Simons Theory Research Articles

Chern–Simons theory and the R-matrix

It has been a long-standing problem how to relate Chern–Simons theory to the quantum groups. In this paper we recover the classical r-matrix directly from a three-dimensional Chern–Simons theory with boundary conditions, thus creating a direct link to the quantum groups. It is known that the Jones polynomials can be constructed using an R-matrix. We show how these constructions can be seen to arise directly from 3-dimensional Chern–Simons theory.

Batalin\u2013Vilkovisky quantization of fuzzy field theories

We apply the modern Batalin–Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogues of the field theories proposed recently through the notion of ‘braided L_infty -algebras’. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern–Simons theories on the fuzzy 2-sphere, as well as for braided scalar field theories on the fuzzy 2-torus.

Intersecting defects and supergroup gauge theory

We consider 5d supersymmetric gauge theories with unitary groups in the Ω-background and study codim-2/4 BPS defects supported on orthogonal planes intersecting at the origin along a circle. The intersecting defects arise upon implementing the most generic Higgsing (geometric transition) to the parent higher dimensional theory, and they are described by pairs of 3d supersymmetric gauge theories with unitary groups interacting through 1d matter at the intersection. We explore the relations between instanton and generalized vortex calculus, pointing out a duality between intersecting defects subject to the Ω-background and a deformation of supergroup gauge theories, the exact supergroup point being achieved in the self-dual or unrefined limit. Embedding our setup into refined topological strings and in the simplest case when the parent 5d theory is Abelian, we are able to identify the supergroup theory dual to the intersecting defects as the supergroup version of refined Chern–Simons theory via open/closed duality. We also discuss the BPS/CFT side of the correspondence, finding an interesting large rank duality with super-instanton counting.

Narain to Narnia

We generalize the holographic correspondence between topological gravity coupled to an abelian Chern-Simons theory in three dimensions and an ensemble average of Narain's family of massless free bosons in two dimensions, discovered by Afkhami-Jeddi et al. and by Maloney and Witten. We find that the correspondence also works for toroidal orbifolds but not for K3 or Calabi-Yau sigma-models and not always for the minimal models. We conjecture that the correspondence requires that the central charge is equal to the critical central charge defined by the asymptotic density of states of the chiral algebra. For toroidal orbifolds, we extend the holographic correspondence to correlation functions of twist operators by using topological properties of rational tangles in the three-dimensional ball, which represent configurations of vortices associated to a discrete gauge symmetry.

Holographic Schwinger effect in the confining background with D-instanton

Using the gauge-gravity duality, we study the holographic Schwinger effect by performing the potential analysis on the confining D3- and D4-brane background with D-instantons then evaluate the pair production/decay rate by taking account into a fundamental string and a single flavor brane respectively. The two confining backgrounds with D-instantons are obtained from the black D(-1)–D3 and D0–D4 solution with a double Wick rotation. The total potential and pair production/decay rate in the Schwinger effect are calculated numerically by examining the NG action of a fundamental string and the DBI action of a single flavor brane all in the presence of an electric field. In both backgrounds our numerical calculation agrees with the critical electric field evaluated from the DBI action and shows the potential barrier is increased by the presence of the D-instantons, thus the production/decay rate is suppressed by the D-instantons. The interpretation is that particles in the dual field theory could acquire an effective mass through the Chern-Simons interaction or the theta term due to the presence of D-instantons so that the pair production/decay rate in Schwinger effect is suppressed since it behaves as e^{-m^{2}}. This conclusion is in agreement with the previous results obtained in the deconfined D(-1)–D3 background at zero temperature limit and from the approach of the flavor brane in the D0–D4 background. In this sense, this work may be also remarkable to study the phase transition in Maxwell–Chern–Simons theory and observable effects by the theta angle in QCD.

From D = 3 to D = 2 dimensions: A note on topological order

We construct, by a procedure involving a dimensional reduction from a Chern–Simons theory with borders, an effective theory for (1 + 1)-dimensional superconductor. That system can be in either an ordinary phase or a topological one, depending on the value of two phases, associated to complex order parameters.

Routes to realize the axion-insulator phase in MnBi2Te4(Bi2Te3)n family

Axion, first postulated as a hypothetical particle in high-energy physics, is now extended to describe a novel topological magnetoelectric effect derived from the Chern–Simons theory in condensed matter systems. The recent discovered intrinsic magnetic topological insulators (MTIs) MnBi2Te4 and its derivatives have attracted great attention because of their potential as a material platform to realize such a quantized axion field. Since the magnetic exchange gap can bring the “half-quantized” anomalous Hall effect at the surface, an axion insulator manifests as quantum anomalous Hall and zero Hall plateau effects in the thin films. However, many puzzles about this material family remain elusive yet, such as the gapless surface state and the direct experimental evidence of the axion insulator. In this Perspective, we discuss the preconditions, manifestations, and signatures of the axion-insulator phase, in the context of the development of the natural magnetic topological heterostructure MnBi2Te4(Bi2Te3)n family with various intriguing quantum phenomena. Recent theoretical and experimental efforts regarding the intrinsic magnetic topological insulators are summarized here to pave the way for this phenomenally developing field.

Tunable dynamical magnetoelectric effect in antiferromagnetic topological insulator MnBi2Te4 films

Axion was postulated as an elementary particle to solve the strong charge conjugation and parity puzzle, and later axion was also considered to be a possible component of dark matter in the universe. However, the existence of axions in nature has not been confirmed. Interestingly, axions arise out of pseudoscalar fields derived from the Chern–Simons theory in condensed matter physics. In antiferromagnetic insulators, the axion field can become dynamical due to spin-wave excitations and exhibits rich exotic phenomena, such as axion polariton. However, antiferromagnetic dynamical axion insulator has yet been experimentally identified in realistic materials. Very recently, MnBi2Te4 was discovered to be an antiferromagnetic topological insulator with a quantized static axion field protected by inversion symmetry {mathcal{P}} and magnetic-crystalline symmetry {mathcal{S}}. Here, we studied MnBi2Te4 films in which both the {mathcal{P}} and {mathcal{S}} symmetries are spontaneously broken and found that substantially enhanced dynamical magnetoelectric effects could be realized through tuning the thickness of MnBi2Te4 films, temperature, or element substitutions. Our results show that thin films of MnBi2Te4 and related compounds could provide a promising material platform to experimentally study axion electrodynamics.

Scalarization of slowly rotating black holes

It is interesting to note that most black holes are born very slowly rotating. We investigate scalarization of slowly rotating black holes in the Einstein-scalar-Chern–Simons (EsCS) theory. In the slow rotation approximation, the CS term takes a linear form of rotation parameter [Formula: see text] which determines the tachyonic instability. The tachyonic instability for slowly rotating black holes represents the onset of spontaneous scalarization. It is shown that the slowly rotating black holes are unstable against a spherically symmetric scalar-mode perturbation for positive coupling [Formula: see text], whereas these black holes are unstable for negative coupling without any [Formula: see text]-bound.

Quantum theory of k(phi )-FLRW-metrics its connection to Chern-Simons-models and the theta vacuum structure of quantum gravity

We show how a foliated 4-dimensional FLRW-metric becomes a gravitational instanton, if the spatial metric minimizes a three-dimensional Einstein–Hilbert action with positive cosmological constant, which is equal to the demand, that the scale factor satisfies the Bogomolny-equation, where the curvature parameter varies over the one-parameter family of hyperslices and takes the role of a potential depending on the scale factor. Additionally, we draw the connection to SO(4)-Chern–Simons theory and show how the established interpolating solutions describe the gradient flow between the minima of the vacuums of the Einstein–Hilbert action, as well as how they can be used to calculate tunnelling-amplitudes of gravitons and trivialize the calculations of path integrals in quantum gravity. All the calculations are carried out particularly for k admitting a {mathbb {Z}}_{2}-symmetry.

The Hamilton–Jacobi analysis for higher-order Maxwell–Chern–Simons gauge theory

By using the Hamilton–Jacobi [HJ] framework, the higher-order Maxwell–Chern–Simons theory is analyzed. The complete set of HJ Hamiltonians and a generalized HJ differential are reported, from which all symmetries of the theory are identified. In addition, we complete our study by performing the higher-order Gitman–Lyakhovich–Tyutin [GLT] framework and compare the results of both formalisms.

On the Zakharov\u2013Mikhailov action: 4hbox {d} Chern\u2013Simons origin and covariant Poisson algebra of the Lax connection

We derive the 2hbox {d} Zakharov–Mikhailov action from 4hbox {d} Chern–Simons theory. This 2hbox {d} action is known to produce as equations of motion the flatness condition of a large class of Lax connections of Zakharov–Shabat type, which includes an ultralocal variant of the principal chiral model as a special case. At the 2hbox {d} level, we determine for the first time the covariant Poisson bracket r-matrix structure of the Zakharov–Shabat Lax connection, which is of rational type. The flatness condition is then derived as a covariant Hamilton equation. We obtain a remarkable formula for the covariant Hamiltonian in terms of the Lax connection which is the covariant analogue of the well-known formula “H={{,mathrm{Tr},}}L^2”.

Nambu brackets, Chern–Simons theories, quantum curves and M2-branes

Abstract A non-technical review on recent progress in M2-branes is given. We focus especially on (1) historical progress in searching for the worldvolume theory of M2-branes using Nambu brackets and supersymmetric Chern–Simons theories, (2) studies of the ABJM matrix model obtained from localization using various techniques and (3) new insights of quantum curves derived from the studies of the ABJM matrix model.

Hopf algebra gauge theory on a ribbon graph

We generalize gauge theory on a graph so that the gauge group becomes a finite-dimensional ribbon Hopf algebra, the graph becomes a ribbon graph, and gauge-theoretic concepts such as connections, gauge transformations and observables are replaced by linearized analogs. Starting from physical considerations, we derive an axiomatic definition of Hopf algebra gauge theory, including locality conditions under which the theory for a general ribbon graph can be assembled from local data in the neighborhood of each vertex. For a vertex neighborhood with [Formula: see text] incoming edge ends, the algebra of non-commutative ‘functions’ of connections is dual to a two-sided twist deformation of the [Formula: see text]-fold tensor power of the gauge Hopf algebra. We show these algebras assemble to give an algebra of functions and gauge-invariant subalgebra of ‘observables’ that coincide with those obtained in the combinatorial quantization of Chern–Simons theory, thus providing an axiomatic derivation of the latter. We then discuss holonomy in a Hopf algebra gauge theory and show that for semisimple Hopf algebras this gives, for each path in the embedded graph, a map from connections into the gauge Hopf algebra, depending functorially on the path. Curvatures — holonomies around the faces canonically associated to the ribbon graph — then correspond to central elements of the algebra of observables, and define a set of commuting projectors onto the subalgebra of observables on flat connections. The algebras of observables for all connections or for flat connections are topological invariants, depending only on the topology, respectively, of the punctured or closed surface canonically obtained by gluing annuli or discs along edges of the ribbon graph.

Classification of solutions for self-dual Chern–Simons CP(1) model

In this paper, we consider the nonlinear equation arising from the Chern–Simons theory of planar matter fields interacting with the Chern–Simons gauge field in a CP(1) invariant fashion. Then, we establish the sharp region of flux for non-topological solutions and prove the classification of solutions of all types in the case of one vortex point. Moreover, we also give the complete result of Theorem 1.3 in the work by Choe et al. [J. Differ. Equations, 255, 2136 (2013)] from Theorem 1.4(ii) as follows.

Roman Jackiw and Chern–Simons theories

I recount my personal experience interacting with Roman Jackiw in the 1980s, when we both worked on Chern-Simons theories in three dimensions.

3d spectral networks and classical Chern–Simons theory

3d spectral networks and classical Chern–Simons theory

Abelian Topological Order on Lattice Enriched with Electromagnetic Background

In topological phases of matter, the interplay between intrinsic topological order and global symmetry is an interesting task. In the study of topological orders with discrete global symmetry, an important systematic approach is the construction of exactly soluble lattice models. However, for continuous global symmetry, in particular the electromagnetic U(1), the lattice approach has been less systematically developed. In this paper, we introduce a systematic construction of effective theories for a large class of abelian topological orders on three-dimensional spacetime lattice with electromagnetic background. We discuss the associated topological properties, including the Hall conductivity and the spin-c nature of the electromagnetic background. Some of these effective spacetime lattice theories can be readily mapped to microscopic Hamiltonians on spatial lattice; others may also shed light on their possible microscopic Hamiltonian realizations. Our approach is based on the gauging of 1-form \(\mathbb {Z}\) symmetries. Our construction is naturally related to the continuum path integral of (doubled) U(1) Chern–Simons theory, through the latter’s formal description in terms of Deligne–Beilinson cohomology; when the global symmetry is dropped, our construction can be reduced to the Dijkgraaf–Witten model of associated abelian topological orders, as expected.

Four-dimensional Brane\u2013Chern\u2013Simons Gravity and Cosmology

From the field equations corresponding to a four-dimensional brane embedded in the five-dimensional spacetime of the Einstein–Chern–Simons theory for gravity, we find cosmological solutions that describe an accelerated expansion for a flat universe. Apart from a quintessence-type evolution scheme, we obtain a transient phantom evolution, which is not ruled out by the current observational data. Additionally, a bouncing solution is shown. The introduction of a kinetic term in the action shows a de Sitter behavior although the energy density is not constant. A quintessence behavior is also found. We conjecture on a possible geometric origin of dark energy coming from this action.

In and around abelian anyon models * * Z.W. is partially supported by NSF grant FRG-1664351 and ARO MURI contract W911NF-20-1-0082. We thank M. Bischoff, T. Gannon, C. Meng, and E. Rowell for helpful comments and discussions.

Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K-matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.