non-Abelian Theories Research Articles

Flux quantization on M5-branes

We highlight the need for global completion of the field content in the M5-brane sigma-model analogous to Dirac’s charge/flux quantization, and we point out that the superspace Bianchi identities on the worldvolume and on its ambient supergravity background constrain the M5’s flux-quantization law to be in a non-abelian cohomology theory rationally equivalent to a twisted form of co-Homotopy. In order to clearly bring out this subtle point we give a streamlined re-derivation of the worldvolume 3-flux via M5 “super-embeddings”. Finally, assuming the flux-quantization law to actually be in co-Homotopy (“Hypothesis H”) we show how this implies Skyrmion-like solitons on general M5-worldvolumes and (abelian) anyonic solitons on the boundaries of “open M5-branes” in heterotic M-theory.

Digital Quantum Simulation of a (1+1)D SU(2) Lattice Gauge Theory with Ion Qudits

We present a quantum simulation strategy for a (1+1)-dimensional SU(2) non-Abelian lattice gauge theory with dynamical matter, a hardcore-gluon Hamiltonian Yang-Mills, tailored to a six-level trapped-ion-qudit quantum processor, as recently experimentally realized [Nat. Phys. 18, 1053 (2022)]. We employ a qudit encoding fulfilling gauge invariance, an SU(2) Gauss’s law. We discuss the experimental feasibility of generalized Mølmer-Sørensen gates used to efficiently simulate the dynamics. We illustrate how a shallow circuit with these resources is sufficient to implement scalable digital quantum simulation of the model. We also numerically show that this model, albeit simple, can dynamically manifest physically relevant properties specific to non-Abelian field theories, such as baryon excitations. Published by the American Physical Society 2024

Froggatt-Nielsen meets the SMEFT

We study the matching of Froggatt-Nielsen theories of flavour onto the Standard Model Effective Field Theory (SMEFT), upon integrating out a heavy Beyond-the-Standard-Model (BSM) scalar ‘flavon’ whose vacuum expectation value breaks an Abelian flavour symmetry at energies ΛFN well above the electroweak scale, ΛFN> ΛSM. We include matching contributions to the infrared dSM = 6 (Warsaw basis) SMEFT sourced from ultraviolet contact terms suppressed up to order 1/ΛUV2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Lambda}_{\ extrm{UV}}^2 $$\\end{document} in the Froggatt-Nielsen Lagrangian, where ΛUV> ΛFN is an arbitrary ultraviolet scale where further unspecified BSM particles are dynamical. This includes tree-level (one-loop) ultraviolet diagrams with dFN = 6 (5) effective vertices. We first do so with a toy model, but then generalize our findings to arbitrary Frogatt-Nielsen charges. Our results indicate a rich and non-trivial signature of Froggatt-Nielsen theories on the (otherwise) model-independent operators of the SMEFT, and we briefly speculate on extending our analysis to broader classes of BSM flavour models, e.g. non-Abelian and/or gauged theories. We thus take an important step towards determining how to use rapidly developing theoretical and experimental SMEFT technologies to gain unambiguous insight into the SM’s longstanding fermion flavour puzzle.

Chromoelectric flux tubes within non-Abelian Proca theory

Flux tube solutions within non-Abelian SU(3) Proca theory with external sources are obtained. It is shown that such tubes have a longitudinal chromoelectric field possessing two components (nonlinear and gradient), as well as a transverse chromomagnetic field whose force lines create concentric circles with the center on the axis of the tube. The scenario of a possible relationship between non-Abelian Proca theory and quantum chromodynamics is considered. In such scenario: (a) the components of color fields have different behavior: those which are almost classical, and those which are purely quantum; (b) the second components create a gluon condensate that is a source of the field for the almost classical components of the Proca field; (c) Proca masses may appear as a result of an approximate description of the gluon condensate; (d) the question of gauge invariance is considered. It is shown that the results obtained are in good agreement with the results of lattice calculations. We make an assumption that an approximate description of a flux tube in quantum chromodynamics can be carried out using classical Proca equations but with a mandatory account of a gluon condensate.

The Dirac–Higgs Complex and Categorification of (BBB)-Branes

Abstract Let ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$. Following Gaiotto’s suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence.

Local Non Abelian Class Field Theory

The “local class field theory”, which can be defined as the description of the extensions of a given local field K in terms of the algebraic and analytic objects depending ony on the base K is one of the central problems of modern number theory. The theory developed for the abelian extensions, around the fundamental works of Artin and Hasse in the first quarter of the 20th century. It is natural to ask if one could construct this theory including the non-abelian extensions of the base field. There are two approches to this problem. One approach is based on the ideas of Langlands, and the other on Koch. Koch’s method was later generalized by Fesenko and Koch-de Shalit for some non-abelian extensions of the base field. On the other hand, Ikeda and Serbest extended Fesenko’s works to construct a non-abelian local class field theory. But there is a restriction for the base field K. In this study, we extended Ikeda-Serbest’s construction of the local reciprocity map for a certain local field to any local field. Also we have shown that the extended map satisfies the certain functoriality and ramification theoretic properties.

A geometric p-adic Simpson correspondence in rank one

For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb {Q}_p$ we give a geometrisation of the $p$ -adic Simpson correspondence of rank one in terms of analytic moduli spaces: the $p$ -adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$ -line bundles. As an application, we study a major open question in $p$ -adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the $p$ -adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $\pi ^{{\mathrm {\acute {e}t}}}_1(X)\to K^\times$ in terms of moduli spaces: for projective $X$ over $K=\mathbb {C}_p$ , it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group $\operatorname {Pic}(X)$ .

Quantization of non-Abelian Yang-Mills theories

A non-Abelian theory of fermions interacting with gauge bosons, the constrained system, is studied. The equations of motion for a singular system are obtained as total differential equations in many variables. The integrability conditions are investigated, and the set of equations of motion is integrable. The Senjanovic and the canonical methods are used to quantize the system, and the integration is taken over the canonical phase space coordinates.

Topological phase transition and surface states in a non-Abelian charged nodal line photonic crystal

Abstract Topological charges of nodal lines in a multigap system are represented by non-Abelian numbers, and the Euler class, a topological invariant, can be used to explain their topological phase transitions, such as pair-annihilation of nodal lines. Up until now, no discussion of phase transitions of nodal lines in photonic crystals using the Euler class has been reported, despite the fact that the Euler class and topological phase transition have recently been addressed in metallic or acoustic crystals. Here, we show how the deformation of a photonic crystal causes topological phase transitions in the nodal lines, and the Euler class can be used to theoretically predict the nodal lines’ stability based on the non-Abelian topological charge theory. Specifically, by manipulating the separation between the two single diamond structures and the extent of structural distortion, we numerically demonstrate the topological transition of nodal lines, e.g., from nodal lines to nodal rings. We then demonstrate that the range of surface states is strongly influenced by the topological phase transition of nodal lines. Moreover, the Zak phase was used to explain the surface states’ existence.

Cogroupoid structures on the circle and the Hodge degeneration

Abstract We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$ -fold delooping of the filtered loop space $E_2$ -groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle $S^1_{fil}$ of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an $E_2$ -cogroupoid object in the $\infty $ -category of spaces. We relate this cogroupoid structure with the more commonly studied ‘pinch map’ on $S^1$ , as well as the Todd class of the Lie algebroid $\mathbb {T}_{X}$ ; this is an invariant of a smooth and proper scheme X that arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontrivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally, we record some consequences of this bit of structure at the level of Hochschild cohomology.

A geometric approach to the Yang-Mills mass gap

I provide a new idea based on geometric analysis to obtain a positive mass gap in pure non-abelian renormalizable Yang-Mills theory. The orbit space, that is the space of connections of Yang-Mills theory modulo gauge transformations, is equipped with a Riemannian metric that naturally arises from the kinetic part of reduced classical action and admits a positive definite sectional curvature. The corresponding regularized Bakry-Émery Ricci curvature (if positive) is shown to produce a mass gap for 2+1 and 3+1 dimensional Yang-Mills theory assuming the existence of a quantized Yang-Mills theory on (ℝ1+2, η) and (ℝ1+3, η), respectively. My result on the gap calculation, described at least as a heuristic one, applies to non-abelian Yang-Mills theory with any compact semi-simple Lie group in the aforementioned dimensions. In 2+1 dimensions, the square of the Yang-Mils coupling constant gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} has the dimension of mass, and therefore the spectral gap of the Hamiltonian is essentially proportional to gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} with proportionality constant being purely numerical as expected. Due to the dimensional restriction on 3+1 dimensional Yang-Mills theory, it seems one ought to introduce a length scale to obtain an energy scale. It turns out that a certain ‘trace’ operation on the infinite-dimensional geometry naturally introduces a length scale that has to be fixed by measuring the energy of the lowest glu-ball state. However, this remains to be understood in a rigorous way.

Non-Abelian Landau-Ginzburg theory of ferromagnetic superconductivity and mixing between U(1) and SU(2) gauge bosons

We propose an effective theory of non-Abelian superconductivity, an SU(2)xU(1) extension of the Abelian Landau-Ginzburg theory, which could be viewed as an effective theory of ferromagnetic superconductivity made of spin-up and spin-down doublet Cooper pair. Just like the Abelian Landau-Ginzburg theory it has the U(1) electromagnetic interaction, but the new ingredient is the non-Abelian SU(2) gauge interaction between the spin doublet Cooper pairs. A remarkable feature of the theory is the mixing between the U(1) gauge boson and the diagonal part of the SU(2) gauge boson. After the mixing it has a massless non-Abelian gauge boson (the massless s-boson) and massive gauge boson (the massive photon), in addition to the massive off-diagonal non-Abelian gauge bosons (the massive s-bosons) which induce the spin-flip interaction between the spin up and down components of the Cooper pair. So, unlike the ordinary Landau-Ginzburg theory it has a long range interaction mediated by the massless s-boson, which could be responsible for the long range magnetic order and spin waves observed in ferromagnetic superconductors. The theory is characterized by three scales. In addition to the correlation length fixed by the mass of the Higgs field it has two different penetration lengths, the one fixed by the mass of the photon (which generates the well known Meissner effect) and the other fixed by the mass of the off-diagonal gauge bosons (which determines the range of the spin flip interaction). The non-Abelian structure of the theory naturally accommodates new topological objects, the non-Abrikosov quantized spin vortex (as well as the well known Abrikosov vortex) and non-Abelian spin monopole. We discuss the physical implications of the non-Abelian Landau-Ginzburg theory.

Non-perturbative Lee-Wick gauge theory: Towards Confinement & RGE with strong couplings

We consider a non-Abelian Lee–Wick gauge theory and discuss Becchi-Rouet-Stora-Tyutin invariance. It contains fourth-order derivative as extensions of the kinetic term, leading to massive ghosts in the theory upon quantization. We particularly provide essential clues towards confinement conditions in strongly-coupled regimes, using the Kugo–Ojima approach, and obtain the functions in the non-perturbative regimes. This is achieved using a set of exact solutions of the corresponding local theory in terms of Jacobi elliptical functions. We obtain a similar function just as for the ordinary Yang-Mills theory but the main differences are that now, the cut-off arises naturally from the Lee–Wick heavy mass scales (M). We show that the fate of the ghosts are fixed in these regimes: they are no more the propagating degrees of freedom in the infrared-limit. As it also happens for the ordinary case, confinement is due to the non-Abelian nature of the theory. In the limit , one recovers the standard results for the local non-Abelian Yang-Mills theory.

Non-Abelian Hyperbolic Band Theory from Supercells

Non-Abelian Hyperbolic Band Theory from Supercells

Integrability and BRST invariance from BF topological theory

We consider the Becchi, Rouet, Stora and Tyutin (BRST) invariant effective action of the non-abelian BF topological theory in two dimensions with gauge group . By considering different gauge fixing conditions, the zero-curvature field equation gives rise to several well known integrable equations. We prove that each integrable equation together with the associated ghost field evolution equation, obtained from the BF theory, is a BRST invariant system with an infinite sequence of BRST invariant conserved quantities. We construct explicitly the systems and the BRST transformation laws for the Korteweg-de Vries (KdV) sequence (including the KdV, mKdV and CKdV equations) and Harry Dym integrable equation.

A Chern-Simons theory for dipole symmetry

We present effective field theories for dipole symmetric topological matters that can be described by the Chern-Simons theory. Unlike most studies using higher-rank gauge theory, we develop a framework with both U(1)U(1) and dipole gauge fields. As a result, only the highest multipole symmetry can support the ’t Hooft anomaly. We show that with appropriate point group symmetries, the dipolar Chern-Simons theory can exist in any dimension and, moreover, the bulk-edge correspondence can depend on the boundary. As two applications, we draw an analogy between the dipole anomaly and the torsional anomaly and generalize particle-vortex duality to dipole phase transitions. All of the above are in the flat spacetime limit, but our framework is able to systematically couple dipole symmetry to curved spacetime. Based on that, we give a proposal about anomalous dipole hydrodynamics. Moreover, we show that the fracton-elasticity duality arises naturally from a non-abelian Chern-Simons theory in 3D.

On Amitsur cohomology of monads

We introduce and study zero and one-dimensional non-abelian Amitsur cohomology theory for monads equipped with involutive distributive law satisfying the Yang–Baxter equation. Some applications to extensions of (non-commutative) rings and Banach algebras are given.

BPS cohomology for rank 2 degree 0 Higgs bundles (and more)

We give a formula comparing the E-series of the moduli stacks of rank 2 degree 0 semistable Higgs bundles in genus g≥2 to intersection E-polynomials of its coarse moduli space. A parallel formula holds in various 2-Calabi–Yau settings, for example for sheaves on K3 surfaces, or preprojective algebras of g-loop quivers. As a consequence we provide evidence for a conjecture of Davison on the BPS cohomology of Higgs bundles, which has implications for non-abelian Hodge theory for stacks. We apply the formula to cohomological χ-independence tests for BPS cohomology of Higgs bundles and K3 surfaces.

Mini-Workshop: Quantization of Complex Symplectic Varieties

The mini-workshop featured two main series of lectures: Functoriality in non-abelian Hodge theory by Tony Pantev, and Quantization of the Hitchin system and the analytic Langlands program by Jörg Teschner. In addition, four senior mathematicians and physicists gave two talks each on their recent mysterious discoveries related to the theme of the workshop. Three junior mathematicians also gave a talk based on their fresh results. All talks by mathematicians and physicists were coordinated to form a common ground of understanding. The smallness of the size of workshop promoted deeper discussions and helped to create friendly and inclusive atmosphere.

On Instabilities Caused by Magnetic Background Fields

We review instabilities that appear from the coupling of spin-one fields to a magnetic background in a non-Abelian theory. Such coupling results, due to asymptotic freedom in a negative quantum, contribute to the effective potential. In QCD, the Savvidy vacuum results. However, due to the tachyonic mode, such a state is not stable, and the question about the true ground state of QCD is still open. In the electroweak model, the corresponding instability is postponed to very large background fields and may be of relevance in the early universe, at best. We start with an introduction to the topic and display the necessary formulas and methods. Then, we consider the one-particle spectra of the fields in a magnetic background and the related Euler–Heisenberg Lagrangians. In addition, we discuss the potential instability connected with the anomalous moment of the electron. The main part is on the quantum correction to the energy in non-Abelian fields, including massive ones. Here, the focus is on so-called electroweak magnetism and the search for a classical solution of the field equations and their approximations by a lattice of flux tubes. Finally, we review approaches with non-homogeneous background fields and the background of an A0-field.