Generators Of Symmetry Group Research Articles

Soliton generation and conservation laws for vector light pulses propagating in weakly birefringent waveguides

The principal algebra Lp of coupled nonlinear Schrödinger equations describing the propagation of polarized optical pulses and involving four-wave mixing terms is obtained in terms of the three arbitrary labelling parameters of the system of equations. This algebra is found to be five-dimensional, indicating the richness in symmetries of the system. The most general symmetry group transformation by generators of Lp is found and it is shown that this transformation preserves the boundedness of solutions, and an example of transformation of a soliton solution into a completely new soliton of a different type is presented. Moreover, it is explained how an infinite sequence of bounded solutions can thus be generated, yielding new solitons. Several other important properties of solutions and symmetry group transformations of the system are also demonstrated. As the system of Schrödinger equations under study turns out to be of Lagrange type, conservation laws associated with all variational symmetries of Lp are constructed and interpreted. Some symmetry reductions of the system are also derived.

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In this paper, the A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A} $$\\end{document}-theory, an extension of F-theory, is described as a fully U-duality covariant brane theory. This theory has some distinguishing features not known from world-sheet models. In particular, seen as a sigma model, both world-volume and target space coordinates are specific representations of the same group (the U-duality group). The U-duality group in question is an exceptional group (a split form of the Ed series). The structure of this group allows it to encompass both the T-duality group of string theory as well as the general linear symmetry group of M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document}-theory. A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A} $$\\end{document}-theory is defined by the current algebras in Hamiltonian formalism, or by world-volume actions in Lagrangian formalism. The spacetime coordinates are selfdual gauge fields on the world-volume, requiring the Gauß law constraints tying the world-volume to spacetime. Solving the Gauß law constraints/the Virasoro constraints gives the world-volume/spacetime sectioning from A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A} $$\\end{document}-theory to T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document}-theory/M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document}-theory respectively. The A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A} $$\\end{document}-theory Lagrangian admits extended symmetry which has not been observed previously in the literature, where the background fields include both the spacetime and the world-volume gravitational fields. We also constructed the four-point amplitude of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A} $$\\end{document}-theory in the low energy limit. The amplitude is written in a way that the U-duality symmetry is manifest, but after solving the section condition, it reduces to the usual four-graviton amplitude.In the previous papers, we have referred to this model as F-theory, however, F-theory initiated by Vafa is now a big branch of string theory as the study of elliptic fibrations, so we refer to these constructions as generalized models of theory for all dimensions with all duality symmetries as A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{A} $$\\end{document}-theory.

Anomaly of $(2+1)$-dimensional symmetry-enriched topological order from $(3+1)$-dimensional topological quantum field theory

Symmetry acting on a (2+1)DD topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)DD on-site symmetry. In this paper, we develop a (3+1)DD topological quantum field theory to calculate the anomaly indicators of a (2+1)DD topological order with a general symmetry group GG, which may be discrete or continuous, Abelian or non-Abelian, contain anti-unitary elements or not, and permute anyons or not. These anomaly indicators are partition functions of the (3+1)DD topological quantum field theory on a specific manifold equipped with some GG-bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Our framework is applied to derive the anomaly indicators for various symmetry groups, including \mathbb{Z}_2\times\mathbb{Z}_2ℤ2×ℤ2, \mathbb{Z}_2^T\times\mathbb{Z}_2^Tℤ2T×ℤ2T, SO(N)SO(N), O(N)^TO(N)T, SO(N)\times \mathbb{Z}_2^TSO(N)×ℤ2T, etc, where \mathbb{Z}_2ℤ2 and \mathbb{Z}_2^Tℤ2T denote a unitary and anti-unitary order-2 group, respectively, and O(N)^TO(N)T denotes a symmetry group O(N)O(N) such that elements in O(N)O(N) with determinant -1−1 are anti-unitary. In particular, we demonstrate that some anomaly of O(N)^TO(N)T and SO(N)\times \mathbb{Z}_2^TSO(N)×ℤ2T exhibit symmetry-enforced gaplessness, i.e., they cannot be realized by any symmetry-enriched topological order. As a byproduct, for SO(N)SO(N) symmetric topological orders, we derive their SO(N)SO(N) Hall conductance.

Classification of (2+1) D invertible fermionic topological phases with symmetry

Integer quantum Hall states and their generalization to topological insulators and superconductors are paradigmatic examples of invertible fermionic topological states of matter. Here, the authors develop a comprehensive characterization and classification of invertible fermionic topological phases of matter in two spatial dimensions for general symmetry groups. The results are nonperturbative and apply to strongly interacting systems. In particular, they extend previous classification results to account for the possible chiral nature of invertible phases.

Anomaly cascade in ( 2+1 )-dimensional fermionic topological phases

We develop a theory of anomalies of fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group $G_f$. In general, $G_f$ can be a non-trivial central extension of the bosonic symmetry group $G_b$ by fermion parity $(-1)^F$. We encounter four layers of obstructions to gauging the $G_f$ symmetry, which we dub the anomaly cascade: (i) An $\mathcal{H}^1(G_b,\mathbb{Z}_{\bf T})$ obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An $\mathcal{H}^2(G_b, \ker r)$ obstruction to extending the $G_b$ group structure of the symmetry permutations to the fermion parity gauged theory, where $r$ is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An $\mathcal{H}^3(G_b, \mathbb{Z}_2)$ obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known $\mathcal{H}^4(G_b, U(1))$ obstruction to developing a consistent theory of $G_b$ symmetry defects for the fermion parity gauged theory. We describe how the $\mathcal{H}^2$ obstruction can be canceled by anomaly inflow from a bulk (3+1)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is non-trivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.

Fermionic symmetry fractionalization in (2+1) dimensions

We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group $G_f$. In general $G_f$ is a central extension of the bosonic symmetry group $G_b$ by fermion parity, $(-1)^F$, characterized by a non-trivial cohomology class $[\omega_2] \in \mathcal{H}^2(G_b, \mathbb{Z}_2)$. We show how the presence of local fermions places a number of constraints on the algebraic data that defines the action of the symmetry on the super-modular tensor category that characterizes the anyon content. We find two separate obstructions to defining symmetry fractionalization, which we refer to as the bosonic and fermionic symmetry localization obstructions. The former is valued in $\mathcal{H}^3(G_b, K(\mathcal{C}))$, while the latter is valued in either $\mathcal{H}^3(G_b,\mathcal{A}/\{1,\psi\})$ or $Z^2(G_b, \mathbb{Z}_2)$ depending on additional details of the theory. $K(\mathcal{C})$ is the Abelian group of functions from anyons to $\mathrm{U}(1)$ phases obeying the fusion rules, $\mathcal{A}$ is the Abelian group defined by fusion of Abelian anyons, and $\psi$ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over $\mathcal{H}^2(G_b, \mathcal{A}/\{1,\psi\})$. We study a number of examples in detail; in particular we provide a characterization of fermionic Kramers degeneracy arising in symmetry class DIII within this general framework, and we discuss fractional quantum Hall and $\mathbb{Z}_2$ quantum spin liquid states of electrons.

Parallel Solving Method for the Variable Coefficient Nonlinear Equation

In view of the inaccuracy of traditional methods for solving nonlinear equations with variable coefficients in parallel, a new method for solving nonlinear equations with variable coefficients is proposed. Using the generalized symmetry group, the variable coefficient of the equation is taken as a new variable which is the same as the state of the original actual physical field. Some relations between variable coefficient equations and their solutions are found. This paper analyzes the meaning of linear differential equation and nonlinear differential equation, the difference between linear differential equation and nonlinear differential equation and their role in physics, and the necessity of solving nonlinear differential equation. By solving the nonlinear equation with variable coefficients, it can be seen that the general methods to solve the nonlinear equation include scattering inversion, Backlund transform and traveling wave solution. Based on the existing methods for solving nonlinear equations with variable coefficients, the homogeneous balance method is combined with the improved truncated expansion method, truncated expansion method and function reduction method, and the Hopf Cole transform and trial function are combined respectively. The above three methods are used to solve nonlinear equations with variable coefficients. Based on KdV Painleve principle, a parallel method for solving nonlinear equations with variable coefficients is proposed. Finally, it is proved that the method is accurate and effective for the parallel solution of nonlinear equations with variable coefficients.

Symmetry classification and invariance of the Reynolds equation

In this essay, extensions to the results of Lie symmetry classification of Reynolds equation are proposed. The infinitesimal technique is used to derive symmetry groups of the Reynolds equation. Onedimensional optimal system is constructed for symmetry sub-algebras gained through Lie point symmetry. At the end, the general symmetry group of the non-conservative generalized thin-film equation are determined.

Exploring the landscape for soft theorems of nonlinear sigma models

We generalize soft theorems of the nonlinear sigma model beyond the mathcal{O} (p2) amplitudes and the coset of SU(N) × SU(N)/SU(N). We first discuss the universal flavor ordering of the amplitudes for the Nambu-Goldstone bosons, so that we can reinterpret the known mathcal{O} (p2) single soft theorem for SU(N) × SU(N)/SU(N) in the context of a general symmetry group representation. We then investigate the special case of the fundamental representation of SO(N), where a special flavor ordering of the “pair basis” is available. We provide novel amplitude relations and a Cachazo-He-Yuan formula for such a basis, and derive the corresponding single soft theorem. Next, we extend the single soft theorem for a general group representation to mathcal{O} (p4), where for at least two specific choices of the mathcal{O} (p4) operators, the leading non-vanishing pieces can be interpreted as new extended theory amplitudes involving bi-adjoint scalars, and the corresponding soft factors are the same as at mathcal{O} (p2). Finally, we compute the general formula for the double soft theorem, valid to all derivative orders, where the leading part in the soft momenta is fixed by the mathcal{O} (p2) Lagrangian, while any possible corrections to the subleading part are determined by the mathcal{O} (p4) Lagrangian alone. Higher order terms in the derivative expansion do not contribute any new corrections to the double soft theorem.

Quantum reference frames for general symmetry groups

A fully relational quantum theory necessarily requires an account of changes of quantum reference frames, where quantum reference frames are quantum systems relative to which other systems are described. By introducing a relational formalism which identifies coordinate systems with elements of a symmetry groupG, we define a general operator for reversibly changing between quantum reference frames associated to a groupG. This generalises the known operator for translations and boosts to arbitrary finite and locally compact groups, including non-Abelian groups. We show under which conditions one can uniquely assign coordinate choices to physical systems (to form reference frames) and how to reversibly transform between them, providing transformations between coordinate systems which are `in a superposition' of other coordinate systems. We obtain the change of quantum reference frame from the principles of relational physics and of coherent change of reference frame. We prove a theorem stating that the change of quantum reference frame consistent with these principles is unitary if and only if the reference systems carry the left and right regular representations ofG. We also define irreversible changes of reference frame for classical and quantum systems in the case where the symmetry groupGis a semi-direct productG=N⋊Por a direct productG=N×P, providing multiple examples of both reversible and irreversible changes of quantum reference system along the way. Finally, we apply the relational formalism and changes of reference frame developed in this work to the Wigner's friend scenario, finding similar conclusions to those in relational quantum mechanics using an explicit change of reference frame as opposed to indirect reasoning using measurement operators.

Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Certain patterns of symmetry fractionalization in (2+1)D topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological (SET) states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group $G$, one can define a (3+1)D topologically invariant path integral in terms of a state sum for a $G$ symmetry-protected topological (SPT) state. We present an exactly solvable Hamiltonian for the system and demonstrate explicitly a (2+1)D $G$ symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. We present concrete algorithms that can be used to compute anomaly indicators in general. Our approach applies to general symmetry groups, including anyon-permuting and anti-unitary symmetries. In addition to providing a general way to compute the anomaly, our result also shows, by explicit construction, that every symmetry fractionalization class for any UMTC can be realized at the surface of a (3+1)D SPT state. As a byproduct, this construction also provides a way of explicitly seeing how the algebraic data that defines symmetry fractionalization in general arises in the context of exactly solvable models. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the $\mathcal{H}^4(G, U(1))$ obstruction that arises in the theory of $G$-crossed braided tensor categories, for which no general method has been presented to date.

Invariant Solutions of Two Dimensional Heat Equation

The symmetry group of a given differential equation is the group of transformations that translate the solutions of the equation into solutions. If the infinitesimal generators of symmetry groups of systems of partial differential equations are known, the symmetry group can be used to explicitly find particular types of solutions that are invariant with respect to the symmetry group of the system. The class of invariant solutions includes exact solutions that have direct mathematical or physical meaning. In this paper, using the well-known infinitesimal generators of some symmetry groups of the two-dimensional heat conduction equation, solutions are found that are invariant with respect to these groups. It is considered cases when conductivity coefficients of the two-dimensional heat conduction equation are power functions of temperature and conductivity coefficients are exponential functions of temperature. In first case invariant solutions contain well known self-similar solutions which are widely used in applications.

Solving puzzles of spontaneously broken spacetime symmetries

We establish a classical analog of the Nambu-Goldstone theorem for spontaneous breaking of spacetime symmetries. It provides a counting rule for independent Nambu-Goldstone fields and states which of them are gapped. We demonstrate that only those symmetry group generators give rise to independent Nambu-Goldstone fields that act non-trivially on a vacuum at the origin of coordinates. Other generators give rise to auxiliary fields that must be excluded from a theory by the means of inverse Higgs constraints. The physical meaning of the inverse Higgs phenomenon and an application of our results to theories of massive gravity are discussed.

Lie Algebra of Infinitesimal Generators of the Symmetry Group of the Heat Equation

Last time symmetry methods have been recognized to be of great importance for the study of the differential equations arising in mathematics and physics. The purpose of this paper is to provide some application of Lie groups to heat equation. In this example, we determine Lie algebra of infinitesimal generators of symmetry group of heat equation and construct group-invariant solutions of this equation. The some computational methods are presented so that researchers in other fields can readily learn to use them.

Anyon condensation and a generic tensor-network construction for symmetry-protected topological phases

We present systematic constructions of tensor-network wavefunctions for bosonic symmetry protected topological (SPT) phases respecting both onsite and spatial symmetries. From the classification point of view, our results show that in spatial dimensions $d=1,2,3$, the cohomological bosonic SPT phases protected by a general symmetry group $SG$ involving onsite and spatial symmetries are classified by the cohomology group $H^{d+1}(SG,U(1))$, in which both the time-reversal symmetry and mirror reflection symmetries should be treated as anti-unitary operations. In addition, for every SPT phase protected by a discrete symmetry group and some SPT phases protected by continous symmetry groups, generic tensor-network wavefunctions can be constructed which would be useful for the purpose of variational numerical simulations. As a by-product, our results demonstrate a generic connection between rather conventional symmetry enriched topological phases and SPT phases via an anyon condensation mechanism.

Symmetry analysis of reaction diffusion equation with distributed delay

This paper study the reaction–diffusion equation with distributed delay from the Lie group theoretic point of view. We give at first the evolutionary infinitesimal vector field v and a number of group invariant solutions corresponding to v by general symmetry group theory.

Existence Theorems for Vortices in the Aharony–Bergman–Jaferis–Maldacena Model

A series of sharp existence and uniqueness theorems are established for the multiple vortex solutions in the supersymmetric Chern--Simons--Higgs theory formalism of Aharony, Bergman, Jaferis, and Maldacena, for which the Higgs bosons and Dirac fermions lie in the bifundamental representation of the general gauge symmetry group $U(N)\times U(N)$. The governing equations are of the BPS type and derived by Kim, Kim, Kwon, and Nakajima in the mass-deformed framework labeled by a continuous parameter.

Modified orbital branching for structured symmetry with an application to unit commitment

The past decade has seen advances in general methods for symmetry breaking in mixed-integer linear programming. These methods are advantageous for general problems with general symmetry groups. Some important classes of mixed integer linear programming problems, such as bin packing and graph coloring, contain highly structured symmetry groups. This observation has motivated the development of problem-specific techniques. In this paper we show how to strengthen orbital branching in order to exploit special structures in a problem's symmetry group. The proposed technique, to which we refer as modified orbital branching, is able to solve problems with structured symmetry groups more efficiently. One class of problems for which this technique is effective is when the solution variables can be expressed as 0/1 matrices where the problem's symmetry group contains all permutations of the columns. We use the unit commitment problem, an important problem in power systems, to demonstrate the strength of modified orbital branching.

Lie symmetry analysis of the two-dimensional generalized Kuramoto-Sivashinsky equation

Purpose: In this paper, a detailed analysis of an important nonlinear model system, the two dimensional generalized Kuramoto-Sivashinsky (2D gKS) equation, is presented by group analysis. Methods: The basic Lie symmetry method is applied in order to determine the general symmetry group of our analyzed nonlinear model. Results: The symmetry group of the equation and some results related to the algebraic structure of the Lie algebra of symmetries are obtained. Also, a complete classification of the subalgebras of the symmetry algebra is resulted. Conclusions: It is proved that the Lie algebra of symmetries admits no three dimensional subalgebra. Mainly, all the group invariant solutions and the similarity reduced equations associated to the infinitesimal symmetries are obtained.

Equivalence of domains for hyperbolic Hubbard-Stratonovich transformations

We settle a long standing issue concerning the traditional derivation of non-compact non-linear sigma models in the theory of disordered electron systems: the hyperbolic Hubbard-Stratonovich (HS) transformation of Pruisken-Schäfer type. Only recently the validity of such transformations was proved in the case of U(p, q) (non-compact unitary) and O(p, q) (non-compact orthogonal) symmetry. In this article we give a proof for general non-compact symmetry groups. Moreover, we show that the Pruisken-Schäfer type transformations are related to other variants of the HS transformation by deformation of the domain of integration. In particular we clarify the origin of surprising sign factors which were recently discovered in the case of orthogonal symmetry.