Topological order and conformal quantum critical points

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary two-dimensional free boson, the two-dimensional Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously varying critical exponents separating phases with long-range order from a deconfined topologically ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang–Mills gauge theory with a Chern–Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.     

Topological Field Theory Interpretation of String Topology

The string bracket introduced by Chas and Sullivan is reinterpreted from the point of view of topological field theories in the Batalin–Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern–Simons and BF theories) are considered together with generalized Wilson loops. The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the S¹-equivariant cohomology of the loop space of the manifold on which the theory is defined. It is proved that, in the case of GL(n,) with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles. Generalizations to other groups are briefly described.     

Multiple Existence of the Multivortex Solutions of the Self-Dual Chern–Simons CP(1) Model on a Doubly Periodic Domain

In this Letter, we investigate multiple existence of the multivortex solutions of the self-dual Chern–Simons CP(1) model on a flat torus for each range of the symmetry breaking parameter. We also study asymptotics of solutions as the Chern–Simons coupling constant goes to zero.     

The coupling of Chern–Simons theory to topological gravity

We couple Chern–Simons gauge theory to 3-dimensional topological gravity with the aim of investigating its quantum topological invariance. We derive the relevant BRST rules and Batalin–Vilkovisky action. Standard BRST transformations of the gauge field are modified by terms involving both its anti-field and the super-ghost of topological gravity. Beyond the obvious couplings to the metric and the gravitino, the BV action includes hitherto neglected couplings to the super-ghost. We use this result to determine the topological anomalies of certain higher ghost deformations of SU(N)$\mathit{SU}(N)$ Chern–Simons theory, introduced years ago by Witten. In the context of topological strings these anomalies, which generalize the familiar framing anomaly, are expected to be cancelled by couplings of the closed string sector. We show that such couplings are obtained by dressing the closed string field with topological gravity observables.     

Polynomial Invariants for Torus Knots¶and Topological Strings

We make a precision test of a recently proposed conjecture relating Chern–Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern–Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU(N) for torus knots. We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes. We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern–Simons vevs. Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems. Received: 1 May 2000 / Accepted: 6 November 2000     

Broken symmetries in the reconstruction of ν=1 quantum Hall edges

Spin-polarized reconstruction of the ν=1 quantum Hall edge is accompanied by a spatial modulation of the charge density along the edge. We find that this is also the case for finite quantum Hall droplets: current spin density functional calculations show that the so-called Chamon–Wen edge forms a ring of apparently localized electrons around the maximum density droplet (MDD). The boundaries of these different phases qualitatively agree with recent experiments. For very soft confinement, Chern–Simons Ginzburg–Landau theory indicates formation of a non-translational invariant edge with vortices (holes) trapped in the edge region.     

A note on Chern–Simons solitons—A type III vortex from the wall vortex

We study some properties of topological Chern–Simons vortices in 2+1 dimensions. As has already been understood in the past, in the large magnetic flux limit, they are well described by a Chern–Simons domain wall, which has been compactified on a circle with the symmetric phase inside and the asymmetric phase on the outside.Our goal is two-fold. First we want to explore how the tension depends on the magnetic flux discretized by the integer n. The BPS case is already known, but not much has been explored about the non-BPS potentials. A generic renormalizable potential has two dimensionless parameters that can be varied. Variation of only one of them leads to a type I and type II vortex, very similar to the Abrikosov–Nielsen–Olesen (ANO) case. Variation of both the parameters leads to a much richer structure. In particular we have found a new type of vortex, which is type I-like for small flux and then turns type II-like for larger flux. We could tentatively denote it a type III vortex. This results in a stable vortex with number of fluxes which can be greater than one.Our second objective is to study the Maxwell–Chern–Simons theory and understand how the large n limit of the CS vortex is smoothly connected with the large n limit of the ANO vortex.     

Three-Manifold Invariants¶from Chern–Simons Field Theory¶with Arbitrary Semi-Simple Gauge Groups

Invariants for framed links in S ³ obtained from Chern–Simons gauge field theory based on an arbitrary gauge group (semi-simple) have been used to construct a three-manifold invariant. This is a generalization of a similar construction developed earlier for SU(2) Chern–Simons theory. The procedure exploits a theorem of Lickorish and Wallace and also those of Kirby, Fenn and Rourke which relate three-manifolds to surgeries on framed unoriented links. The invariant is an appropriate linear combination of framed link invariants which does not change under Kirby calculus. This combination does not see the relative orientation of the component knots. The invariant is related to the partition function of Chern–Simons theory. This thus provides an efficient method of evaluating the partition function for these field theories. As some examples, explicit computations of these manifold invariants for a few three-manifolds have been done. Received: 24 July 2000 / Accepted: 19 September 2000     

A Modular Functor Which is Universal¶for Quantum Computation

We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern–Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation has topological implications which will be considered elsewhere. Received: 4 May 2001 / Accepted: 18 February 2002     

Second-Order Five-Dimensional Chern–Simons Gravity

Continuing our previous discussion of the canonical covariant formalism (Zandron, O. S. (in press). International Journal of Theoretical Physics), the second-order canonical fünfbein formalism of the topological five-dimensional Chern–Simons gravity is constructed. Since this gravity model naturally contains a Gauss–Bonnet term quadratic in curvature, the second-order formalism requires the implementation of the Ostrogradski transformation in order to introduce canonical momenta. This is due to the presence of second time-derivatives of the fünfbein field. By performing the space–time decomposition of the manifold M ⁵, the set of first-class constraints that determines all the Hamiltonian gauge symmetries can be found. The total Hamiltonian as generator of time evolution is constructed, and the apparent gauge degrees of freedom are unambiguously removed, leaving only the physical ones.     

Solitons of the Self-dual Chern–Simons Theory on a Cylinder

We study the self-dual Chern–Simons Higgs theory on an asymptotically flat cylinder. A topological multivortex solution is constructed and the fast decaying property of solutions is proved.     

From Weak to Strong Coupling in ABJM Theory

The partition function of N=6{\mathcal{N}=6} supersymmetric Chern–Simons-matter theory (known as ABJM theory) on \mathbbS³{\mathbb{S}^3} , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super–matrix model is closely related to a matrix model describing topological Chern–Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on AdS₄×\mathbbC\mathbbP³{{\rm AdS}_4\times\mathbb{C}\mathbb{P}^3} and gives the correct N ^3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in \mathbbC\mathbbP³{\mathbb{C}\mathbb{P}^3} . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.     

Three-Manifold Invariants and Their Relation with the Fundamental Group

We consider the 3-manifold invariant I(M) which is defined by means of the Chern–Simons quantum field theory and which coincides with the Reshetikhin–Turaev invariant. We present some arguments and numerical results supporting the conjecture that for nonvanishing I(M), the absolute value |I(M)| only depends on the fundamental group π₁ (M) of the manifold M. For lens spaces, the conjecture is proved when the gauge group is SU(2). In the case in which the gauge group is SU(3), we present numerical computations confirming the conjecture. Received: 15 November 1996 / Accepted: 17 June 1997     

The Chern–Simons term induced at high temperature and the quantization of its coefficient

By perturbative calculations of the high-temperature ground-state axial vector current of fermion fields coupled to gauge fields, an anomalous Chern–Simons topological mass term is induced in the three-dimensional effective action. The anomaly in three dimensions appears just in the ground-state current rather than in the divergence of ground-state current. In the Abelian case, the contribution comes only from the vacuum polarization graph, whereas in the non-Abelian case, contributions come from the vacuum polarization graph and the two triangle graphs. The relation between the quantization of the Chern–Simons coefficient and the Dirac quantization condition of magnetic charge is also obtained. It implies that in a (2+1)-dimensional QED with the Chern–Simons topological mass term and a magnetic monopole with magnetic charge g present, the Chern–Simons coefficient must be also quantized, just as in the non-Abelian case. Received: 7 April 1999 / Published online: 3 November 1999     

Vortex Condensates¶for the SU(3) Chern–Simons Theory

We investigate SU(3)-periodic vortices in the self-dual Chern–Simons theory proposed by Dunne in [13, 15]. At the first admissible non-zero energy level E= 2 π, and for each (broken and unbroken) vacuum state φ₍₀₎ of the system, we find a family of periodic vortices asymptotically gauge equivalent to φ₍₀₎, as the Chern–Simons coupling parameter k→ 0. At higher energy levels, we show the existence of multiple gauge distinct periodic vortices with at least one of them asymptotically gauge equivalent to the (broken) principal embedding vacuum, when k→ 0. Received: 23 October 1999 / Accepted: 14 March 2000     

Non-minimally coupled scalar fields, Holst action and black hole mechanics

The paper deals with the extension of the Weak Isolated Horizon (WIH) formulation of black hole horizons to the non-minimally coupled scalar fields. In the early part of the paper, we introduce an appropriate Holst type action to incorporate scalar fields non-minimally coupled to gravity and construct the covariant phase space of the theory. Using this phase space, we proceed to prove the laws of black hole mechanics. Further, we show that with a gauge fixing, the symplectic structure on the horizon reduces to that of a U(1) Chern–Simons theory. The level of the Chern–Simons theory is shown to depend on the non-minimally coupled scalar field.     

No parity anomaly in massless QED3: A BPHZL approach

In this Letter we call into question the perturbatively parity breakdown at 1-loop for the massless QED₃ frequently claimed in the literature. As long as perturbative quantum field theory is concerned, whether a parity anomaly owing to radiative corrections exists or not shall be definitely proved by using a renormalization method independent of any regularization scheme. Such a problem has been investigated in the framework of BPHZL renormalization method, by adopting the Lowenstein–Zimmermann subtraction scheme. The 1-loop parity-odd contribution to the vacuum-polarization tensor is explicitly computed in the framework of the BPHZL renormalization method. It is shown that a Chern–Simons term is generated at that order induced through the infrared subtractions — which violate parity. We show then that, what is called “parity anomaly”, is in fact a parity-odd counterterm needed for restauring parity.