Narrow Results

Motivated by the computation of loop space quantum mechanics as indicated in [14], here we seek a better understanding of the tubular geometry of loop space ${ℒ}{ℳ}$ corresponding to a Riemannian manifold ${ℳ}$ around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of ${({{ℳ}}^{2N+1})}_C$ around the diagonal submanifold, where ${({{ℳ}}^N)}_C$ is the Cartesian product of $N$ copies of ${ℳ}$ with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to ${ℒ}{ℳ}$ can be obtained by taking the limit $N\to \infty$. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [21] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-$N$ limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of ${ℳ}$ which, for ${ℳ}\times {ℳ}$, is the tangent bundle $T{ℳ}$.

Explicit solutions to the conifold equations with complex dimension n = 3, 4 in terms of complex coordinates (fields) are employed to construct the Ricci-flat Kähler metrics on these manifolds. The Kähler two-forms are found to be closed. The complex realization of these conifold metrics are used in the construction of two-dimensional nonlinear sigma model with the conifolds as target spaces. The action for the sigma model is shown to be bounded from below. By a suitable choice of the "integration constants", arising in the solution of Ricci flatness requirement, the metric and the equations of motion are found to be non-singular. As the target space is Ricci-flat, the perturbative one-loop counterterms being absent, the model becomes topological. The inherent U(1) fiber over the base of the conifolds is shown to correspond to a gauge connection in the sigma model.

The same procedure is employed to construct the metric for the resolved conifold, in terms of complex coordinates and the action for a nonlinear sigma model with resolved conifold as target space, is found to have a minimum value, which is topological. The metric is expressed in terms of the six real coordinates and compared with earlier works. The harmonic function, which is the warp factor in Type II-B string theory, is obtained and the ten-dimensional warped metric has the AdS₅ × X₅ geometry.

The derivative expansion of the Wilsonian renormalization group generates additional terms in the effective β-functions not present in the perturbative approach. Applied to the nonlinear σ-model, to lowest order the vanishing of the β-function for the tachyon field generates an equation analogous to that found in open string field theory. Although the nonlinear term depends on the cutoff function, this arbitrariness can be removed by a rescaling of the tachyon field.

The so-called Stueckelberg transformation is introduced as a prescription of minimal coupling following the procedure of the Utiyama theory, for both internal and spacetime symmetry Lie groups. As a natural example, the theory is applied to the Weyl group and the corresponding gauge gravitational theory is developed. This context appears to be a natural source to account for some sort of dark matter intrinsically related to the gauge-group parameter associated with scale transformations.

We study the topological defects in the nonlinear O(3) sigma model in terms of the decomposition of U(1) gauge potential. Time-dependent baby skyrmions are discussed in the (2+1)-dimensional spacetime with the CP¹ field. Furthermore, we show that there are three kinds of topological defects–vortex lines, point defects and knot exist in the (3+1)-dimensional model, and their topological charges, locations and motions are determined by the ϕ-mapping topological current theory.

We explore the O(N)-invariant Non-Linear Sigma Model (NLSM) in a different perturbative regime from the usual relativistic-free-field one, by using non-canonical basic commutation relations adapted to the underlying O(N) symmetry of the system, which also account for the nontrivial (nonflat) geometry and topology of the target manifold.

We show that the noncommutativity of space–time destroys the renormalizability of the 1/N expansion of the O(N) Gross–Neveu model. A similar statement holds for the noncommutative nonlinear sigma model. However, we show that, up to the subleading order in 1/N expansion, the noncommutative supersymmetric O(N) nonlinear sigma model becomes renormalizable in D=3. We also show that dynamical mass generation is restored and there is no catastrophic UV/IR mixing. Unlike the commutative case, we find that the Lagrange multiplier fields, which enforce the supersymmetric constraints, are also renormalized. For D=2 the divergence of the four-point function of the basic scalar field, which in D=3 is absent, cannot be eliminated by means of a counterterm having the structure of a Moyal product.

Following a previous work on 2-dimensional Abelian (2,0)-gauge theories, one reacesses the analysis of the consistency of the dynamics of (2,0) "relaxed" Yang-Mills super-potentials and their coupling to (2,0)-nonlinear σ-model, by gauging the isotropy or the isometry group of the latter. One pays special attention to the extra "chiral-like" component-field gauge potential that comes out from the relaxation of superspace constraints.

We show that the least energy conditions in the gauged nonlinear sigma model with Chern–Simons term lead to exact soliton-like solutions which look like domain walls. In fact, they are string-like solutions on the 2D plane. We will derive and discuss the corresponding solutions, and compute their energy and charge per unit length. The spin of the solutions is shown to vanish.

We consider the pp-wave limit of an exactly solvable gauged WZWN model proposed by Sfetsos and Tseytlin. The monodromy matrix is constructed for the model by adopting the methods utilized by us earlier. The pole structure of the monodromy matrix is analyzed and their properties are discussed in the context of the pp-wave limit of the gauged WZWN model.

Gauge-invariant systems in unconstrained configuration and phase spaces, equivalent to second-class constraints systems upon a gauge-fixing, are discussed. A mathematical pendulum on an (n-1)-dimensional sphere S^n-1 as an example of a mechanical second-class constraints system and the O(n) nonlinear sigma model as an example of a field theory under second-class constraints are discussed in details and quantized using the existence of underlying dilatation gauge symmetry and by solving the constraint equations explicitly. The underlying gauge symmetries involve, in general, velocity dependent gauge transformations and new auxiliary variables in extended configuration space. Systems under second-class holonomic constraints have gauge-invariant counterparts within original configuration and phase spaces. The Dirac's supplementary conditions for wave functions of first-class constraints systems are formulated in terms of the Wigner functions which admit, as we show, a broad set of physically equivalent supplementary conditions. Their concrete form depends on the manner the Wigner functions are extrapolated from the constraint submanifolds into the whole phase space.

Recently a perturbative theory has been constructed, starting from the Feynman rules of the nonlinear sigma model at the tree level in the presence of an external vector source coupled to the flat connection and of a scalar source coupled to the nonlinear sigma model constraint (flat connection formalism). The construction is based on a local functional equation, which overcomes the problems due to the presence (already at one loop) of nonchiral symmetric divergences. The subtraction procedure of the divergences in the loop expansion is performed by means of minimal subtraction of properly normalized amplitudes in dimensional regularization. In this paper we complete the study of this subtraction procedure by giving the formal proof that it is symmetric to all orders in the loopwise expansion. We provide further arguments on the issue that, within our subtraction strategy, only two parameters can be consistently used as physical constants.

We consider a (4+N)-dimensional Einstein gravity coupled to a nonlinear sigma model. This theory admits a solution in which the N extra dimensions contract exponentially while the ordinary space expand exponentially. Physically, the nonlinear sigma fields induce the dynamical compactification of the extra dimensions, which in turn drives inflation. No inflatons are required.

We have embedded the $O(N)$ nonlinear sigma model in a non-Abelian gauge theory. After that as a first class-system, it was quantized using two different approaches: the functional Schrödinger method and the nonlocal field–antifield procedure. First, the quantization was performed with the functional Schrödinger method, for $N=2$, obtaining the wave functionals for the ground and excited states. Second, using the well-known BV formalism, we have computed the one-loop anomaly. This result shows that the classical gauge symmetries, which appear due to the conversion via BFFT method, are broken at the quantum level.

We make progress towards a derivation of a low energy effective theory for SU(2) Yang–Mills theory. This low energy action is computed to 1-loop using the renormalization group technique, taking proper care of the Slavnov–Taylor identities in the Maximal Abelian Gauge. After that, we perform the Spin-Charge decomposition in a way proposed by Faddeev and Niemi. The resulting action describes a pair of nonlinear O(3) and $\sigma$-models interacting with a scalar field. The potential of the scalar field is a Mexican hat and the location of the minima sets the energy scale of solitonic configurations of the $\sigma$-model fields whose excitations correspond to glueball states.

It is shown how the Landau–Lifshitz model can be related to a nonlinear Schrödinger equation. In two space dimensions, the continuous Heisenberg model and more generally, the nonlinear sigma models are described by gauged Cherns–Simons field theories.

A new framework for quantum Hall skyrmions in O(4) nonlinear sigma model is studied here. The size and energy of the skyrmions are determined incorporating the quartic stability term in the Lagrangian. Moreover, the introduction of a θ-term determines the spin and statistics of these skyrmions.

We find approximate solutions for the two-dimensional nonlinear $\mathrm{\Sigma }$-model with Dzyalioshinkii–Moriya term, representing magnetic Skyrmions. They are built in an analytic form, by pasting different approximate solutions found in different regions of space. We verify that our construction reproduces the phenomenology known from numerical solutions and Monte Carlo simulations, giving rise to a Skyrmion lattice at an intermediate range of magnetic field, flanked by spiral and spin-polarized phases for low and high magnetic fields, respectively.

We consider a nonlinear sigma model coupled to the metric of a conic space. We obtain restrictions for a nonlinear sigma model to be a source of the conic space. We then study a nonlinear sigma model in the conic space background. We find coordinate transformations which reduce the chiral fields equations in the conic space background to field equations in Minkowski space–time. This enables us to apply the same methods for obtaining exact solutions in Minkowski space–time to the case of a conic space–time. In the case the solutions depend on two spatial coordinates we employ Ivanov's geometrical ansatz. We give a general analysis and also present classes of solutions in which there is dependence on three and four coordinates. We discuss with special attention the intermediate, instanton and meron solutions and their analogous in the conic space. We find differences in the total actions and topological charges of these solutions and discuss the role of the deficit angle.

We generalize the idea of quantum clock time to quantum spacetime reference frame via physical realization of a reference system by quantum rulers and clocks. Omitting the internal degrees of freedom (such as spins) of the physical rulers and clocks, only considering their metric properties, the spacetime reference frame is described by a bosonic nonlinear sigma model (NLSM). We study the quantum behavior of the system under approximations, and obtain (1) a cosmological constant valued $(2/\pi ){\rho }_{c0}$ (${\rho }_{c0}$ the critical density at near current epoch) which is very close to the observations; (2) an effective Einstein–Hilbert term; (3) the ratio of variance to mean-squared of spacetime interval tends to a universal constant $2/\pi$ in the infrared region. This effect is testable by observing a linear dependence between the inherent quantum variance and mean-squared of the redshifts from cosmic distant spectral lines. The proportionality is expected to be the observed percentage of the dark energy. The equivalence principle is also generalized to the quantum level.