non-Abelian Generalization Research Articles

Novel knotted non-Abelian gauge fields

In analogy to null electromagnetic fields we define null Yang-Mills (YM) fields. We show that the null non-Abelian SU(N) gauge fields admit a set of 2N2 conserved “helicities.” We derive null YM solutions that carry finite helicities by uplifting the Abelian Hopfion solution and their generalizations. Another method that we implement is to deform YM solutions which do not carry helicities into ones that have nontrivial helicities. A nontrivial non-Abelian solution with helicities is found as a wave of infinite energy. We also discuss non-Abelian generalizations of the Bateman parameterization for null Abelian gauge fields. Published by the American Physical Society 2024

Ground states of Class S theory on ADE singularities and dual Chern-Simons theory

In radial quantization, the ground states of a gauge theory on ADE singularities ℝ4/Γ are characterized by flat connections that are maps from Γ to the gauge group. We study Class S theory of type a1 = su2 on a Riemann surface of genus g > 1, without punctures. The fundamental building block of Class S theory is the trifundamental Trinion theory — a low energy limit of two M5 branes compactified on the three-punctured Riemann sphere. We show, through the superconformal index, that the supersymmetric Casimir energy of the trifundamental theory imposes a constraint on the set of allowed flat connections, which agrees with the prediction of a duality relating the ground state Hilbert space of Class S on ADE singularities to the Hilbert space of a certain dual Chern-Simons theory whose gauge group is given by the McKay correspondence. The conjecture is shown to hold for Γ = ℤk, agreeing with the previous results of Benini et al. and Alday et al. A non-abelian generalization of this duality is analyzed by considering the example of the dicyclic group Γ = Dic2, corresponding to Chern-Simons gauge group SO(8).

What can abelian gauge theories teach us about kinematic algebras?

The phenomenon of BCJ duality implies that gauge theories possess an abstract kinematic algebra, mirroring the non-abelian Lie algebra underlying the colour information. Although the nature of the kinematic algebra is known in certain cases, a full understanding is missing for arbitrary non-abelian gauge theories, such that one typically works outwards from well-known examples. In this paper, we pursue an orthogonal approach, and argue that simpler abelian gauge theories can be used as a testing ground for clarifying our understanding of kinematic algebras. We first describe how classes of abelian gauge fields are associated with well-defined subalgebras of the diffeomorphism algebra. By considering certain special subalgebras, we show that one may construct interacting theories, whose kinematic algebras are inherited from those already appearing in a related abelian theory. Known properties of (anti-)self-dual Yang-Mills theory arise in this way, but so do new generalisations, including self-dual electromagnetism coupled to scalar matter. Furthermore, a recently obtained non-abelian generalisation of the Navier-Stokes equation fits into a similar scheme, as does Chern-Simons theory. Our results provide useful input to further conceptual studies of kinematic algebras.

The (1,0) tensor and hypermultiplets in loop space

We show that the (1,0) tensor and hypermultiplet supersymmetry variations can be uplifted to loop space. Upon dimensional reduction we make contact with abelian five-dimensional super Yang-Mills, which has a nonabelian generalization that we subsequently uplift back to loop space where we conjecture a nonabelian generalization of the (1,0) supersymmetry variations and demonstrate their on-shell closure.

Different linearizations of non-abelian second Painlevé systems and related monodromy surfaces

In this paper, we discuss a connection between different linearizations for non-abelian analogs of the second Painlevé equation. For each of the analogs, we listed the pairs of the Harnard–Tracy–Widom (HTW), Flaschka–Newell (FN), and Jimbo–Miwa (JM) types. A method for establishing the HTW-JM correspondence is suggested. For one of the non-abelian analogs, we derive the corresponding non-abelian generalizations of the monodromy surfaces related to the FN- and JM-type linearizations. A natural Poisson structure associated with these monodromy surfaces is also discussed.

Kac-Moody symmetry in the light front of gauge theories

We discuss the emergence of a new symmetry generator in a Hamiltonian realisation of four-dimensional gauge theories in the flat space foliated by retarded (advanced) time. It generates an asymptotic symmetry that acts on the asymptotic fields in a way different from the usual large gauge transformations. The improved canonical generators, corresponding to gauge and asymptotic symmetries, form a classical Kac-Moody charge algebra with a non-trivial central extension. In particular, we describe the case of electromagnetism, where the charge algebra is the U(1) current algebra with a level proportional to the coupling constant of the theory, κ = 4π2/e2. We construct bilinear generators yielding Virasoro algebras on the null boundary. We also provide a non-Abelian generalization of the previous symmetries by analysing the evolution of Yang-Mills theory in Bondi coordinates.

On classification of non-abelian Painlevé type systems

We find all non-abelian generalizations of P1−P6 Painlevé systems such that the corresponding autonomous system obtained by freezing the independent variable is integrable. All these systems have isomonodromic Lax representations.

Noninvertible Chiral Symmetry and Exponential Hierarchies

We elucidate the fate of classical symmetries which suffer from Abelian Adler-Bell-Jackiw anomalies. Instead of being completely destroyed, these symmetries survive as noninvertible topological global symmetry defects with world volume anyon degrees of freedom that couple to the bulk through a magnetic 1-form global symmetry as in the fractional Hall effect. These noninvertible chiral symmetries imply selection rules on correlation functions and arise in familiar models of massless quantum electrodynamics and models of axions (as well as their non-Abelian generalizations). When the associated bulk magnetic 1-form symmetry is broken by the propagation of dynamical magnetic monopoles, the selection rules of the noninvertible chiral symmetry defects are violated nonperturbatively. This leads to technically natural exponential hierarchies in axion potentials and fermion masses.Received 3 August 2022Accepted 2 February 2023DOI:https://doi.org/10.1103/PhysRevX.13.011034Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAnomaliesPhysical SystemsAxionsPropertiesChiral symmetryTechniquesMonopolesParticles & Fields

Homotopy double copy and the Kawai–Lewellen–Tye relations for the non-abelian and tensor Navier–Stokes equations

Recently, a non-abelian generalization of the Navier–Stokes equation that exhibits a manifest duality between color and kinematics has been proposed by Cheung and Mangan. In this paper, we offer a new perspective on the double copy formulation of this equation based on the homotopy algebraic picture suggested by Borsten, Kim, Jurčo, Macrelli, Saemann, and Wolf. In the process, we describe precisely how the double copy can be realized at the level of perturbiner expansions. Specifically, we will show that the color-dressed Berends–Giele currents for the non-abelian version of the Navier–Stokes equation can be used to construct the Berends–Giele currents for the double copied equation by replacing the color factors with a second copy of kinematic numerators. We will also show a Kawai–Lewellen–Tye relation stating that the full tree-level scattering amplitudes in the latter can be written as a product of tree-level color ordered partial amplitudes in the former.

Artificial Non-Abelian Lattice Gauge Fields for Photons in the Synthetic Frequency Dimension.

Non-Abelian gauge fields give rise to nontrivial topological physics. Here we develop a scheme to create an arbitrary SU(2) lattice gauge field for photons in the synthetic frequency dimension using an array of dynamically modulated ring resonators. The photon polarization is taken as the spin basis to implement the matrix-valued gauge fields. Using a non-Abelian generalization of the Harper-Hofstadter Hamiltonian as a specific example, we show that the measurement of the steady-state photon amplitudes inside the resonators can reveal the band structures of the Hamiltonian, which show signatures of the underlying non-Abelian gauge field. These results provide opportunities to explore novel topological phenomena associated with non-Abelian lattice gauge fields in photonic systems.

Nonabelian embedding tensors

In this paper, first we introduce the notion of a nonabelian embedding tensor, which is a nonabelian generalization of an embedding tensor. Then, we introduce the notion of a Leibniz–Lie algebra, which is the underlying algebraic structure of a nonabelian embedding tensor, and can also be viewed as a nonabelian generalization of a Leibniz algebra. Next using the derived bracket, we construct a differential graded Lie algebra, whose Maurer–Cartan elements are exactly nonabelian embedding tensors. Consequently, we obtain the differential graded Lie algebra that governs deformations of a nonabelian embedding tensor. Finally, we define the cohomology of a nonabelian embedding tensor and use the second cohomology group to characterize linear deformations.

Cohomology of nonabelian embedding tensors on Hom-Lie algebras

<abstract><p>In this paper, we generalize known results of nonabelian embedding tensor to the Hom setting. We introduce the concept of Hom-Leibniz-Lie algebra, which is the basic algebraic structure of nonabelian embedded tensors on Hom-Lie algebras and can also be regarded as a nonabelian generalization of Hom-Leibniz algebra. Moreover, we define a cohomology of nonabelian embedding tensors on Hom-Lie algebras with coefficients in a suitable representation. The first cohomology group is used to describe infinitesimal deformations as an application. In addition, Nijenhuis elements are used to describe trivial infinitesimal deformations.</p></abstract>

Real Spinors and Real Dirac Equation

We reexamine the minimal coupling procedure in the Hestenes' geometric algebra formulation of the Dirac equation, where spinors are identified with the even elements of the real Clifford algebra of spacetime. This point of view, as we argue, leads naturally to a non-Abelian generalisation of the electromagnetic gauge potential.

Dimensional reduction of M5 branes

We study dimensional reduction of M5 branes on a circle bundle when the supersymmetry parameter is not constant along the circle. When the gauge group is Abelian and the fields appear quadratically in the Lagrangian, we can always obtain a supersymmetric five-dimensional theory by keeping fermionic nonzero modes that match with the corresponding nonzero modes of the supersymmetry parameter, and by keeping the zero modes for the bosonic fields as usual. But a supersymmetric non-Abelian generalization can be found only under special circumstances. One instance where we find a non-Abelian supersymmetric generalization is when we perform dimensional reduction along a null direction.

Canonical transformations generated by the boundary volume: Unimodular and non-Abelian teleparallel gravity

Recently, a new choice of variables was identified to understand how the quantum group structure appeared in three-dimensional gravity [1]. These variables are introduced via a canonical transformation generated by a boundary term. We show that this boundary term can actually be taken to be the volume of the boundary and that the new variables can be defined in any dimension greater than three. In addition, we study the associated metric and teleparallel formalisms. The former is a variant of the Henneaux--Teitelboim model for unimodular gravity. The latter provides a non-Abelian generalization of the usual Abelian teleparallel formulation.

Superspace BRST/BV Operators of Superfield Gauge Theories

We consider the superspace BRST and BV description of 4D,N=1 super-Maxwell theory and its non-abelian generalization Super Yang–Mills. By fermionizing the superspace gauge transformation of the gauge superfields, we define the nilpotent superspace BRST symmetry transformation (𝓈). After introducing an appropriate set of anti-superfields and defining the superspace antibracket, we use it to construct the BV-BRST nilpotent differential operator (s) in terms of superspace covariant derivatives. The anti-superfield independent terms of s provide a superspace generalization of the Koszul–Tate resolution (δ). In the linearized limit, the set of superspace differential operators that appear in s satisfy a nonlinear algebra which can be used to construct a BRST charge Q, without requiring pure spinor variables. Q acts on the Hilbert space of superfield states, and its cohomology generates the expected superspace equations of motion.

Local abelian Kato-Parshin reciprocity law: A survey

Let $K$ denote an $n$-dimensional local field. The aim of this expository paper is to survey the basic arithmetic theory of the $n$-dimensional local field $K$ together with its Milnor $K$-theory and Parshin topological $K$-theory; to review Kato's ramification theory for finite abelian extensions of the $n$-dimensional local field $K$, and to state the local abelian higher-dimensional $K$-theoretic generalization of local abelian class field theory of Hasse, which is developed by Kato and Parshin. The paper is geared toward non-abelian generalization of this theory.

Three-dimensional non-Abelian generalizations of the Hofstadter model: Spin-orbit-coupled butterfly trios

We theoretically introduce and study a three-dimensional Hofstadter model with linearly varying non-Abelian gauge potentials along all three dimensions. The model can be interpreted as spin-orbit coupling among a trio of Hofstadter butterfly pairs since each Cartesian surface ($xy$, $yz$, or $zx$) of the model reduces to a two-dimensional non-Abelian Hofstadter problem. By evaluating the commutativity among arbitrary loop operators around all axes, we derive its genuine (necessary and sufficient) non-Abelian condition, namely, at least two out of the three hopping phases should be neither 0 nor $\pi$. Under different choices of gauge fields in either the Abelian or the non-Abelian regime, both weak and strong topological insulating phases are identified in the model.

Non-Abelian Alice strings in two-flavor dense QCD

Quark-hadron continuity with two-flavor quarks that was proposed recently connects hadronic matter with neutron $^3P_2$ superfluidity and two-flavor dense quark matter. This two-flavor dense quark phase consists of the coexistence of the 2SC condensates and the $P$-wave diquark condensates of $d$-quarks, which gives rise to color superconductivity as well as superfluidity. We classify vortices in this phase. The most stable vortices are what we call the non-Abelian Alice strings, which are superfluid vortices with non-Abelian color magnetic fluxes therein, exhibiting so-called topological obstruction, or a non-Abelian generalization of the Alice property. We show that a single Abelian superfluid vortex is unstable against decay into three non-Abelian Alice strings. We discover that a non-Abelian Alice string carries orientational moduli of the real projective space $\mathbb{R}P^2$ corresponding to the color flux therein in the presence of the $P$-wave condensates alone. We calculate Aharanov-Bohm (AB) phases around the non-Abelian Alice string, and find that the 2SC condensates and string's orientational moduli must be aligned with each other because of single-valuedness of the AB phases of the 2SC condensates.

The volume of the quiver vortex moduli space

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vacuum expectation value of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey–Kirwan residue formula) leads to the Bradlow bounds ( upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss the properties of the moduli space volume in these theories. Our formulae are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with $\mathbb{C} P^N$ target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.