauge fixing at the action level

Regardless of the long history of gauge theories, it is not well recognized under which condition gauge fixing at the action level is legitimate. We address this issue from the Lagrangian point of view, and prove the following theorem on the relation between gauge fixing and Euler-Lagrange equations: In any gauge theory, if a gauge fixing is complete, i.e., the gauge functions are determined uniquely by the gauge conditions, the Euler-Lagrange equations derived from the gauge-fixed action are equivalent to those derived from the original action supplemented with the gauge conditions. Otherwise, it is not appropriate to impose the gauge conditions before deriving Euler-Lagrange equations as it may in general lead to inconsistent results. The criterion to check whether a gauge fixing is complete or not is further investigated. We also provide applications of the theorem to scalar-tensor theories and make comments on recent relevant papers on theories of modified gravity, in which there are confusions on gauge fixing and counting physical degrees of freedom.

BRST quantization of cosmological perturbations

BRST quantization is an elegant and powerful method to quantize theories with local symmetries. In this article we study the Hamiltonian BRST quantization of cosmological perturbations in a universe dominated by a scalar field, along with the closely related quantization method of Dirac. We describe how both formalisms apply to perturbations in a time-dependent background, and how expectation values of gauge-invariant operators can be calculated in the in-in formalism. Our analysis focuses mostly on the free theory. By appropriate canonical transformations we simplify and diagonalize the free Hamiltonian. BRST quantization in derivative gauges allows us to dramatically simplify the structure of the propagators, whereas Dirac quantization, which amounts to quantization in synchronous gauge, dispenses with the need to introduce ghosts and preserves the locality of the gauge-fixed action.

Comparing Poisson Sigma Model with A-model

We discuss the A-model as a gauge fixing of the Poisson Sigma Model with target a symplectic structure. We complete the discussion in [arXiv:0706.3164], where a gauge fixing defined by a compatible complex structure was introduced, by showing how to recover the A-model hierarchy of observables in terms of the AKSZ observables. Moreover, we discuss the off-shell supersymmetry of the A-model as a residual BV symmetry of the gauge-fixed PSM action.

Green-Schwarz superstring on the lattice

We consider possible discretizations for a gauge-fixed Green-Schwarz action of Type IIB superstring. We use them for measuring the action, from which we extract the cusp anomalous dimension of planar $\mathcal{N}=4$ SYM as derived from AdS/CFT, as well as the mass of the two $AdS$ excitations transverse to the relevant null cusp classical string solution. We perform lattice simulations employing a Rational Hybrid Monte Carlo (RHMC) algorithm and two Wilson-like fermion discretizations, one of which preserves the global $SO(6)$ symmetry of the model. We compare our results with the expected behavior at various values of $g=\frac{\sqrt{\lambda}}{4\pi}$. For both the observables, we find a good agreement for large $g$, which is the perturbative regime of the sigma-model. For smaller values of $g$, the expectation value of the action exhibits a deviation compatible with the presence of quadratic divergences. After their non-perturbative subtraction the continuum limit can be taken, and suggests a qualitative agreement with the non-perturbative expectation from AdS/CFT. Furthermore, we detect a phase in the fermion determinant, whose origin we explain, that for very small $g$ leads to a sign problem not treatable via standard reweigthing. The continuum extrapolations of the observables in the two different discretizations agree within errors, which is strongly suggesting that they lead to the same continuum limit. Part of the results discussed here were presented earlier in arXiv:1601.04670.

A systematic study of finite BRST-BV transformations within W–X formulation of the standard and the Sp(2)-extended field–antifield formalism

Finite BRST-BV transformations are studied systematically within the W-X formulation of the standard and the Sp(2)-extended field-antifield formalism. The finite BRST-BV transformations are introduced by formulating a new version of the Lie equations. The corresponding finite change of the gauge-fixing master action X and the corresponding Ward identity are derived.

Quark-gluon vertex from the Landau gauge Curci-Ferrari model

We investigate the quark-gluon three-point correlation function within a one-loop computation performed in the Curci-Ferrari massive extension of the Faddeev-Popov gauge-fixed action. The mass term is used as a minimal way for taking into account the influence of the Gribov ambiguity. Our results, with renormalization-group improvement, are compared with lattice data. We show that the comparison is in general very satisfactory for the functions which are compatible with chiral symmetry, except for one. We argue that this may be due to large systematic errors {when extracting this function from} lattice simulations. The quantities which break chiral symmetry are more sensitive to the details of the renormalization scheme. We however manage to reproduce some of them with good precision. The chosen parameters allow to simultaneously fit the quark mass function coming from the quark propagator with a reasonably agreement.

Infrared Abelian dominance without Abelian projection

Maximal Abelian gauge has been a particular choice to study dynamical generation of off-diagonal gluon masses in QCD. This gauge is a special case of an Abelian projection. Abelian dominance is characterized by off-diagonal gluons acquiring masses in the relevant phase. Here we propose a gauge condition which is quadratic in fields and which does not fall in the class of an Abelian projection. We explore the possible vacua of the gauge-fixed effective action of the theory and find evidence that ghost bilinears may be subject to condensation, which would signal acquisition of masses by off-diagonal gluons. Such a vacuum satisfies the requirement of Abelian dominance, providing an example of the hypothesis through a mechanism other than Abelian projection.

Covariant gauges without Gribov ambiguities in Yang-Mills theories

We propose a one-parameter family of nonlinear covariant gauges which can be formulated as an extremization procedure that may be amenable to lattice implementation. At high energies, where the Gribov ambiguities can be ignored, this reduces to the Curci-Ferrari-Delbourgo-Jarvis gauges. We further propose a continuum formulation in terms of a local action which is free of Gribov ambiguities and avoids the Neuberger zero problem of the standard Faddeev-Popov construction. This involves an averaging over Gribov copies with a nonuniform weight, which introduces a new gauge-fixing parameter. We show that the proposed gauge-fixed action is perturbatively renormalizable in four dimensions and we provide explicit expressions of the renormalization factors at one loop. We discuss the possible implications of the present proposal for the calculation of Yang-Mills correlators.

A Renormalizable Theory of Quantum Gravity: Renormalization Proof of the Gauge Theory of Volume Preserving Diffeomorphisms

Inertial and gravitational mass or energy-momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory are shown to obey a BRST-type symmetry. The related Zinn-Justin-type equation restricting the corresponding quantum effective action is established. This equation limits the infinite parts of the quantum effective action to have the same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. The inner space integrals occurring in the quantum effective action which are divergent due to the gauge group's infinite volume are shown to be regularizable in a way consistent with the symmetries of the theory demonstrating as a byproduct that viable quantum gauge field theories are not limited to finite-dimensional compact gauge groups as is commonly assumed.