V. Rubakov, M. Osipov
Within recently proposed scenario which explains flatness of the spectrum of scalar cosmological perturbations by a combination of conformal and global symmetries, we discuss the effect of weak breaking of conformal invariance. We find that the scalar power spectrum obtains a small tilt which depends on both the strength of conformal symmetry breaking and the law of evolution of the scale factor.
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We derive consistency conditions for the CFT data, which systems with exact but spontaneously broken conformal invariance must satisfy.
We consider a relativistic effective field theory of vector boson whose vacuum exhibits spontaneous breaking of Lorentz invariance. We argue that a simple model of this type, considered recently by Kraus and Tomboulis, is obstructed from having a consistent ultraviolet completion according to the diagnostic recently suggested by Adams, Arkani-Hamed, Dubovsky, Nicolis, and Rattazzi.
We propose a mechanism to break the translational invariance of compactified space spontaneously. As a simple model, we study a real $ϕ^4$ model compactified on $M^{D-1}\otimes S^1$ in detail, where we impose a nontrivial boundary condition on $ϕ$ for the $S^1$-direction. It is shown that the translational invariance for the $S^1$-direction is spontaneously broken when the radius $R$ of $S^1$ becomes larger than a critical radius $R^*$ and also that the model behaves like a $ϕ^4$ model on a single kink background for $R \to \infty$. It is pointed out that spontaneous breakdown of translational…
In this work, we investigate a theory of linear Weyl gravity coupled to a scalar field and study the scenario in which Lorentz symmetry is broken by a non-vanishing vacuum expectation value of the Weyl field in the flat space limit after Weyl symmetry breaking. We show that a $CPT$-odd Lorentz-violating interaction is generated after symmetry breaking. Features of different symmetry-broken phases and their dependence on the spacetime character of the generated Lorentz-violating background are discussed. Also, we analyze the naturalness of the theory by showing that the light mass scale is prot…
We argue that when conformal symmetry is spontaneously broken the trace anomalies in the broken and unbroken phases are matched. This puts strong constraints on the various couplings of the dilaton. Using the uniqueness of the effective action for the Goldstone supermultiplet for broken ${\cal N}=1$ superconformal symmetry the dilaton effective action is calculated.
The first model of Foot, Kobakhidze and Volkas described in their work in arXiv:0704.1165 [hep-ph] is a tailored version of our model on broken scale invariance in the standard model presented in hep-th/0403039.
The Ward identities involving the currents associated to the spontaneously broken scale and special conformal transformations are derived and used to determine, through linear order in the two soft-dilaton momenta, the double-soft behavior of scattering amplitudes involving two soft dilatons and any number of other particles. It turns out that the double-soft behavior is equivalent to performing two single-soft limits one after the other. We confirm the new double-soft theorem perturbatively at tree-level in a $D$-dimensional conformal field theory model, as well as nonperturbatively by using …
The majority of renormalizable field theories possessing the scale invariance at the classical level exhibits the trace anomaly once quantum corrections are taken into account. This leads to the breaking of scale and conformal invariance. At the same time any realistic theory must contain gravity and is thus non-renormalizable. We show that discarding the renormalizability it is possible to construct viable models allowing to preserve the scale invariance at the quantum level. We present explicit one-loop computations for two toy models to demonstrate the main idea of the approach. Constructin…
In a wide class of three-dimensional Abelian gauge theories with a bare Chern-Simons term, the Lorentz invariance is spontaneously broken by dynamical generation of a non-vanishing magnetic field. A detailed computation of an energy density of the true vacuum is given. The originally massive photon becomes massless, fulfilling the role of a Nambu-Goldstone boson associated with the spontaneous breaking of the Lorentz invariance.
We introduce Weyl's scale invariance as an additional local symmetry in the standard model of electroweak interactions. An inevitable consequence is the introduction of general relativity coupled to scalar fields a la Dirac and an additional vector particle we call the Weylon. We show that once Weyl's scale invariance is broken, the phenomenon (a) generates Newton's gravitational constant G_N and (b) triggers spontaneous symmetry breaking in the normal manner resulting in masses for the conventional fermions and bosons. The scale at which Weyl's scale symmetry breaks is of order Planck mass. I…
In a class of three-dimensional abelian gauge theory the Lorentz invariance is spontaneously broken by dynamical generation of a magnetic field. An originally topologically massive photon becomes gapless, i.e. p_0=0 at {\vec p}=0. Indeed, the photon is the Nambu-Goldstone boson associated with the spontaneous breaking of the Lorentz invariance. Although symmetry generated by two Lorentz boost generators is broken, there seems to appear only one physical Nambu-Goldstone boson, namely a photon. We argue that the Ward identities in the Nambu-Goldstone theorem are saturated by the photon pole.
We point out that it is possible to associate the electroweak Higgs boson with the pseudo-Goldstone boson of broken scale invariance, thus resolving the hierarchy problem in a technically natural way. We illustrate this idea with two specific gauge models. Besides being consistent with all currently available experimental data, both models maintain the predictive power of the standard model, since the first model has only one additional parameter beyond the standard model, and the second has the same number of free parameters as the standard model.
We investigate five-dimensional braneworld setups with broken Lorentz invariance continuing the developments of our previous paper (arXiv:0712.1136), where a family of static self-tuning braneworld solutions was found. We show that several known braneworld models can be embedded into this family. Then we give a qualitative analysis of spectra of field fluctuations in backgrounds with broken Lorentz invariance. We also elaborate on one particular model and study spectra of scalar and spinor fields in it. It turns out that the spectra we have found possess very peculiar and unexpected properties…
In this paper, based on the basic principles of thermodynamics, we explore the hydrodynamic regime of interacting Lifshitz field theories in the presence of broken rotational invariance. We compute the entropy current and discover new dissipative effects those are consistent with the principle of local entropy production in the fluid. In our analysis, we consider both the parity even as well as the parity odd sector upto first order in the derivative expansion. Finally, we argue that the present construction of the paper could be systematically identified as that of the hydrodynamic descriptio…
Here we show that local scale invariance -- invariance under Weyl rescalings -- may safely coexist with broken electroweak symmetry if assume the Weyl geometric theory to govern the affine structure of spacetime. We find that within the resulting scale invariant theory of gravity cosmological inflation is enough to explain the large hierarchy between the Higgs and Planck masses.
Conformal invariance is spontaneously broken in many physical systems leading to the appearance of a single massless Goldstone mode in the spectrum, the dilaton. The dilaton soft limit is shown to generically encode the action of both the dilatation and the special conformal transformation on observables. For massive on-shell legs these take the form of sub-leading soft theorems. At loop level, we show how anomalous dimensions for coupling constants and fields can be included. We illustrate the general analysis with a variety of formal and phenomenological applications.
Standard inflationary models yield a characteristic signature of a primordial power spectrum with a red tensor and scalar tilt. Nevertheless, Cannone et al recently suggested that, by breaking the assumption of spatial diffeomorphism invariance in the context of the effective field theory of inflation, a blue tensor spectrum can be achieved without violating the Null Energy Condition. In this context, we explore in which cases a blue tensor tilt can be obtained along with a red tilt in the scalar spectrum. Ultimately, we analyze under which conditions this model can reproduce the specific cons…
The cosmological consequences of a simple scalar field model for the generation of Newton's constant through the spontaneous breaking of scale invariance in a curved space-time are again presented and discussed. Such a model leads to a consistent description wherein the introduction of matter introduces a small perturbation on a de Sitter universe and a time dependence of the gravitational coupling.
We argue that scale invariance is not anomalous in quantum field theory, provided it is broken cosmologically. We consider a locally scale invariant extension of the Standard Model of particle physics and argue that it fits both the particle and cosmological observations. The model is scale invariant both classically and quantum mechanically. The scale invariance is broken cosmologically producing all the dimensionful parameters. The cosmological constant or dark energy is a prediction of the theory and can be calculated systematically order by order in perturbation theory. It is expected to b…
We construct a model of inflation based on a low-energy effective theory of spontaneously broken global scale invariance. This provides a shift symmetry that protects the inflaton potential from quantum corrections. Since the underlying scale invariance is non-compact, arbitrarily large inflaton field displacements are readily allowed in the low-energy effective theory. A weak breaking of scale invariance by almost marginal operators provides a non-trivial inflaton minimum, which sets and stabilizes the final low-energy value of the Planck scale. The underlying scale invariance ensures that th…
We study gravity coupled to a cosmological constant and a scale but not conformally invariant sector. In Minkowski vacuum, scale invariance is spontaneously broken. We consider small fluctuations around the Minkowski vacuum. At the linearised level we find that the trace of metric perturbations receives a positive or negative mass squared contribution. However, only for the Fierz-Pauli combination the theory is free of ghosts. The mass term for the trace of metric perturbations can be cancelled by explicitly breaking scale invariance. This reintroduces fine-tuning. Models based on four form fi…
Broken blue-tilted inflationary gravitational waves: a joint analysis of NANOGrav 15-year and BICEP/Keck 2018 data
本文联合NANOGrav 15年与BICEP/Keck 2018数据，对破缺幂律暴胀引力波谱进行马尔可夫链蒙特卡洛分析，发现PTA与CMB频段谱倾斜显著不同，并探讨了其对暴胀模型的约束意义。
If scale invariance is exact, unparticles are unlikely to be probed in colliders since there are stringent constraints from astrophysics and cosmology. However these constraints are inapplicable if scale invariance is broken at a scale mu >~ 1 GeV. The case 1 GeV <~ mu < M_Z is particularly interesting since it allows unparticles to be probed at and below the Z pole. We show that mu can naturally be in this range if only vector unparticles exist, and briefly remark on implications for Higgs phenomenology. We then obtain constraints on unparticle parameters from e+ e- -> mu+ mu- cross-section a…
In this paper, we show the existence of static and rotating universal horizons and black holes in gravitational theories with the broken Lorentz invariance. We pay particular attention on the ultraviolet regime, and show that universal horizons and black holes exist not only in low energy scales but also in the UV scales. This is realized by presenting various static and stationary exact solutions of the full theory of the projectable Hořava gravity with an extra U(1) symmetry in (2+1)-dimensions, which, by construction, is power-counting renormalizable.
There has been a recent interest in considering Quantum Field Theories in which Lorentz Invariance is broken in the UV sector. However attention has been mostly limited to dispersive theories. In this work we provide the generalized settings for studying dissipation. Unitarity is preserved by coupling the original fields to additional (heavy) fields which induce the dissipation. Starting with Lagrangians breaking LI in the UV, we learn that dissipative effects unavoidably develop in the effective theory. We then covariantize these Lagrangians in order to address the trans-Planckian question of…
We derive positivity bounds on EFT coefficients in theories where boosts are spontaneously broken. We employ the analytic properties of the retarded Green's function of conserved currents (or of the stress-energy tensor) and assume the theory becomes conformal in the UV. The method is general and applicable to both cosmology and condensed matter systems. As a concrete example, we look at the EFT of conformal superfluids which describes the universal low-energy dynamics of CFT's at large chemical potential and we derive inequalities on the coefficients of the operators, in three dimensions, at …
Recently a family of solutions of Einstein equations in backgrounds with broken Lorentz invariance was found ArXiv:0712.1136. We show that the gravitational solution recently obtained by Kachru, Liu and Mulligan in ArXiv:0808.1725 is a part of the former solution which was derived earlier in the framework of extra dimensional theories. We show how the energy-momentum and Einstein tensors are related and establish a correspondence between parameters which govern Lorentz invariance violation. Then we demonstrate that scaling behavior of two point correlation functions of local operators in scala…
We study the cosmological evolution of an induced gravity model with a self-interacting scalar field $σ$ and in the presence of matter and radiation. Such model leads to Einstein Gravity plus a cosmological constant as a stable attractor among homogeneous cosmologies and is therefore a viable dark-energy (DE) model for a wide range of scalar field initial conditions and values for its positive $γ$ coupling to the Ricci curvature $γσ^{2}R$.
Non-analyticity of the $S$-matrix with spontaneously broken Lorentz invariance
本文研究有限电荷、Lorentz 不变性自发破缺的 $U(1)$ 复标量可重整理论，证明 Goldstone 粒子的 $S$-矩阵因径向与 Goldstone 模式的能量依赖混合而在复化四动量上呈现非解析性。
Wiggly dilaton: a landscape of spontaneously broken scale invariance
本文利用五维全息模型构造含多个真空解的扭动伸缩子势，表明边界类轴子标量场可根据3-膜能量密度实现relaxion势或轻伸缩子。
It is shown that in the quadratic gravity based on Weyl's conformal geometry, the Planck mass scale can be generated from quantum effects of the gravitational field and the Weyl gauge field via the Coleman-Weinberg mechanism where a local scale symmetry, that is, conformal symmetry, is broken. At the same time, the Weyl gauge field acquires mass less than the Planck mass by absorbing the dilaton. The shape of the effective potential is almost flat owing to a gravitational character and high symmetries, so our model would provide an attractive model for the inflationary universe. We also presen…
Reheating surface volume renormalization and eternal inflation under broken scale invariance
本研究探讨标度不变性轻微破缺（$\varepsilon=n_s-1<0$）如何通过两种独立方法修改重加热表面体积重整化，进而缓和无永恒暴胀条件。
We consider a class of spatially flat cold dark matter (CDM) models, with a cosmological constant and a broken-scale-invariant (BSI) steplike primordial spectrum of adiabatic perturbations, previously found to be in very good agreement with observations. Performing a Fisher matrix analysis, we show that in case of a large gravitational waves (GW) contribution some free parameters (defining the step) of our BSI model can be extracted with remarkable accuracy by the Planck satellite, thanks to the polarisation anisotropy measurements. Further, cosmological parameters can still be found with very…
Lorentz invariance violation and the CPT-odd electromagnetic response of a tilted anisotropic Weyl semimetal
本研究将标准模型扩展的费米子扇区与倾斜各向异性外尔半金属对应，通过热场论和手征动力学推导其电磁响应，发现倾斜与各向异性会在有限密度下引入保留轴子场论结构的新修正，并应用于两种实际材料。
We consider 5D brane world models with broken global 4D Poincar\a'{e} invariance (4D part of the spacetime metric is not conformal to the Minkowski spacetime). The bulk is filled with the negative cosmological constant and may contain a perfect fluid. In the case of empty bulk (the perfect fluid is absent), it is shown that one brane solution always has either a physical or a coordinate singularity in the bulk. We cut off these singularities in the case of compact two brane model and obtain regular exact solutions for both 4D Poincar\a'{e} broken and restored invariance. When the perfect fluid…
While general relativity possesses local Lorentz invariance, both canonical quantum gravity and string theory suggest that Lorentz invariance may be broken at high energies. Broken Lorentz invariance has also been postulated as an explanation for astrophysical anomalies such as the missing GZK cutoff. Therefore, we seek an effective field theory description of gravity where Lorentz invariance is broken. We will construct a candidate theory and then briefly discuss some of the implications.
We explore the possibility that scale symmetry is a quantum symmetry that is broken only spontaneously and apply this idea to the Standard Model (SM). We compute the quantum corrections to the potential of the higgs field ($ϕ$) in the classically scale invariant version of the SM ($m_ϕ=0$ at tree level) extended by the dilaton ($σ$). The tree-level potential of $ϕ$ and $σ$, dictated by scale invariance, may contain non-polynomial effective operators, e.g. $ϕ^6/σ^2$, $ϕ^8/σ^4$, $ϕ^{10}/σ^6$, etc. The one-loop scalar potential is scale invariant, since the loop calculations manifestly preserve t…
We show that in a quadratic gravity based on Weyl's conformal geometry, the Planck mass scale can be generated from quantum effects of the gravitational field and the Weyl gauge field via the Coleman-Weinberg mechanism where a local scale symmetry is broken. At the same time, the Weyl gauge field acquires a mass less than the Planck mass by absorbing the scalar graviton. The shape of the effective potential is almost flat owing to a gravitational character and high symmetries, so our model would provide for an attractive model for the inflationary universe. We also present a toy model showing …
In this paper, we revisit the issue of static hairs of black holes in gravitational theories with broken Lorentz invariance in the case that the speed $c_ϕ$ of the khronon field becomes infinitely large, $c_ϕ = \infty$, for which the sound horizon of the khronon field coincides with the universal horizon, and the boundary conditions at the sound horizon reduce to those given normally at the universal horizons. As a result, less boundary conditions are present in this extreme case in comparison with the case $c_ϕ = $ finite. Then, it would be expected that static hairs might exist. However, we …
We show that if the $α$-attractor model is realized by the spontaneous breaking of the scale symmetry, then the stability and the dynamics of the vector field that gauges the scale symmetry can severely constrain the $α$-parameter as $5/6 < α< 1$ restricting the inflationary predictions in a very tiny region in the $n_s - r$ plane that are in great agreement with the latest Planck data. Although the different values of $α$ do not make a tangible difference for $n_s$ and $r$, they provide radically different scenarios for the post-inflationary dynamics which determines the standard BBN processe…
We re-examine three-dimensional gauge theory with a Chern-Simons term in which the Lorentz invariance is spontaneously broken by dynamical generation of a magnetic field. A non-vanishing magnetic field leads, through the Nambu-Goldstone theorem, to the decrease of zero-point energies of photons, which accounts for a major part of the mechanism. The asymmetric spectral flow plays an important role. The instability in pure Chern-Simons theory is also noted.
We consider a generic scale invariant scalar quantum field theory and its symmetry breakdown. Based on the dimension counting identity, we give a concise proof that dilaton is exactly massless at the classical level if scale invariance is broken spontaneously. On the other hand, on the basis of the generalized dimension counting identity, we prove that the dilaton becomes massive at the quantum level if scale invariance is explicitly broken by quantum anomaly. It is pointed out that a subtlety occurs when scale invariance is spontaneously broken through a scale invariant regularization method …
Employing the operator algebra of the conformal group and the conformal Ward identities, we derive the constraints for the anomalies of dilatation and special conformal transformations of the local twist-2 operators in Quantum Chromodynamics. We calculate these anomalies in the leading order of perturbation theory in the minimal subtraction scheme. From the conformal consistency relation we derive then the off-diagonal part of the anomalous dimension matrix of the conformally covariant operators in the two-loop approximation of the coupling constant in terms of these quantities. We deduce corr…
We study the IIB matrix model, which is conjectured to be a nonperturbative definition of superstring theory, by introducing an integer deformation parameter `nu' which couples to the imaginary part of the effective action induced by fermions. The deformed IIB matrix model continues to be well-defined for arbitrary `nu', and it preserves gauge invariance, Lorentz invariance, and the cluster property. We study the model at `nu' = infinity using a saddle-point analysis, and show that ten-dimensional Lorentz invariance is spontaneously broken at least down to an eight-dimensional one. We argue th…
The relationship between thermodynamics and the Lloyd bound on the holographic complexity for a black hole has been of interest. We consider $D$ dimensional anti-de Sitter black holes with hyperbolic geometry as well as black holes with momentum relaxation that have a minimum for temperature and mass. We show that the singular points of the thermodynamic curvature of the black holes, as thermodynamic systems, correspond to the zero points of the action and volume complexity at the Lloyd bound. For such black holes with a single horizon, the complexity of volume and the complexity of action at …
Classical scale invariance represents a promising framework for model building beyond the Standard Model. However, once coupled to gravity, any scale-invariant microscopic model requires an explanation for the origin of the Planck mass. In this paper, we provide a minimal example for such a mechanism and show how the Planck mass can be dynamically generated in a strongly coupled gauge sector. We consider the case of hidden SU(N_c) gauge interactions that link the Planck mass to the condensation of a scalar bilinear operator that is nonminimally coupled to curvature. The effective theory at ene…
We review the formalism of primordial black holes (PBHs) production and show that the mass variance at horizon crossing has been systematically overestimated in previous studies. We derive the correct expression. The difference is maximal at the earliest formation times and still very significant for PBH masses $\sim 10^{15}$g, an accurate estimate requiring numerical calculations. In particular, this would lead to weaker constraints on the spectral index $n$. We then derive constraints on inflationary models from the fact that primordial black holes must not overclose the Universe. This is do…
Lorentz invariance is broken for the non-Abelian monopoles. Here we will consider the case of 't Hooft-Polyakov monopole and show that the Lorentz invariance of its field will be restored using Dirac quantization.
A promising candidate for cold dark matter is primordial black holes (PBH) formed from strong primordial quantum fluctuations. A necessary condition for the formation of PBH's is a change of sign in the tilt governing the anomalous scale invariance of the power spectrum from red at large scales into blue at small scales. Non-perturbative information on the dependence of the power spectrum tilt on energy scale can be extracted from the quantum Fisher information measuring the energy dependence of the quantum phases defining the de Sitter vacua. We show that this non-perturbative quantum tilt go…
The emergent gauge theories are reconsidered in light of supersymmetry and an appropriate emergence conjecture is formulated. Accordingly, it might be expected that only global symmetries are fundamental symmetries of Nature, whereas local symmetries and associated massless gauge fields could solely emerge due to spontaneous breaking of underlying spacetime symmetries involved, such as relativistic invariance and supersymmetry. We further argue that this breaking, taken in the form of the nonlinear sigma-model type pattern for vector fields or superfields, puts essential restrictions on geomet…
We study the scalar sector of the Two Measures Field Theory (TMT) model in the context of cosmological dynamics. The scalar sector includes the inflaton ϕand the Higgs \upsilon fields. The model possesses gauge and scale invariance. The latter is spontaneously broken due to intrinsic features of the TMT dynamics. In the model with the inflaton ϕalone, in different regions of the parameter space the following different effects can take place without fine tuning of the parameters and initial conditions: a) Possibility of resolution of the old cosmological constant problem: this is done in a cons…
Do Anomalies Break Momentum Routing Invariance?
本文证明手征反常具有动量路由不变性，并利用该不变性研究非最小CPT和Lorentz破坏量子电动力学中何种对称性被破坏，为尚无实验确定动量路由的理论框架提供判据。
Burgers-Kardar-Parisi-Zhang (KPZ) scaling has recently (re-) surfaced in a variety of physical contexts, ranging from anharmonic chains to quantum systems such as open superfluids, in which a variety of random forces may be encountered and/or engineered. Motivated by these developments, we here provide a generalization of the KPZ universality class to situations with long-ranged temporal correlations in the noise, which purposefully break the Galilean invariance that is central to the conventional KPZ solution. We compute the phase diagram and critical exponents of the KPZ equation with $1/f$-…
In a previous work [1], we have argued that the algebra of non-abelian superselection rules is spontaneously broken to its maximal abelian subalgebra, that is, the algebra generated by its completing commuting set (the two Casimirs and a basis of its Cartan subalgebra). In this paper, alternative arguments confirming these results are presented. In addition, Lorentz invariance is shown to be broken in QCD, just as it is in QED. The experimental consequences of these results include fuzzy mass and spin shells of coloured particles like quarks, and decay life times which depend on the frame of o…
Field theories with global continuous symmetries may admit configurations in which time translation invariance is broken by the movement of homogeneous background fields evolving along the flat directions implied by the symmetries. In this context, the field fluctuations along the broken symmetry are well parametrized by a Goldstone boson field that may non-trivially interact with other fields present in the theory. These interactions violate Lorentz invariance as a result of the broken time translation invariance of the background, producing a mixing between the field content and the particle…
The presence of a light vector field coupled to a scalar field during inflation makes a distinct prediction: the observed correlation functions of the cosmic microwave background (CMB) become statistically anisotropic. We study the implications of the current bound on statistical anisotropy derived from the Planck 2013 CMB temperature data for such a model. The previous calculations based on the attractor solution indicate that the magnitude of anisotropy in the power spectrum is proportional to $N^2$, where $N$ is the number of $e$-folds of inflation counted from the end of inflation. In this…
We study the spectral tilt of primordial perturbations in Horava-Lifshitz cosmology. The uniform approximation, which is a generalization of the familiar Wentzel-Kramers-Brillouin (WKB) method, is employed to compute the spectral index both numerically and analytically in a closed form. We clarify how the spectral index depends on the inflation model and parameters in the modified dispersion relation.
Starting with a simple two scalar field system coupled to a modified measure that is independent of the metric, we, first, find a Born-Infeld dynamics sector of the theory for a scalar field and second, show that the initial scale invariance of the action is dynamically broken and leads to a scale charge nonconservation, although there is still a conserved dilatation current.
We consider the lightlike spontaneous Lorentz invariance violation (SLIV) appearing through the zero "length-fixing" constraint put on a gauge vector field, $A_{μ}A^{μ}=0$, and discuss its physical consequences in the framework of a conventional QED and beyond. Again, as in the timelike and spacelike SLIV cases, $A_{μ}A^{μ}=\pm M_{A}^{2}$ ($M_{A}$ is a scale of SLIV), while this constraint leads to an emergence of the Nambu-Goldstone modes collected in physical photon, the SLIV itself is still left unobservable unless gauge invariance in the theory is broken. At the same time, a crucial differ…
Tilts from 2-Groups
该研究分析了受2-群1-形式子群荷电的线的普适约束，发现其通常破缺0-形式对称性，由族反常强制执行，利用Wess-Zumino条件等工具研究了不同类型2-群的族反常及相关算符。
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