Non-contractible Loops Research Articles

Semi-local cosmic strings and the cosmological constant problem

We study the cosmological constant problem in a three-dimensional N = 2 supergravity theory with gauge group SU(2) global × U(1) local. The model we consider is known to admit string-like configurations, the so-called semi-local cosmic strings. We show that the stability of these solitonic solutions is provided by supersymmetry through the existence of a lower bound for the energy, even though the manifold of the Higgs vacuum does not contain non-contractible loops. Charged Killing spinors do exist over configurations that saturate the Bogomol'nyi bound, as a consequence of an Aharonov-Bohmlike effect. Nevertheless, there are no physical fermionic zero modes on these backgrounds. The exact vanishing of the cosmological constant does not imply, then, Bose-Fermi degeneracy. This provides a non-trivial example of the recent claim made by Witten on the vanishing of the cosmological constant in three dimensions without unphysical degeneracies.

WZW models of general simple groups

It is shown that a WZW model corresponding to a general simple group possesses in general different quantisations which are parametrised by Hom( π 1( G), Hom( π 1( G), U(1))). The quantum theories are generically neither monodromy nor modular invariant, but all the modular invariant theories of Felder et al. are contained among them. A formula for the transformation of the Sugawara expression for L 0 under conjugation with respect to non-contractible loops in LG is derived. This formula is then used to analyse the monodromy properties of the various quantisations. It turns out that for π 1(G) ⋟ Z N , with N even, there are two monodromy invariant theories, one of which is modular invariant, and for π 1(G) ⋟ Z 2 × Z 2 there are eight monodromy invariant theories, two of which are modular invariant. A few specific examples are worked out in detail to illustrate the results.

Painlevé III and 2D polymers

Recently, the scaling function of the dilute non-contractible self-avoiding 2D polymer loop on a cylinder was the related to the Painlevé III transcendent. Using perturbation theory, the thermodynamic Bethe ansatz and numerical calculations we argue a similar relation for the contractible self-avoiding loop.

An explicit description of the symplectic structure of moduli spaces of flat connections

The moduli space of flat connections on a principal G-bundle over a compact oriented surface of genus g≥1 is considered herein. Using the holonomies around noncontractible loops, the moduli space is described as a quotient of a submanifold of G2g. An explicit expression is obtained for the symplectic form on the smooth part of moduli space, and several properties of this form are established.

Topological aspects of an antisymmetric background field on orbifolds

We study string theory on orbifolds in the presence of an antisymmetric constant background field in detail and discuss some of the new aspects of the theory. It is pointed out that the term with the antisymmetric background field can be regarded as a topological term like a Chern-Simons term. Detailed analysis in the operator formalism shows that orbifold models with topologically nontrivial twists exhibit various anomalous behavior: Zero-mode variables obey nontrivial quantization conditions. Coordinate transformations which define orbifolds are modified at quantum level. Wave functions of twisted strings in general acquire nontrivial phases when they move around non-contractible loops orbifolds. Zero-mode eigenvalues are shifted from naively expected values, in favor of modular invariance.

Existence of a new instanton in constrained Yang-Mills-Higgs theory

Our goal is to discover possible new four-dimensional euclidean solutions (instantons) in fundamental SU(2) Yang-Mills-Higgs theory, with a constraint added to prevent collapse of the scale. We show that, most likely, there exists one particular new constrained instanton (I∗) with vanishing Pontryagin index. This is based on a topological argument that involves the construction of a non-contractible loop of four-dimensional configurations with a certain upper bound on the action, which we establish numerically. We expect I∗ to be the lowest action non-trivial solution in the vacuum sector of the theory. There also exists a related static, but unstable, solution, the new sphaleron S∗. Possible applications of I∗ to the electroweak interactions include the asymptotics of perturbation theory and the high-energy behaviour of the total cross section.

A Counterexample to the Equivariant Simple Loop Conjecture

In 1985 Gabai gave a complete proof of the Simple Loop Conjecture, which states that any map between closed surfaces, which does not induce an injection on the level of ${\pi _1}$, takes some noncontractible simple loop in the domain surface to a contractible loop in the target surface. In this paper we study an analogous result for the category of surfaces equipped with finite group actions and the maps which commute with the group structures (the "equivariant" maps). We find a counterexample to the equivariant analog of the Simple Loop Conjecture for the cyclic group of order $3$. The proof uses an equivariant analog of a theorem of Edmonds which gives a standard geometric representative for any homotopy class of surface maps of nonzero degree.

A counterexample to the equivariant simple loop conjecture

In 1985 Gabai gave a complete proof of the Simple Loop Conjecture, which states that any map between closed surfaces, which does not induce an injection on the level of π 1 {\pi _1} , takes some noncontractible simple loop in the domain surface to a contractible loop in the target surface. In this paper we study an analogous result for the category of surfaces equipped with finite group actions and the maps which commute with the group structures (the "equivariant" maps). We find a counterexample to the equivariant analog of the Simple Loop Conjecture for the cyclic group of order 3 3 . The proof uses an equivariant analog of a theorem of Edmonds which gives a standard geometric representative for any homotopy class of surface maps of nonzero degree.

N = 2 supersymmetry, Painlevé III and exact scaling functions in 2D polymers

We discuss in this paper various aspects of the off-critical O( n) model in two dimensions. We find the ground-state energy conjectured by Zamolodchikov for the unitary minimal models, and extend the results to some non-unitary minimal cases. These results are applied to the discussion of scaling functions for polymers on a cylinder. Using the underlying N = 2 supersymmetry, the scaling function for one non-contractible polymer loop around the cylinder is shown to be simply related to the solution of the Painlevé III differential equation. We also find the ground-state energy for a single polymer on the cylinder. These results are checked by numerically simulating the polymer system. The flow to the dense polymer phase is also analyzed numerically. We find there surprising results, with a c eff function that is not monotonically decreasing and seems to have a roaming behavior, getting very close to the value 81 70 and 7 10 between its UV and IR values of 1.

Semi-local strings and monopoles

A variation on the abelian Higgs model, with SU(2) global × U(1) local symmetry broken to U(1) global, was recently shown by Vachaspati and Achúcarro to admit stable, finite-energy cosmic string solutions, even though the manifold of minima of the potential energy does not have non-contractible loops. This new and unexpected feature motivates a full investigation of the properties of the model. Here we exploit the existence of first-order Bogomol'nyi equations to classify all static finite-energy vortex solutions in the Bogomol'nyi limit. We find a 4 n-dimensional moduli space for the nth topological ( n-vortex) sector. Single-vortex configurations depend on a position coordinate and on an additional complex parameter and may be regarded as hybrids of Nielsen-Olesen vortices and C P 1 lumps. The model is also shown to obey Bogomol'nyi equations in curved space, and these allow a simple calculation of the gravitational field of the above configurations. Finally, monopole-like solutions interpolating between a Dirac monopole and a global monopole are found. These must be sorrounded by an event horizon as isolated solutions, but may also arise as unstable end points of semi-local strings.

On the existence of a new constrained instanton and high-energy electroweak interactions

We construct, in the SU(2) Yang-Mills-Higgs theory of the electroweak interactions, a non-contractible loop of four-dimensional configurations. From this we argue that there may exist a new constrained instanton I ★ with action ∼ 16 π 2/ g 2 and zero topological charge. We also discuss the spectral flow of the Dirac operator along the loop and expect I ★ to have fermion zero-modes. As a possible application for I ★ we consider the total ( B + L)-violating cross section, evaluated semiclassically from the imaginary part of the forward elastic scattering amplitude. We find a threshold-like behaviour for non-perturbative effects at a parton center-of-mass energy close to the sphaleron energy E S ∼ 10 TeV.

The sphaleron at finite mixing angle

The exact sphaleron solution in the bosonic sector of the electro-weak standard model is only known for weak mixing angle θw=0. To investigate the behaviour closer to the experimental value θWexp≈30° we construct a non-contractible loop in configuration space that is sensitive to θW. Our numerical results for the energy and the magnetic dipole moment, combined with previous perturbative results, indicate that the dependence on θW is weak.

Topological lower bound on the energy of a twisted rod

If one end of an elastic rod is rotated by an angle of 2π relative to the other, the “body frame” along the rod traces out a noncontractible loop in SO(3). This is not the case for a rotation by 4π. A lower bound is derived for the energy of a thin elastic rod whose body frame traces out a noncontractible loop in SO(3).

Anyons on a torus: Braid group, Aharonov-Bohm period, and numerical study.

We present a careful construction of anyons on a torus starting with the braid-group analysis. The rules of Wen, Dagotto, and Fradkin for putting anyons on a torus are reproduced with some minor improvements. The existence of noncontractible loops leads to braid-group representations characterized not only by anyon statistics \ensuremath{\theta} but also by the magnetic fluxes ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ threading through the holes of the torus. The three parameters are tangled with each other. We explore the symmetries of the torus to separate the effects of ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ from those of \ensuremath{\theta}. It is shown that the anyon system always has a smaller period \ensuremath{\theta}/\ensuremath{\pi} in ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ than the natural period 1. We perform several numerical calculations to investigate the spectral flow and consistency of the method and find interesting features in the spectral flow, which are relevant in understanding the fractional quantum Hall effect.

Braid group and anyons on a cylinder.

In this paper we present a careful reexamination of anyons on a cylinder (or annulus), starting from the braid-group analysis. Proper attention is paid to the topological features arising from the existence of noncontractible loops. The rule for putting anyons on a square lattice has to be modified when the periodic boundary condition is imposed on one direction. In contrast to the annulus, one extra restriction is needed for the cylinder geometry to recover its symmetry between the two edges. We have performed some finite-system calculations. The consistency of our results has been checked by the agreement from two sets of seemingly different rules and also by that with free fermions. We have also calculated the spectral flow of the excited states with varying statistics or flux and have seen a lot of level crossings, which seem to be a general feature of anyon systems. The mean-field treatment is found to be good until level crossing occurs, and to be better if one starts with the hard-core boson rather than the fermion.

PHASE TRANSITION DUE TO NON-CONTRACTIBLE LOOP IN HETEROTIC STRING D<10

The heterotic string of the uncompactified dimensions less than ten, D<10, without space-time supersymmetry is considered. The states which have excitations only in the instanton sectors appear in the spectrum. These states become tachyonic below or above a certain scale of the compactified space and make the vacuum unstable. These phenomena are understood as phase transitions due to noncontractible loop on the compactified space. The investigation into the effective potential tells that the phase transition is of first order. The properties of the new phase are studied both in the field theoretic and the stringy manners. The heterotic string at finite temperature in D<10 is also discussed.

Sphalerons, deformed sphalerons and configuration space

The SU(2)-Higgs theory has, in addition to the sphaleron, other spherically symmetric solutions (“deformed sphalerons”). We discuss their position in configuration space. In particular, we show that these sphalerons satisfy a Morse equality and present some non-contractible loops in configuration space passing through them.

ANOMALOUS FERMION NUMBER NON-CONSERVATION AT HIGH TEMPERATURES: TWO DIMENSIONAL EXAMPLE

We consider γ5 version of Abelian Higgs model in 1 + 1 dimensions and find one loop expression for the rate of fermion number non-conservation at high temperatures. We discuss the construction of the noncontractible loop in the configuration space and an analog of the sphaleron solution in the theory.

Hadroids

Weinberg's chiral theory of ϱ-π interactions has variety of non-perturbative excitations. They devide into two classes: those built on the Wu-Yang form and those on the Witten form. It is shown that in the Weinberg model the hadroid in the Witten form, though stable against variations along a non-contractible loop is unstable in the pion direction. This instability can be overcome by a stabilization term which is consistent with known phenomenology and does not change the mass of the hadroid. Also, the hadroid in the Wsu-Yang form is a hadronic sibling of the nucleon. Some physical properties of the hadroid are discussed.

Radially separated classical lumps in non-Abelian gauge models

We search for smooth and time-independent finite-energy solutions to Yang–Mills–Higgs theory with an arbitrary compact gauge group. Excluding the monopole solutions which have been studied before, we concentrate on configurations with no long-range fields, which include the saddle points corresponding to noncontractible (hyper-) loops. It is shown that if the radial dependence of the fields is factorized, only one solution satisfies all these conditions. This solution is the one which has been studied before by Dashen, Hasslacher, and Neveu and by Boguta, and whose existence has recently been proved rigorously. Formulas for the asymptotic behavior of this solution are given.