Global Symmetry Transformation Research Articles

Colour-twist operators. Part I. Spectrum and wave functions

We introduce a new class of operators in any theory with a ’t Hooft large-N limit that we call colour-twist operators. They are defined by twisting the colour-trace with a global symmetry transformation and are continuously linked to standard, untwisted single-trace operators. In particular, correlation functions between operators that are twisted by an R-symmetry of mathcal{N} = 4 SYM extend those in the γ-deformed theory. The most general deformation also breaks the Lorentz symmetry but preserves integrability in the examples we consider. In this paper, we focus on colour-twist operators in the fishnet model. We exemplify our approach for the simplest colour-twist operators with one and two scalar fields, which we study non-perturbatively using field-theoretical as well as integrability methods, finding a perfect match. We also propose the quantisation condition for the Baxter equation appearing in the integrability calculation in the fishnet model. The results of this paper constitute a crucial step towards building the separation of variable construction for the correlation functions by means of the Quantum Spectral Curve approach.

BRST-BV quantization of gauge theories with global symmetries

We consider the general gauge theory with a closed irreducible gauge algebra possessing the non-anomalous global (super)symmetry in the case when the gauge fixing procedure violates the global invariance of classical action. The theory is quantized in the framework of BRST-BV approach in the form of functional integral over all fields of the configuration space. It is shown that the global symmetry transformations are deformed in the process of quantization and the full quantum action is invariant under such deformed global transformations in the configuration space. The deformed global transformations are calculated in an explicit form in the one-loop approximation.

Global Symmetries, Volume Independence, and Continuity in Quantum Field Theories.

We discuss quantum field theories with global SU(N) and O(N) symmetries for which temporal direction is compactified on a circle of size L with periodicity of fields up to a global symmetry transformation, i.e., twisted boundary conditions. Such boundary conditions correspond to an insertion of the global symmetry operator in the partition function. We argue in general and prove in particular for CP(N-1) and O(N) nonlinear sigma models that large-N volume independence holds. Further we show that the CP(N-1) theory is free from the Affleck phase transition confirming the Ünsal-Dunne continuity conjecture.

Dynamical flavor origin ofZNsymmetries

Discrete Abelian symmetries ($\mathbb{Z}_N$) are a common "artifact" of beyond the standard model physics models. They provide different avenues for constructing consistent scenarios for lepton and quark mixing patterns, radiative neutrino mass generation as well as dark matter stabilization. We argue that these symmetries can arise from the spontaneous breaking of the Abelian $U(1)$ factors contained in the global flavor symmetry transformations of the gauge invariant kinetic Lagrangian. This will be the case provided the ultra-violet completion responsible for the Yukawa structure involves scalar fields carrying non-trivial $U(1)$ charges. Guided by minimality criteria, we demonstrate the viability of this approach with two examples: first, we derive the "scotogenic" model Lagrangian, and second, we construct a setup where the spontaneous symmetry breaking pattern leads to a $\mathbb{Z}_3$ symmetry which enables dark matter stability as well as neutrino mass generation at the 2-loop order. This generic approach can be used to derive many other models, with residual $\mathbb{Z}_N$ or $\mathbb{Z}_{N_1}\times \cdots \times \mathbb{Z}_{N_k}$ symmetries, establishing an intriguing link between flavor symmetries, neutrino masses and dark matter.

Formula omitted] global SUSY: R-matrix approach

R-matrix method is used to construct supersymmetric extensions of θ-Euclidean group preserving N=12 supersymmetry and its three-parameter generalization. These quantum symmetry supergroups can be considered as global counterparts of appropriately twisted Euclidean superalgebras. The corresponding generalized global symmetry transformations act on deformed superspaces as the usual ones do on undeformed spaces. However, they depend on non(anti)commuting parameters satisfying (anti)commutation relations defined by relevant R matrix.

Fractional Spin in Non-Abelian Chern-Simons Theories at the Quantum level

Based on the quantization of constrained Hamiltonian system, the quantal conserved laws can be established under the global symmetry transformation. The application of the results to non-Abelian Chern-Simons theory, we can show that property of fractional spin is still preserved at the quantum level in the non-Abelian Chern-Simons theory.

Tensor Product Structures, Entanglement, and Particle Scattering

Particle systems admit a variety of tensor product structures (TPSs) depending on the complete system of commuting observables chosen for the analysis. Different notions of entanglement are associated with these different TPSs. Global symmetry transformations and dynamical transformations factor into products of local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs and corresponding measures of entanglement are defined for particle scattering systems.

Galilean and Dynamical Invariance of Entanglement in Particle Scattering

Particle systems admit a variety of tensor product structures (TPSs) depending on the algebra of observables chosen for analysis. Global symmetry transformations and dynamical transformations may be resolved into local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs are defined and various notions of entanglement are considered for scattering states.

UNFOLDED FORM OF CONFORMAL EQUATIONS IN M DIMENSIONS AND 𝔬(M + 2)-MODULES

A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra 𝔣. Under certain conditions, 𝔣-invariant systems of differential equations are shown to be associated with 𝔣-modules that are integrable with respect to some parabolic subalgebra of 𝔣. The suggested construction is motivated by the unfolded formulation of dynamical equations developed in the higher spin gauge theory and provides a starting point for generalization to the nonlinear case. It is applied to the conformal algebra 𝔬(M, 2) to classify all linear conformally invariant differential equations in the Minkowski space. Numerous examples of conformal equations are discussed from this perspective.

The symmetries in the Maxwell–Chern–Simons theory coupled to matter fields

The Maxwell–Chern–Simons gauge theory coupled to a complex scalar field is quantized in the Becchi–Rouet–Stora–Tyutin path integral formalism. Based on the symmetries of a constrained canonical (Hamiltonian) system, we obtain the quantal conserved angular momentum of the system under the global symmetry transformation. It is shown that fractional spin also appears at the quantum level. The canonical Ward identities for this system are derived under local gauge transformation.

Quantum Symmetries in the Maxwell–Chern–Simons Theory Coupled to Scalar Fields

The Maxwell-Chern-Simons gauge theory coupled to a complex scalar field is quantized in the Becchi-Rouet-Stora-Tyutin (BRST) path integral formalism. On the basis of the symmetries of a constrained canonical (Hamiltonian) system, we get the quantal conserved angular momentum of the system under the global symmetry transformation. It is shown that fractional spin also appears at the quantum level. The canonical Ward identities for this system are derived under local gauge transformation.

Non-BPS D-branes and enhanced symmetry in an asymmetric orbifold

In this paper properties of D-branes in a nine dimensional asymmetric orbifold are discussed, using a (−1)FLσ1/2 projection, where FL is the leftmoving space-time fermion number and σ1/2 is a freely acting shift of order two. There are two types of non BPS D branes, which are stable at R > 2 and R < 2 respectively. At R = 2 there is a perturbative enhancement of gauge symmetry and the two types of branes are related by a global bulk symmetry transformation. At this point in the moduli space the associated boundary states are constructed using a free fermion representation of the theory. Some aspects of the enhancement of gauge symmetry in the S-dual type-Ĩ theory are discussed.

Quantal symmetry for a system with a singular higher-order Lagrangian

Based on the phase-space generating functional of the Green function for a constrained Hamiltonian system with a singular higher-order Lagrangian, the canonical Ward identities for such a system under the local and global transformation have been derived, respectively. The quantal conserved charge (QCC) under the global symmetry transformation is also deduced. In general, these QCCs are different from the Noether charge in classical theory. A comparison of these quantal conservation laws with those deriving from the configuration-space path integral for gauge-invariant theories is discussed. We give a preliminary application of our results to Yang-Mills (YM) theory and Chern-Simons (CS) theory with higher-order derivatives. A new form of gauge-ghost proper vertices and new conserved charges at the quantum level are obtained for the YM theory; the quantal Becchi-Rouet-Stora (BRS) conserved charges and conserved angular momentum are also derived for CS theory. The advantage of our canonical formalism is that we do not carry out the integration over the canonical momenta in the phase-space path integral as usually performed.

Quantal global symmetry for a gauge-invariant system

Based on the configuration-space generating functional obtained by using the Faddeev-Popov trick for a gauge-invariant system, the Ward identities for global transformation are derived. The conservation laws at the quantum level for global symmetry transformation are also deduced. A preliminary application of the present formulation to non-Abelian Chern-Simons (CS) theory is given. The Ward identity and quantal BRS charge under the BRS transformation are deduced. The quantal conserved angular momentum is obtained and the fractional spin for CS theories is discussed.

Canonical Quantal Symmetry and Conserved Charges for a System with Higher-Order Lagrangian11The project supported by National Natural Science Foundation of China under grant 19275009

Based on the phase space generating functional of Green function for a system with regular higher-order Lagrangian, the conserved charge is deduced at quantum level if the canonical action is invariant under the global symmetry transformation in extended phase space. The generalization of our results to a system with singular higher-order Lagrangian is also studied. In general the quantal conserved charges are different from classical ones.

Canonical global symmetry and quantal conservation laws for a system with singular higher order Lagrangian

Starting from the phase-space generating functional of the Green function for a system with singular higher order Lagrangian, the generalized canonical Ward identities under the global symmetry transformation in phase space is deduced. The local transformation connected with this global symmetry transformation is studied, and the quantal conservation laws are obtained for such a system. We give a preliminary application to higher derivative Yang-Mills theory; a generalized quantal BRS conserved quantity is found.

Quantal canonical symmetry for a constrained Hamiltonian system

Starting from the phase-space generating functional of the Green's function for a constrained Hamiltonian system, the canonical Ward identities under the global symmetry transformation in phase space is deduced. The local transformation connected with this global symmetry transformation is studied, the conserved charges are obtained at quantum level if the effective canonical action is symmetric (the constraints are also invariant under the transformation) and, therefore, the canonical Noether theorem in the quantum case is obtained. The generalized canonical Ward identities under the local transformation has been deduced. We give a preliminary application to a system of an interacting polaron with a photon. The conserved charges and Ward identities for proper vertices are obtained, but we do not carry out the integration over the canonical momenta in the phase-space generating functional as usually performed.

Gauge transformations for dynamical systems with first- and second-class constraints.

Gauge theories with second-class constraints are investigated. The relation between primary first-class constraints and gauge degrees of freedom is shown. Next, a method to obtain the generator of the gauge transformation is presented. The generator is expressed in terms of a linear combination of constraints. In the expression, all constraints are employed without distinguishing the first- from the second-class ones. The generator consists of the generator of the pure gauge transformation and that of global symmetry transformations with constant parameters. The method to construct the generator can be applied to a system having only second-class constraints, and extended Noether currents (generators) with constant parameters are obtained. If the first-class constraints and the Hamiltonian are in involution, the generator of pure gauge transformations can be obtained using only the first-class constraints.

Perturbation theory and renormalization of supersymmetric Yang-Mills theories

We solve the problem of introducing, in supersymmetric Yang-Mills Lagrangians, a supersymmetric gauge breaking term and a Faddeev-Popov ghost interaction term. The resulting Lagrangian turns out to be invariant under a global symmetry transformation which is the supersymmetric extension of the Slavnov symmetry. We show that the complete analysis of all primitively divergent supergraphs ensures, in conjunction with the Slavnov identities, the renormalizability of the theory, once a supersymmetric and gauge invariant regularizing procedure has been introduced. We find that the simplest regularizing procedure is a generalization of the higher covariant derivatives method. In the case of interaction with matter fields we prove that no mass counter term is needed, in exact analogy with the model without gauge fields. Finally we show that, in the Abelian situation, a supersymmetric mass term for the vector multiplet can be introduced without spoiling the renormalizability, thus providing the supersymmetric extension of massive vector bosons theories.