Abelian BF Theory Research Articles

Topological BF Description of 2D Accelerated Chiral Edge Modes

In this paper, we consider the topological abelian BF theory with radial boundary on a generic 3D manifold, as we were motivated by the recently discovered accelerated edge modes on certain Hall systems. Our aim was to research if, where, and how the boundary keeps the memory of the details of the background metrics. We discovered that some features were topologically protected and did not depend on the bulk metric. The outcome was that these edge excitations were accelerated, as a direct consequence of the non-flat nature of the bulk spacetime. We found three possibilities for the motion of the edge quasiparticles: same directions, opposite directions, and a single-moving mode. However, requiring that the Hamiltonian of the 2D theory is bounded by below, the case of the edge modes moving in the same direction was ruled out. Systems involving parallel Hall currents (for instance, a fractional quantum Hall effect with ν=2/5) cannot be described by a BF theory with the boundary, independently from the geometry of the bulk spacetime, because of positive energy considerations. Thus, we were left with physical situations characterized by edge excitations moving with opposite velocities (for example, the fractional quantum Hall effect with ν=1−1/n, with the n positive integer, and the helical Luttinger liquids phenomena) or a single-moving mode (quantum anomalous Hall). A strong restriction was obtained by requiring time reversal symmetry, which uniquely identifies modes with equal and opposite velocities, and we know that this is the case of topological insulators. The novelty, with respect to the flat bulk background, is that the modes have local velocities, which correspond to topological insulators with accelerated edge modes.

Towards holography for quantum mechanics

We derive a holographic description of the simplest quantum mechanical system, a 1d free particle. The dual formulation uses a couple of two-dimensional topological abelian BF theories with appropriate boundary conditions, interactions and constraints. The aim of this construction is not to use holography as a tool for quantum mechanics but rather to find the simplest possible setup in order to explore holography.

Two-dimensional abelian BF theory in Lorenz gauge as a twisted N=(2,2) superconformal field theory

We study the two-dimensional topological abelian BF theory in the Lorenz gauge and, surprisingly, we find that the gauged-fixed theory is a free type B twisted N=(2,2) superconformal theory with odd linear target space, with the ghost field c being the pullback of the linear holomorphic coordinate on the target. The QBRST of the gauge-fixed theory equals the total Q of type B twisted theory. This unexpected identification of two different theories opens a way for nontrivial deformations of both of these theories.

Perturbative Quantum Gauge Theories on Manifolds with Boundary

This paper introduces a general perturbative quantization scheme for gauge theories on manifolds with boundary, compatible with cutting and gluing, in the cohomological symplectic (BV-BFV) formalism. Explicit examples, like abelian BF theory and its perturbations, including nontopological ones, are presented.

Exact Computations in Topological Abelian Chern-Simons and BF Theories

We introduce Deligne cohomology that classifies U1 fibre bundles over 3 manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact (nonperturbative) computations in U1 Chern-Simons theory (BF theory, resp.) at the level of functional integrals. The partition functions (and observables) of these theories are strongly related to topological invariants well known to the mathematicians.

Three-dimensional dynamics of four-dimensional topological BF theory with boundary

We consider the four-dimensional (4D) abelian topological BF theory with a planar boundary, following Symanzik's method. We find the most general boundary conditions compatible with the field equations broken by the boundary. The residual gauge invariance is described by means of two Ward identities which generate a current algebra. We interpret this algebra as canonical commutation relations of fields, which we use to construct a 3D Lagrangian. As a remarkable by-product, we find a (unique) boundary condition which can be read as a duality relation between 3D dynamical variables.

DUAL LINEARIZED GRAVITY COUPLED TO BF-TYPE TOPOLOGICAL FIELD THEORIES IN D = 7

All consistent interactions in D = 7 that can be added to one of the dual formulations of linearized gravity via a massless tensor field with the mixed symmetry (4, 1) and an Abelian BF theory with a maximal field spectrum are constructed by means of deforming the solution to the master equation on behalf of specific cohomological techniques. Among the striking features of the coupled model, we mention that the gauge transformations of all fields are deformed and the main ingredients of the gauge structure are strongly modified during the deformation procedure.

HAMILTONIAN BF THEORY AND PROJECTED BORROMEAN RINGS

It is shown that the canonical formulation of the Abelian BF theory in D = 3 allows to obtain topological invariants associated to curves and points in the plane. The method consists on finding the Hamiltonian on-shell of the theory coupled to external sources with support on curves and points in the spatial plane. We explicitly calculate a nontrivial invariant that could be seen as a "projection" of the Milnor's link invariant [Formula: see text], and as such, it measures the entanglement of generalized (or projected) Borromeans Rings in the Euclidean plane.

Topological BF field theory description of topological insulators

Topological phases of matter are described universally by topological field theories in the same way that symmetry-breaking phases of matter are described by Landau–Ginzburg field theories. We propose that topological insulators in two and three dimensions are described by a version of abelian BF theory. For the two-dimensional topological insulator or quantum spin Hall state, this description is essentially equivalent to a pair of Chern–Simons theories, consistent with the realization of this phase as paired integer quantum Hall effect states. The BF description can be motivated from the local excitations produced when a π flux is threaded through this state. For the three-dimensional topological insulator, the BF description is less obvious but quite versatile: it contains a gapless surface Dirac fermion when time-reversal-symmetry is preserved and yields “axion electrodynamics”, i.e., an electromagnetic E · B term, when time-reversal symmetry is broken and the surfaces are gapped. Just as changing the coefficients and charges of 2D Chern–Simons theory allows one to obtain fractional quantum Hall states starting from integer states, BF theory could also describe (at a macroscopic level) fractional 3D topological insulators with fractional statistics of point-like and line-like objects.

Four-dimensional couplings among BF and massless Rarita–Schwinger theories: a BRST cohomological approach

The local and manifestly covariant Lagrangian interactions in four spacetime dimensions that can be added to a free model that describes a massless Rarita-Schwinger theory and an Abelian BF theory are constructed by means of deforming the solution to the master equation on behalf of specific cohomological techniques.

Quantization of four-dimensional Abelian gravity

An Abelian version of standard general relativity in the Cartan–Palatini gauge-like formulation in four dimensions has been introduced. Traditional canonical analysis utilizing similarities to the akin Husain–KuchařSU(2) version of gravity has been performed. The model has been next quantized in the canonical path-integral Faddeev–Popov formalism yielding Abelian BF theory.

Four-dimensional couplings among BF and matter theories from BRST cohomology

The local and manifestly covariant Lagrangian interactions in four spacetime dimensions that can be added to a “free” model that describes a generic matter theory and an abelian BF theory are constructed by means of deforming the solution to the master equation on behalf of specific cohomological techniques.

Four‐dimensional couplings among BF and matter theories from BRST cohomology

AbstractThe local and manifestly covariant Lagrangian interactions in four spacetime dimensions that can be added to a “free” model that describes a generic matter theory and an abelian BF theory are constructed by means of deforming the solution to the master equation on behalf of specific cohomological techniques.

The direct link model for polymer rings using topological field theory and the second topological moment in dense systems

Polymer rings in solution are either permanently entangled or not. Permanent topological restrictions give rise to additional entropic interactions apart from the ones arising due to mere chain flexibility or excluded volume. Conversely, entangled polymer rings systems may be formed by closing randomly entangled flexible linear chains. The dependance of linking numbers between randomly entangled rings on the chain length, more specifically the second topological moment $<n^2>$, i.e. the average squared linking number, may be determined. In this paper, an approach recently discussed in mathematical physics and called abelian BF theory, is presented which allows to express the linking constraint in its simplest form, the Gauss integral, in terms of two gauge fields. The model of Brereton and Shah for a single ring entangled with many other surrounding rings is rederived. The latter model is finally used to calculate the second topological moment, in agreement with a recent result by Ferrari, Kleinert, and Lazzizzera obtained by using ${\rm n}$-component $\phi^4$ theory in the limit ${\rm n}\to 0$.

Abelian BF-theory and spherically symmetric electromagnetism

Three different methods to quantize the spherically symmetric sector of electromagnetism are presented: First, it is shown that this sector is equivalent to Abelian BF-theory in four spacetime dimensions with suitable boundary conditions. This theory, in turn, is quantized by both a reduced phase space quantization and a spin network quantization. Finally, the outcome is compared with the results obtained in the recently proposed general quantum symmetry reduction scheme. In the magnetically uncharged sector, where all three approaches apply, they all lead to the same quantum theory.

Abelian BF Theories and Knot Invariants

In the context of the Batalin-Vilkovisky formalism, a new observable for the Abelian BF theory is proposed whose vacuum expectation value is related to the Alexander-Conway polynomial. The three-dimensional case is analyzed explicitly, and it is proved to be anomaly free. Moreover, at the second order in perturbation theory, a new formula for the second coefficient of the Alexander-Conway polynomial is obtained. An account on the higher-dimensional generalizations is also given.

BF topological theories and infinitely reducible systems

We present a rigorous discussion for abelian BF theories in which the base manifold of the U(1) bundle is homeomorphic to a Hilbert space. The theory has an infinite number of stages of reducibility. We specify conditions on the base manifold under which the covariant quantization of the system can be performed unambiguously. Applications of the formulation to the superparticle and the supertstring are also discussed.

ℵ 0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO( N) symmetry. We call these ℵ 0 (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ℵ 0 supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ℵ 0 globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ℵ 0 supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.

The Partition Function for Topological Field Theories

We use a Hodge decomposition and its generalization to non-abelian flat vector bundles to calculate the partition function for abelian and non-abelian BF theories in n dimensions. This enables us to provide a simple proof that the partition function is related to the Ray-Singer torsion defined on flat vector bundles for all odd-dimensional manifolds and is equal to unity for even dimensions.

Twisting to abelian BF/Chern-Simons theories

Starting from a D = 3, N = 4 supersymmetric theory for matter fields, a twist with a Grassmann parity change is defined which maps the theory into a gauge-fixed, abelian BF theory on curved 3-manifolds. After adding surface terms to this theory, the twist is seen to map the resulting supersymmetric action to two uncoupled copies of the gauge-fixed Chern-Simons action. In addition, we give a map which takes the BF and Chern-Simons theories into Donaldson-Witten TQFTs. A similar construction, but with N = 2 supersymmetry, is given in two dimensions.