Instanton-like Solutions Research Articles

’t Hooft–Polyakov Monopole and Instanton-like Topological Solution in Gauge-Higgs Unification

We consider 't Hooft-Polyakov monopole (TPM) as a topological soliton in the 5-dimensional (5D) theory of gauge-Higgs unification. This scenario provides a very natural framework to incorporate the TPM, since the adjoint scalar is builtin as the extra-space component of higher dimensional gauge field. In the process of the analysis, we realize that the condition to be satisfied by the Bogomolny-Prasad-Sommerfield (BPS) state of TPM, the BPS monopole, is equivalent to an (anti-)self-dual condition for the higher dimensional gauge field. This observation, in turn, suggests the presence of instanton-like topological soliton living in the 4D space (including the extra dimension), instead of the 4D space-time in the case of ordinary instanton, say "space-like instanton" with finite energy, instead of finite action. We construct the field configuration for the space-like instanton and calculate its mass, handled by the compactification scale. Next we discuss the BPS monopole, as an anti-self-dual gauge field. We start by constructing a hedgehog-type solution as what is obtained by a local gauge transformation from a trivial vacuum. We also argue in some detail that the relation between these two types of topological solitons becomes manifest through the unified description by use of the ansatz adopted by 't Hooft.

On Exact Solutions and Perturbative Schemes in Higher Spin Theory

We review various methods for finding exact solutions of higher spin theory in four dimensions, and survey the known exact solutions of (non)minimal Vasiliev’s equations. These include instanton-like and black hole-like solutions in (A)dS and Kleinian spacetimes. A perturbative construction of solutions with the symmetries of a domain wall is also described. Furthermore, we review two proposed perturbative schemes: one based on perturbative treatment of the twistor space field equations followed by inverting Fronsdal kinetic terms using standard Green’s functions; and an alternative scheme based on solving the twistor space field equations exactly followed by introducing the spacetime dependence using perturbatively defined gauge functions. Motivated by the need to provide a higher spin invariant characterization of the exact solutions, aspects of a proposal for a geometric description of Vasiliev’s equation involving an infinite dimensional generalization of anti de Sitter space are revisited and improved.

Fermion zero mode associated with instantonlike self-dual solution to lattice Euclidean gravity

We prove the existence of the lattice fermion zero mode associated with the self-dual lattice gravity solution.Received 2 June 2017DOI:https://doi.org/10.1103/PhysRevD.96.054512© 2017 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasLattice field theoryQuantum field theoryQuantum gravityParticles & Fields

A note on the possible existence of an instanton-like self-dual solution to lattice Euclidean gravity

The self-dual solution to lattice Euclidean gravity is constructed. In contrast to the well known Eguchi-Hanson solution to continuous Euclidean Gravity, the lattice solution is asymptotically globally Euclidean, i.e., the boundary of the space as r −→ ∞ is S3 = SU(2).

Instanton-like self-dual solution to lattice Euclidean Gravity: Difference from Eguchi–Hanson solution to continuous gravity

The self-dual solution to lattice Euclidean gravity is constructed. In contrast to the well-known Eguchi–Hanson solution to continuous Euclidean Gravity, the lattice solution is asymptotically globally Euclidean, i.e., the boundary of the space as r → ∞ is S 3 = SU(2)

Computing Transition Rates for the 1-D Stochastic Ginzburg–Landau–Allen–Cahn Equation for Finite-Amplitude Noise with a Rare Event Algorithm

In this paper we compute and analyse the transition rates and duration of reactive trajectories of the stochastic 1-D Allen-Cahn equations for both the Freidlin-Wentzell regime (weak noise or temperature limit) and finite-amplitude white noise, as well as for small and large domain. We demonstrate that extremely rare reactive trajectories corresponding to direct transitions between two metastable states are efficiently computed using an algorithm called adaptive multilevel splitting. This algorithm is dedicated to the computation of rare events and is able to provide ensembles of reactive trajectories in a very efficient way. In the small noise limit, our numerical results are in agreement with large-deviation predictions such as instanton-like solutions, mean first passages and escape probabilities. We show that the duration of reactive trajectories follows a Gumbel distribution like for one degree of freedom systems. Moreover, the mean duration growths logarithmically with the inverse temperature. The prefactor given by the potential curvature grows exponentially with size. The main novelty of our work is that we also perform an analysis of reactive trajectories for large noises and large domains. In this case, we show that the position of the reactive front is essentially a random walk. This time, the mean duration grows linearly with the inverse temperature and quadratically with the size. Using a phenomenological description of the system, we are able to calculate the transition rate, although the dynamics is described by neither Freidlin--Wentzell or Eyring--Kramers type of results. Numerical results confirm our analysis.

The functional measure for the in–in path integral

The in–in path integral of a scalar field propagating in a fixed background is formulated in a suitable function space. The free kinetic operator, whose inverse gives the propagators of the in–in perturbation theory, becomes essentially self adjoint after imposing appropriate boundary conditions. An explicit spectral representation is given for the scalar in the flat space, and the standard propagators are rederived using this representation. In this way the subtle boundary path integral over the field configurations at the return time is handled straightforwardly. It turns out that not only the values of the forward (+) and the backward (−) evolving fields but also their time derivatives must be matched at the return time, which is mainly overlooked in the literature. This formulation also determines the field configurations that are included in the path integral uniquely. We show that some of the recently suggested instanton-like solutions corresponding to the stationary phases of the cosmological in–in path integrals can be rigorously identified as limits of sequences in the function space.

2n-dimensional at Fujii model instanton-like solutions and coupling constant's role between instantons with higher derivatives

An instanton-like solution is considered for a class of two-dimensional conformal model with Liouville term. The model expressed here is a new higher-dimensional model with scalar-spinor interaction on even-dimensional spaces, and is a generalization of the even-dimensional Fujii model. A recurrence relation for the Fujii model and coupling constant's indices between relations with interaction constant is extracted. The behavior of Gürsey, Akdeniz-Smailagi' c and Fujii coupling constants in 4-dimensional conformal models are examined.

FUZZY INSTANTONS

We present a series of instanton-like solutions to a matrix model which satisfy a self-duality condition and possess an action whose value is, to within a fixed constant factor, an integer l2. For small values of the dimension n2 of the matrix algebra the integer resembles the result of a quantization condition but as n → ∞ the ratio l/n can tend to an arbitrary real number between zero and one.

A study of a non-Abelian generalization of the Born-Infeld action

A new type of non-Abelian generalization of the Born-Infeld action is proposed, in which the spacetime indices and group indices are combined. The action is manifestly Lorentz and gauge invariant. In its power expansion, the lowest order term is the Yang-Mills action. Solutions of the Euler-Lagrange equation for the SU(2) case are considered and we show that there exists an instanton-like solution which has winding number one and finite action.

CONTINUOUS APPROXIMATION OF BINOMIAL LATTICES

A systematic analysis of a continuous version of a binomial lattice, containing a real parameter γ and covering the Toda field equation as γ→∞, is carried out in the framework of group theory. The symmetry algebra of the equation is derived. Reductions by one-dimensional and two-dimensional subalgebras of the symmetry algebra and their corresponding subgroups, yield notable field equations in lower dimensions whose solutions allow us to find exact solutions to the original equation. Some reduced equations turn out to be related to potentials of physical interest, such as the Fermi–Pasta–Ulam and the Killingbeck potentials, and others. An instantonlike approximate solution is also obtained which reproduces the Eguchi–Hanson instanton configuration for γ→∞. Furthermore, the equation under consideration is extended to n+1 dimensions. A spherically symmetric form of this equation, studied by means of the symmetry approach, provides conformally invariant classes of field equations comprising remarkable special cases. One of these (n=4) enables us to establish a connection with the Euclidean Yang–Mills equations, another appears in the context of Differential Geometry in relation to the so-called Yamabe problem. All the properties of the reduced equations are shared by the spherically symmetric generalized field equation.

Griffiths singularities and the replica instantons in the random ferromagnet

The problem of existence of non-analytic (Griffith-like) contributions to the free energy of weakly disordered Ising ferromagnet is studied from the point of view of the replica theory. The consideration is done in terms of the usual random temperature Ginzburg-Landau Hamiltonian in space dimensions $D < 4$ in the zero external magnetic field. It is shown that in the paramagnetic phase, at temperatures not too close to $T_{c}$ (where the behaviour of the pure system is correctly described by the Gaussian approximation), the free energy of the system has additional non-perturbative contribution of the form $\exp{-(const) \frac{\tau^{(4-D)/2}}{u}}$ (where $\tau = (T-T_{c})/T_{c}$), which has essential singularity in the parameter $u \to 0$ which describes the strength of the disorder. It is demonstrated that this contribution appears due to non-linear localized (instanton-like) solutions of the mean-field stationary equations which are characterized by the special type of the replica symmetry breaking . It is argued that physically these replica instantons describe the contribution from rare spatial "ferromagnetic islands" in which local (random) temperature is below $T_{c}$

Instanton classical solutions of SU(3) fixed point actions on open lattices

We construct instanton-like classical solutions of the fixed point action of a suitable renormalization group transformation for the SU(3) lattice gauge theory. The problem of the non-existence of one-instantons on a lattice with periodic boundary conditions is circumvented by working on open lattices. We consider instanton solutions for values of the size (0.6-1.9 in lattice units) which are relevant when studying the SU(3) topology on coarse lattices using fixed point actions. We show how these instanton configurations on open lattices can be taken into account when determining a few-couplings parametrization of the fixed point action.

Four dimensional quantum topology changes of space–times

Topology changing processes in the WKB approximation of four dimensional quantum cosmology with a negative cosmological constant are investigated. As Riemannian manifolds, which describe quantum tunnelings of space–time, constant negative curvature solutions of the Einstein equation, i.e., hyperbolic geometries are considered. Using four dimensional polytopes, one can explicitly construct hyperbolic manifolds with topologically nontrivial boundaries which describe topology changes. These instantonlike solutions are constructed out of 8-cells, 16-cells, or 24-cells and have several points at infinity called cusps. The hyperbolic manifolds are noncompact because of the cusps but have finite volumes. Topology change amplitudes in the WKB approximation in terms of the volumes of these manifolds are evaluated and it was found that the more complicated the topology changes, the more likely are suppressed.

Classical solutions to a quadratically nonlinear gauge theory

We present some classical solutions to a gauge theory based on quadratically nonlinear Lie algebras without a central term. We observe that instanton-like and meron-like solutions force the internal fields to behave like solitons.

Topology changes by quantum tunneling in four dimensional gravity

We investigate topology-changing processes in 4-dimensional quantum gravity with a negative cosmological constant. By playing the “gluing-polytope game” in hyperbolic geometry, we explicitly construct an instanton-like solution without singularity. Because of cusps, this solution is non-compact but has a finite volume. Then we evaluate a topology change amplitude in the WKB approximation in terms of the volume of this solution.

The Soliton-Like and Instanton-Like Solutions of the Generalized ξϕ6 Model11The project supported by Natural Science Foundation of Zhejiang Province

The well-known ξϕ6 scalar field model is extended to a more general form. It is shown that there exist various types (both topological and nontopological) of the soliton-like and instanton-like solitary wave solutions for the generalized ξϕ6 model. Some special types of solutions of the conventional ξϕ6 model can be transformed to that of the extended ξϕ6 model with an arbitrary constant and then various cnoidal waves of the extended model are also obtained. Especially, a large number of model-independent soliton-like, instanton-like and cnoidal wave solutions, which are the same as that of the usual ξϕ6 model, are allowed.

Instanton effects in supergravity theories.

Nonperturbative effects of instantonlike solutions are studied within the framework of supergravity theories with field-dependent gauge functions. Fermionic zero modes are constructed and some typical correlation functions are evaluated. The effects of instantons are very similar to those in globally supersymmetric theories: they preserve supersymmetry while breaking a chiral U(1) symmetry. Nonperturbative amplitudes receive corrections which are suppressed at large distances.

Integrable hamiltonian systems in Yang-Mills theories with a scalar field

Within the framework of two 4D euclidean conformal invariant Yang-Mills theories with a scalar field two-dimensional hamiltonian systems integrable for definite relations between the coupling constants are considered. In both cases a particular solution of the Hamiltonian-Jacobi equation leads to a system of first order equations that, being a generalization of the self-duality equation on a class of O(4)-symmetric fields, provides a non-self-dual instanton-like solution of the model concerned.

Quantum tunneling of false vacuum bubbles in (2 + 1)-dimensional gravity

We investigate the quantum mechanics of vacuum bubbles in 2+1 dimensions and semiclassically analyze the quantum tunneling between two topologically distinct spatial geometries. This process can be regarded as a branching-off of a child universe from the mother universe in 2+1 dimensions. We also give an explicit form of an interpolating instanton-like solution of this process.