We construct a non Abelian model for SU(2) QCD in Euclidean three-dimensional spacetime and study its different phases. The model contains a center vortex sector coupled to a dual effective field encoding information about how the vortices are paired in the ensemble. The possible phases in parameter space are interpreted in terms of the proliferation of either closed center vortices or closed chains, where the endpoints of open vortices are attached in pairs to monopole-like defects.
I explain how conventional topological defects -- the Abrikosov-Nielsen-Olesen (ANO) string and domain walls -- can acquire non-Abelian moduli localized on their world sheets. The set-up is conceptually similar and generalizes that used by Witten for cosmic strings.
We investigate the Laplacian Abelian gauge on the sphere S^4 in the background of a single `t Hooft instanton. To this end we solve the eigenvalue problem of the covariant Laplace operator in the adjoint representation. The ground state wave function serves as an auxiliary Higgs field. We find that the ground state is always degenerate and has nodes. Upon diagonalisation, these zeros induce toplogical defects in the gauge potentials. The nature of the defects crucially depends on the order of the zeros. For first-order zeros one obtains magnetic monopoles. The generic defects, however, arise f…
Magnetically charged Abelian defects are shown to arise on most compact base manifolds and in most Abelian projection gauges. They obey the Dirac quantisation condition and give rise to homogeneous magnetic background fields. The reasons for their occurrence are global failures of the procedure with which gauge covariant operators are diagonalised or their eigenphases extracted. Defects related to the former case are string-like; for the latter case they resemble domain walls. Either configuration forms the generic case and indicates a failure of gauge fixing as continuity and periodicity prop…
We present a new kind of defect in Abelian Projections, stemming from pointlike zeros of second order. The corresponding topological quantity is the Hopf invariant pi_3(S^2) (rather than the winding number pi_2(S^2) for magnetic monopoles). We give a visualisation of this quantity and discuss the simplest non-trivial example, the Hopf map. Such defects occur in the Laplacian Abelian gauge in a non-trivial instanton sector. For general Abelian projections we show how an ensemble of Hopf defects accounts for the instanton number.
\emph{Effective} gauge fields arise in the description of the dynamics of defects in lattices of graphene in condensed matter. The interactions between neighboring nodes of a lattice/spin-network are described by the Hubbard model whose effective field theory at long distances is given by the Dirac equation for an \emph{emergent} gauge field. The spin-networks in question can be used to describe the geometry experienced by a non-inertial observer in flat spacetime moving at a constant acceleration in a given direction. We expect such spin-networks to describe the structure of quantum horizons …
The structure of extrinsic defects in topologically ordered states of matter is host to a rich set of universal physics. Extrinsic defects in 2+1 dimensional topological states include line-like defects, such as boundaries between topologically distinct states, and point-like defects, such as junctions between different line defects. Gapped boundaries in particular can themselves be \it topologically \rm distinct, and the junctions between them can localize topologically protected zero modes, giving rise to topological ground state degeneracies and projective non-Abelian statistics. In this pa…
We consider gauged linear sigma models with gauge group U(1) that exhibit a geometric as well as a Landau Ginzburg phase. We construct defects that implement the transport of D-branes from the Landau-Ginzburg phase to the geometric phase. Through their fusion with boundary conditions these defects in particular provide functors between the respective D-brane categories. The latter map (equivariant) matrix factorizations to coherent sheaves and can be formulated explicitly in terms of complexes of matrix factorizations.
It has recently been realized that a general class of non-abelian defects can be created in conventional topological states by introducing extrinsic defects, such as lattice dislocations or superconductor-ferromagnet domain walls in conventional quantum Hall states or topological insulators. In this paper, we begin by placing these defects within the broader conceptual scheme of extrinsic twist defects associated with symmetries of the topological state. We explicitly study several classes of examples, including $Z_2$ and $Z_3$ twist defects, where the topological state with N twist defects ca…
We study topological defects in the Georgi-Machacek model in a hierarchical symmetry breaking in which extra triplets acquire vacuum expectation values before the doublet. We find a possibility of topologically stable non-Abelian domain walls and non-Abelian flux tubes (vortices) in this model. In the limit of the vanishing $U(1)_{\rm Y}$ gauge coupling in which the custodial symmetry becomes exact, the presence of a vortex spontaneously breaks the custodial symmetry, giving rise to $S^2$ Nambu-Goldstone (NG) modes localized around the vortex corresponding to non-Abelian fluxes. Vortices are c…
Ensembles of magnetic defects represent quantum variables that have been detected and extensively explored in lattice ${\rm SU}(N)$ pure Yang-Mills theory. They successfully explain many properties of confinement and are strongly believed to capture the (infrared) path-integral measure. In this work, we initially motivate the presence of magnetic non-Abelian degrees of freedom in these ensembles. Next, we consider a simple Gaussian model to account for fluctuations. In this case, both center vortices and monopoles become relevant degrees in Wilson loop averages. These physical inputs are then …
We calculate the energetic structure of defect cores and propose controlled methods to imprint a nontrivially entangled vortex pair that undergoes non-Abelian vortex reconnection in a biaxial nematic spin-2 condensate. For a singular vortex, we find three superfluid cores in addition to depletion of the condensate density. These exhibit order parameter symmetries that are different from the discrete symmetry of the biaxial nematic phase, forming an interface between the defect and the bulk superfluid. We provide a detailed analysis of phase mixing in the resulting vortex cores and find an inst…
Grand canonical ensemble of small vortex loops emerging in the London limit of the effective Abelian-projected theory of the SU(3)-gluodynamics is investigated in the dilute gas approximation. An essential difference of this system from the SU(2)-case is the presence of two interacting gases of vortex loops. Two alternative representations for the partition function of such a grand canonical ensemble are derived, and one of them, which is a representation in terms of the integrals over vortex loops, is employed for the evaluation of the correlators of both kinds of loops in the low-energy limi…
In this paper we describe a new type of topological defect, called a homilia string, which is stabilized via interactions with the string network. Using analytical and numerical techniques, we investigate the stability and dynamics of homilia strings, and show that they can form stable electroweak strings. In SU(2)xU(1) models of symmetry breaking the intersection of two homilia strings is identified with a sphaleron. Due to repulsive forces, the homilia strings seperate, resulting in sphaleron annihilation. It is shown that electroweak homilia string loops cannot stabilize as vortons, which c…
We derive non-abelian Toda field theories (NATFTs) from a 4d Chern-Simons (CS) theory with two order defects by employing a certain asymptotic boundary condition. The 4d CS theory is characterized by a meromorphic 1-form $ω$\,. We adopt $ω$ with two simple poles and no zeros, and each of the order defects is located at each pole. As a result, an anisotropy parameter $β^2$ can be identified with the distance between the two defects. As examples, we can derive the (complex) sine-Gordon model and the Liouville theory.
We extend and generalize the seminal work of Brandenberger, Huang and Zhang on the formation of strings during chiral phase transitions(berger) and discuss the formation of abelian and non-abelian topological strings during such transitions in the early Universe and in the high energy heavy-ion collisions. Chiral symmetry as well as deconfinement are restored in the core of these defects. Formation of a dense network of string defects is likely to play an important role in the dynamics following the chiral phase transition. We speculate that such a network can give rise to non-azimuthal distri…
We outline a derivation of area law of the Wilson loop in SU(N) Yang-Mills theory in the maximal Abelian gauge. This is based on a new version of non-Abelian Stokes theorem and the novel reformulation of the Yang-Mills theory. Abelian dominance and monopole dominance of the string tension in SU(N) QCD are immediate consequences of this derivation.
In this work, we propose a $3D$ ensemble measure for center-vortex worldlines and chains equipped with non-Abelian degrees of freedom. We derive an effective field description for the center-element average where the vortices get represented by $N$ flavors of effective Higgs fields transforming in the fundamental representation. This field content is required to accommodate fusion rules where $N$ vortices can be created out of the vacuum. The inclusion of the chain sector, formed by center-vortex worldlines attached to pointlike defects, leads to a discrete set of $Z(N)$ vacua. This type of SS…
This article continues the study of the category of harmonious field models that was recently introduced as a kinetically non-linear generalisation of the well known harmonic category of multiscalar fields over a supporting brane wordsheet in a target space with a curved Riemannian metric. Like the perfectly harmonious case of which a familiar example is provided by ordinary barotropic perfect fluids, another important subcategory is the simply harmonious case, for which it is shown that as well as ``wiggle'' modes of the underlying brane world sheet, and sound type longitudinal modes, there w…
We construct analytical supersymmetric coloured black hole solutions, i.e. non-Abelian black hole solutions that have no asymptotic non-Abelian charge but do have non-Abelian charges on the horizon that contribute to the Bekenstein-Hawking entropy, to two SU(3)-gauged N=2 d= supergravities. The analytical construction is made possible due to the fact that the main ingredient is the Bogomol'nyi equation, which under the assumption of spherical symmetry admits a Lax pair formulation. The Lax matrix needed for the coloured black holes must be defective which, even though it is the non-generic and…
Non-abelian (NA) topological string defects exist in QCD with the spontaneous chiral symmetry breaking $U(N_f)_L \times U(N_f)_R \to U(N_f)_V$. Anomaly effects lead to domain walls connected to these strings. In the framework of linear sigma model we find the configuration of these defects. Also we show that in this model metastable CP violating vacuua exist. Strings as well as extended regions of metastable vacuua may form during the chiral transition which could have interesting effects.
Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes.
In this Letter we use the Julia-Toulouse approach for condensation of defects in order to obtain an effective confinement theory for external chromoelectric probe charges in SU(3) gauge theory in the regime with condensed chromomagnetic monopoles. We use the Cho decomposition of the non-Abelian connection in order to reveal the Abelian sector of the non-Abelian gauge theory and the associated topological defects (monopoles) without resorting to any gauge fixing procedure. Using only the Abelian sector of the theory, we construct a hydrodynamic effective theory for the regime with condensed def…
Non-Abelian defects that bind Majorana or parafermion zero modes are prominent in several topological quantum computation schemes. Underpinning their established understanding is the quantum Ising spin chain, which can be recast as a fermionic model or viewed as a standalone effective theory for the surface-code edge -- both of which harbor non-Abelian defects. We generalize these notions by deriving an effective Ising-like spin chain describing the edge of quantum-double topological order. Relating Majorana and parafermion modes to anyonic strings, we introduce quantum-double generalizations …
We define a class of supersymmetric defect loop operators in N = 2 gauge theories in 2+1 dimensions. We give a prescription for computing the expectation value of such operators in a generic N = 2 theory on the three-sphere using localization. We elucidate the role of defect loop operators in IR dualities of supersymmetric gauge theories, and write down their transformation properties under the SL(2, Z) action on conformal theories with abelian global symmetries.
In the London limit of the Ginzburg-Landau theory (Abelian Higgs model), vortex dipoles (small vortex loops) are treated as a grand canonical ensemble in the dilute gas approximation. The summation over these objects with the most general rotation- and translation invariant measure of integration over their shapes leads to effective sine-Gordon theories of the dual fields. The representations of the partition functions of both grand canonical ensembles are derived in the form of the integrals over the vortex dipoles and the small vortex loops, respectively. By virtue of these representations, …
The relation between defects of Abelian gauges and instantons is discussed for explicit examples in the Laplacian Abelian gauge. The defect coming from an instanton is pointlike and becomes a monopole loop with twist upon perturbation. The interplay between magnetic charge, twist and instanton number - encoded as a Hopf invariant - is investigated with the help of a new method, an auxiliary Abelian fibre bundle.
Puff Field Theory is a low energy decoupling regime of string theory that still retains the non-local attributes of the parent theory - while preserving isotropy for its non-local degrees of freedom. It realizes an extended holographic dictionary at strong coupling and dynamical non-local states akin to defects or the surface operators of local gauge theories. In this work, we probe the non-local features of PFT using D3 branes. We find supersymmetric configurations that end on defects endowed with non-Abelian degrees of freedom. These are 2+1 dimensional defects in the 3+1 dimensional PFT tha…
Gauge theories in the presence of codimension two vortex defects are known to be related to the theories on orbifolds. By using this relation we study the localized path integrals of 2D N=(2,2) SUSY gauge theories with point-like vortex defects. We present a formula for the correlation functions of vortex defects inserted at the north and the south poles of squashed spheres. For Abelian gauge theories the correlators are locally constant as functions of the parameters of the defect, but exhibit discontinuity at some threshold values determined by the deficit angle at the poles and the R-charge…
We consider Yang-Mills theories with general gauge groups $G$ and twists on the four torus. We find consistent boundary conditions for gauge fields in all instanton sectors. An extended Abelian projection with respect to the Polyakov loop operator is presented, where $A_0$ is independent of time and in the Cartan subalgebra. Fundamental domains for the gauge fixed $A_0$ are constructed for arbitrary gauge groups. In the sectors with non-vanishing instanton number such gauge fixings are necessarily singular. The singularities can be restricted to Dirac strings joining magnetically charged defec…
We describe a geometric (or gravitational) analogue of the Laughlin quasiholes in the fractional quantum Hall states. Analogously to the quasiholes these defects can be constructed by an insertion of an appropriate vertex operator into the conformal block representation of a trial wavefunction, however, unlike the quasiholes these defects are extrinsic and do not correspond to true excitations of the quantum fluid. We construct a wavefunction in the presence of such defects and explain how to assign an electric charge and a spin to each defect, and calculate the adiabatic, non-abelian statisti…
Non-invertible Symmetries in Weyl Fermions, and Applications to Fermion-Boundary Scattering Problem
Department of Physics, Nanjing University（南京大学物理系）；School of Quantum and Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学量子学院和理论科学研究所）
This talk discusses two topological features in non-abelian gauge theories, related by the notion of abelian projection and the Hopf invariant. Minimising the energy of the non-linear sigma model with a Skyrme-like term (the Faddeev-Niemi model), can be identified with a non-linear maximal abelian gauge fixing of the SU(2) gauge vacua with a winding number equal to the Hopf invariant. In the context of abelian projection the Hopf invariant can also be associated to a monopole world line, through the Taubes winding, measuring its contribution to topological charge. Calorons with non-trivial hol…
Recently derived results for the exact induced parity-breaking term in 2+1 dimensions at finite temperature are shown to be relevant to the determination of the free energy for fixed-charge ensembles. The partition functions for fixed total charge corresponding to massive fermions in the presence of Abelian and non-Abelian magnetic fields are discussed. We show that the presence of the induced Chern-Simons term manifests itself in that the free energy depends strongly on the relation between the external magnetic flux and the value of the fixed charge.
Starting from an abelian rigid braided monoidal category C we define an abelian rigid monoidal category C_F which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V) and an object in C_F corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment fa…
We revisit the problem of boundary excitations at a topological boundary or junction defects between topological boundaries in non-chiral bosonic topological orders in 2+1 dimensions. Based on physical considerations, we derive a formula that relates the fusion rules of the boundary excitations, and the "half-linking" number between condensed anyons and confined boundary excitations. This formula is a direct analogue of the Verlinde formula. We also demonstrate how these half-linking numbers can be computed in explicit Abelian and non-Abelian examples. As a fundamental property of topological …
Invariants of Sequential Circuits and Generalized Non-Abelian Statistics
The University of Tokyo（东京大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所交叉理论与数学科学中心（iTHEMS））；Department of Applied Physics, The University of Tokyo（东京大学应用物理系）
Topological defects attract much recent interest in high-energy and condensed matter physics because they encode (non-invertible) symmetries and dualities. We study codimension-1 topological defects from a hamiltonian point of view, with the defect location playing the role of `time'. We show that the Weinstein symplectic category governs topological defects and their fusion: each defect is a lagrangian correspondence, and defect fusion is their geometric composition. We illustrate the utility of these ideas by constructing S- and T-duality defects in string theory, including a novel topology-…
