non-Abelian Theories Research Articles

Two-Dimensional Non-abelian BF Theory in Lorenz Gauge as a Solvable Logarithmic TCFT

We study two-dimensional non-abelian BF theory in Lorenz gauge and prove that it is a topological conformal field theory. This opens the possibility to compute topological string amplitudes (Gromov-Witten invariants). We found that the theory is exactly solvable in the sense that all correlators are given by finite-dimensional convergent integrals. Surprisingly, this theory turns out to be logarithmic in the sense that there are correlators given by polylogarithms and powers of logarithms. Furthermore, we found fields with "logarithmic conformal dimension" (elements of a Jordan cell for $L_0$). We also found certain vertex operators with anomalous dimensions that depend on the non-abelian coupling constant. The shift of dimension of composite fields may be understood as arising from the dependence of subtracted singular terms on local coordinates. This generalizes the well-known explanation of anomalous dimensions of vertex operators in the free scalar field theory.

The NSVZ relation and the NSVZ scheme for N = 1 non-Abelian supersymmetric theories, regularized by higher covariant derivatives

We discuss, how the exact NSVZ -function appears in N = 1 supersymmetric non-Abelian gauge theories, regularized by higher covariant derivatives. In particular, we demonstrate that the renormalization group functions defined in terms of the bare couplings satisfy the NSVZ relation in the case of using this regularization. This occurs, because the loop integrals giving the -function are integrals of double total derivatives with respect to loop momenta. It is also shown that for the renormalization group functions standardly defined in terms of the renormalized couplings the NSVZ scheme can be obtained if the theory is regularized by higher covariant derivatives and only powers of ln/µ are included into the renormalization constants. These statements are confirmed by the explicit calculations in the three-loop approximation, where the scheme dependence is essential.

Nature of symmetry breaking in the superconducting ground state

The order parameters which are thought to detect U(1) gauge symmetry breaking in a superconductor are both non-local and gauge dependent. For that reason they are also ambiguous as a guide to phase structure. We point out that a global subgroup of the local U(1) gauge symmetry may be regarded, in analogy to non-abelian theories, as a "custodial" symmetry affecting the matter field alone, and construct, along the lines of our previous work, a new gauge-invariant criterion for breaking symmetries of this kind. It is shown that spontaneous breaking of custodial symmetry is a necessary condition for the existence of spontaneous symmetry breaking of a global subgroup of the (abelian or non-abelian) gauge group in any given gauge, and a sufficient condition for the existence of spontaneous breaking of a global subgroup of the gauge group in some gauge. As an illustration we compute numerically, in the lattice version of the Ginzburg-Landau model, the phase boundaries of the theory and the order parameters associated with various symmetries in each phase.

Solutions to the minimization problem arising in a dark monopole model in gauge field theory

We prove the existence of dark monopole solutions in a recently formulated Yang–Mills–Higgs theory model with technical features similar to the classical monopole problems. The solutions are obtained as energy-minimizing static spherically symmetric field configurations of unit topological charge. We overcome the difficulty of recovering the full set of boundary conditions by a regularization method which may be applied to other more complicated problems concerning monopoles and dyons in non-Abelian gauge field theories. Furthermore we show in a critical coupling situation that an explicit BPS solution may be used to provide energy estimates for non-BPS monopole solutions. Besides, in the limit of infinite Higgs coupling parameter, although no explicit construction is available, we establish an existence and uniqueness result for a monopole solution and obtain its energy bounds.

Covariant momentum map for non-Abelian topological BF field theory

We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the conserved Noether currents. The covariant momentum maps are fundamental to recover, after the space plus time decomposition of the background manifold, not only the extended Hamiltonian of the BF theory but also the generators of the gauge transformations which arise in the instantaneous Dirac–Hamiltonian analysis of the first-class constraint structure that characterizes the BF model under study. To the best of our knowledge, this is the first non-trivial physical model associated to general relativity for which both natural and Noether symmetries have been analyzed at the multisymplectic level. Our study shed some light on the understanding of the manner in which the generators of gauge transformations may be recovered from the multisymplectic formalism for field theory.

Supersymmetric Liouville theory in AdS2 and AdS/CFT

In a series of recent papers, a special kind of AdS2/CFT1 duality was observed: the boundary correlators of elementary fields that appear in the Lagrangian of a 2d conformal theory in rigid AdS2 background are the same as the correlators of the corresponding primary operators in the chiral half of that 2d CFT in flat space restricted to the real line. The examples considered were: (i) the Liouville theory where the operator dual to the Liouville scalar in AdS2 is the stress tensor; (ii) the abelian Toda theory where the operators dual to the Toda scalars are the mathcal{W} -algebra generators; (iii) the non-abelian Toda theory where the Liouville field is dual to the stress tensor while the extra gauged WZW theory scalars are dual to non-abelian parafermionic operators. By direct Witten diagram com- putations in AdS2 one can check that the structure of the boundary correlators is indeed consistent with the Virasoro (or higher) symmetry. Here we consider a supersymmetric generalization: the mathcal{N} = 1 superconformal Liouville theory in AdS2. We start with the super Liouville theory coupled to 2d supergravity and show that a consistent restriction to rigid AdS2 background requires a non-zero value of the supergravity auxiliary field and thus a modification of the Liouville potential from its familiar flat-space form. We show that the Liouville scalar and its fermionic partner are dual to the chiral half of the stress tensor and the supercurrent of the super Liouville theory on the plane. We perform tests supporting the duality by explicitly computing AdS2 Witten diagrams with bosonic and fermionic loops.

SQED3 and SQCD3 : Phase transitions and integrability

We study supersymmetric Yang-Mills theories on the three-sphere, with massive matter and Fayet-Iliopoulos parameter, showing second order phase transitions for the non-Abelian theory, extending a previous result for the Abelian theory. We study both partition functions and Wilson loops and also discuss the case of different $R$-charges. Two interpretations of the partition function as eigenfunctions of the $A_{1} $ and free $A_{N-1}$ hyperbolic Calogero-Moser integrable model are given as well.

Magnetic field and curvature effects on pair production. II. Vectors and implications for chromodynamics

We calculate the pair production rates for spin-$1$ or vector particles on spaces of the form $M \times {\mathbb R}^{1,1}$ with $M$ corresponding to ${\mathbb R}^2$ (flat), $S^2$ (positive curvature) and $H^2$ (negative curvature), with and without a background (chromo)magnetic field on $M$. Beyond highlighting the effects of curvature and background magnetic field, this is particularly interesting since vector particles are known to suffer from the Nielsen-Olesen instability, which can dramatically increase pair production rates. The form of this instability for $S^2$ and $H^2$ is obtained. We also give a brief discussion of how our results relate to ideas about confinement in nonabelian theories.

Non-abelian Toda theory on AdS2 and {mathrm{AdS}}_2/{mathrm{CFT}}_2^{raisebox{1ex}{1}!left/ !raisebox{-1ex}{2}right.} duality

It was recently observed that boundary correlators of the elementary scalar field of the Liouville theory on AdS2 background are the same (up to a non-trivial proportionality coefficient) as the correlators of the chiral stress tensor of the Liouville CFT on the complex plane restricted to the real line. The same relation generalizes to the conformal abelian Toda theory: boundary correlators of Toda scalars on AdS2 are directly related to the correlation functions of the chiral mathcal{W} -symmetry generators in the Toda CFT and thus are essentially controlled by the underlying infinite-dimensional symmetry. These may be viewed as examples of AdS2/CFT1 duality where the CFT1 is the chiral half of a 2d CFT; we shall to this as {mathrm{AdS}}_2/{mathrm{CFT}}_2^{raisebox{1ex}{1}!left/ !raisebox{-1ex}{2}right.} . In this paper we demonstrate that this duality applies also to the non-abelian Toda theory containing a Liouville scalar coupled to a 2d σ-model originating from the SL(2, ℝ)/U(1) gauged WZW model. Here the Liouville scalar is again dual to the chiral stress tensor T while the other two scalars are dual to the parafermionic operators V± of the non-abelian Toda CFT. We explicitly check the duality at the next-to-leading order in the large central charge expansion by matching the chiral CFT correlators of (T, V+, V−) (computed using a free field representation) with the boundary correlators of the three Toda scalars given by the tree-level and one-loop Witten diagrams in AdS2.

Explicit Methods in Number Theory

The aim of the series of Oberwolfach meetings on ‘Explicit methods in number theory’ is to bring together people attacking key problems in number theory via techniques involving concrete or computable descriptions. Here, number theory is interpreted broadly, including algebraic and analytic number theory, Galois theory and inverse Galois problems, arithmetic of curves and higher-dimensional varieties, zeta and L -functions and their special values, modular forms and functions. The meeting provides a forum for presenting new methods and results on concrete aspects of number theory. Considerable attention is paid to computational issues, but the emphasis is on aspects that are of interest to the pure mathematician. In this respect the meetings differ from virtually all other computationally oriented meetings in number theory (most notably the ANTS series), which have a tendency to place actual implementations, numerical results, and cryptographic implications in the foreground. The 2018 meeting featured a mini-course on nonabelian Chabauty theory, so several of the talks were on this topic; the other talks covered a broad range of topics in number theory.

A Brief Survey of Higgs Bundles

Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

Ground-state degeneracy of non-Abelian topological phases from coupled wires

We construct a family of two-dimensional non-Abelian topological phases from coupled wires using a non-Abelian bosonization approach. We then demonstrate how to determine the nature of the non-Abelian topological order (in particular, the anyonic excitations and the topological degeneracy on the torus) realized in the resulting gapped phases of matter. This paper focuses on the detailed case study of a coupled-wire realization of the bosonic $su(2)^{\,}_{2}$ Moore-Read state, but the approach we outline here can be extended to general bosonic $su(2)^{\,}_{k}$ topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases.

Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups

Let k be the algebraic closure of a finite field of odd characteristic p and X a smooth projective scheme over the Witt ring W(k) which is geometrically connected in characteristic zero. We introduce the notion of Higgs–de Rham flow and prove that the category of periodic Higgs–de Rham flows over X/W(k) is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the étale fundamental group \pi_1(X_K) of the generic fiber of X , after Fontaine–Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber X_k of X of rank \leq p initiates a semistable Higgs–de Rham flow and thus those of rank \leq p-1 with trivial Chern classes induce k -representations of \pi_1(X_K) . A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic p , it was constructed by Ogus–Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus–Vologodsky correspondence of Shiho.

Domain Wall Solitons Arising in Classical Gauge Field Theories

Domain wall solitons are basic constructs realizing phase transitions in various field-theoretical models and are solutions to some nonlinear ordinary differential equations descending from the corresponding full sets of governing equations in higher dimensions. In this paper, we present a series of domain wall solitons arising in several classical gauge field theory models. In the context of the Abelian gauge field theory, we unveil the surprising result that the solutions may explicitly be constructed, which enriches our knowledge on integrability of the planar Liouville type equations in their one-dimensional limits. In the context of the non-Abelian gauge field theory, we obtain some existence theorems for domain wall solutions arising in the electroweak type theories by developing some methods of calculus of variations formulated as direct and constrained minimization problems over a weighted Sobolev space.

Quantum Sheaf Cohomology and Duality of Flag Manifolds

We study the quantum sheaf cohomology of flag manifolds with deformations of the tangent bundle and use the ring structure to derive how the deformation transforms under the biholomorphic duality of flag manifolds. Realized as the OPE ring of A/2-twisted two-dimensional theories with (0,2) supersymmetry, quantum sheaf cohomology generalizes the notion of quantum cohomology. Complete descriptions of quantum sheaf cohomology have been obtained for abelian gauged linear sigma models (GLSMs) and for nonabelian GLSMs describing Grassmannians. In this paper we continue to explore the quantum sheaf cohomology of nonabelian theories. We first propose a method to compute the generating relations for (0,2) GLSMs with (2,2) locus. We apply this method to derive the quantum sheaf cohomology of products of Grassmannians and flag manifolds. The dual deformation associated with the biholomorphic duality gives rise to an explicit IR duality of two A/2-twisted (0,2) gauge theories.

Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part I

In this paper, we study gapped edges/interfaces in a 2+1 dimensional bosonic topological order and investigate how the topological entanglement entropy is sensitive to them. We present a detailed analysis of the Ishibashi states describing these edges/interfaces making use of the physics of anyon condensation in the context of Abelian Chern-Simons theory, which is then generalized to more non-Abelian theories whose edge RCFTs are known. Then we apply these results to computing the entanglement entropy of different topological orders. We consider cases where the system resides on a cylinder with gapped boundaries and that the entanglement cut is parallel to the boundary. We also consider cases where the entanglement cut coincides with the interface on a cylinder. In either cases, we find that the topological entanglement entropy is determined by the anyon condensation pattern that characterizes the interface/boundary. We note that conditions are imposed on some non-universal parameters in the edge theory to ensure existence of the conformal interface, analogous to requiring rational ratios of radii of compact bosons.

Gauge field theories with Lorentz-violating operators of arbitrary dimension

The classification of Lorentz- and CPT-violating operators in nonabelian gauge field theories is performed. We construct all gauge-invariant terms describing propagation and interaction in the action for fermions and gauge fields. Restrictions to the abelian, Lorentz-invariant, and isotropic limits are presented. We provide two illustrative applications of the results to quantum electrodynamics and quantum chromodynamics. First constraints on nonlinear Lorentz-violating effects in electrodynamics are obtained using data from experiments on photon-photon scattering, and corrections from nonminimal Lorentz and CPT violation to the cross section for deep inelastic scattering are derived.

Quantum field theory in phase space

The tilde conjugation rule in thermofield dynamics, equivalent to the modular conjugation in a [Formula: see text]-algebra, is used to develop unitary representations of the Poincaré group, where the Hilbert space has the phase space content, a symplectic Hilbert space. The state is described by a quasi-amplitude of probability, which is a sort of wave function in phase space, associated with the Wigner function. The quantum field theory in phase space is then constructed, including the quantization rules for the Klein–Gordon and the Dirac fields, the derivation of the electrodynamics in phase space and elements of a relativistic quantum kinetic theory. Towards a physical interpretation of the theory, propagators are associated with the corresponding Wigner functions. The Feynman rules follow accordingly with vertices similar to those of usual non-Abelian quantum field theories.

AdS asymptotic symmetries from CFT mirrors

We study Kac-Moody asymptotic symmetries and memory effects in AdS4Poincaré gauge theory and (when accompanied by 4D gravity) in its holographic CFT3 dual. While such infinite-dimensional symmetries are absent in standard asymptotic analyses of AdS4, we show how they arise with alternate AdS boundary conditions. In the 3D holographic description, these alternate boundary conditions correspond to a modified {tilde{mathrm{CFT}}}_3 obtained by Chern-Simons gauging of the CFT3 dual defined by standard boundary conditions, so that Kac-Moody symmetries then follow from the familiar Chern-Simons/Wess-Zumino-Witten correspondence. Apart from their own intrinsic interest, in abelian AdS4 gauge theories these alternate boundary conditions are equivalent to standard boundary conditions imposed on electric-magnetic dual variables. In the holographic description this corresponds to 3D “mirror” symmetries connecting the original and modified CFTs. Further, in both abelian and non-abelian theories we show that the alternative/ {tilde{mathrm{CFT}}}_3 theory emerges at leading order in large Chern-Simons level from the correlators of the standard theory, upon incorporating large-wavelength limits in the holographically emergent dimension. We point out similarities and differences between 4D AdS and Minkowski gauge theories in their asymptotic symmetries, “soft” limits and memory effects.

A unified geometric framework for boundary charges and dressings: Non-Abelian theory and matter

Boundaries in gauge theories are a delicate issue. Arbitrary boundary choices enter the calculation of charges via Noether's second theorem, obstructing the assignment of unambiguous physical charges to local gauge symmetries. Replacing the arbitrary boundary choice with new degrees of freedom suggests itself. But, concretely, such boundary degrees of freedom are spurious—i.e. they are not part of the original field content of the theory—and have to disappear upon gluing. How should we fit them into what we know about field-theory? We resolve these issues in a unified and geometric manner, by introducing a connection 1-form, ϖ, in the field-space of Yang–Mills theory. Using this geometric tool, a modified version of symplectic geometry—here called ‘horizontal’—is possible. Independently of boundary conditions, this formalism bestows to each region a physical notion of charge: the horizontal Noether charge. The horizontal gauge charges always vanish, while global charges still arise for reducible configurations characterized by global symmetries. The field-content itself is used as a reference frame to distinguish ‘gauge’ and ‘physical’; no new degrees of freedom, such as group-valued edge modes, are required. Different choices of reference fields give different ϖ's, which are cousins of gauge-fixings like the Higgs-unitary and Coulomb gauges. But the formalism extends well beyond gauge-fixings, for instance by avoiding the Gribov problem. For one choice of ϖ, would-be Goldstone modes arising from the condensation of matter degrees of freedom play precisely the role of the known group-valued edge modes, but here they arise as preferred coordinates in field space, rather than new fields. For another choice, in the Abelian case, ϖ recovers the Dirac dressing of the electron.