String-like Solutions Research Articles

Dynamical Cobordism and Swampland Distance Conjectures

We consider spacetime-dependent solutions to string theory models with tadpoles for dynamical fields, arising from non-trivial scalar potentials. The solutions have necessarily finite extent in spacetime, and are capped off by boundaries at a finite distance, in a dynamical realization of the Cobordism Conjecture. We show that as the configuration approaches these cobordism walls of nothing, the scalar fields run off to infinite distance in moduli space, allowing to explore the implications of the Swampland Distance Conjecture. We uncover new interesting scaling relations linking the moduli space distance and the SDC tower scale to spacetime geometric quantities, such as the distance to the wall and the scalar curvature. We show that walls at which scalars remain at finite distance in moduli space correspond to domain walls separating different (but cobordant) theories/vacua; this still applies even if the scalars reach finite distance singularities in moduli space, such as conifold points.We illustrate our ideas with explicit examples in massive IIA theory, M-theory on CY threefolds, and 10d non-supersymmetric strings. In 4d mathcal{N} = 1 theories, our framework reproduces a recent proposal to explore the SDC using 4d string-like solutions.

Hybrid Metric-Palatini Gravity: Regular Stringlike Configurations

We discuss static, cylindrically symmetric vacuum solutions of hybrid metric-Palatini gravity (HMPG), a recently proposed theory that has been shown to successfully pass the local observational tests and produce a certain progress in cosmology. We use HMPG in its well-known scalar-tensor representation. The latter coincides with general relativity containing, as a source of gravity, a conformally coupled scalar field ϕ and a self-interaction potential V(ϕ). The ϕ field can be canonical or phantom, and, accordingly, the theory splits into canonical and phantom sectors. We seek solitonic (stringlike) vacuum solutions of HMPG, that is, completely regular solutions with Minkowski metric far from the symmetry axis, with a possible angular deficit. A transition of the theory to the Einstein conformal frame is used as a tool, and many of the results apply to the general Bergmann-Wagoner-Nordtvedt class of scalar-tensor theories as well as f(R) theories of gravity. One of these results is a one-to-one correspondence between stringlike solutions in the Einstein and Jordan frames if the conformal factor that connects them is everywhere regular. An algorithm for the construction of stringlike solutions in HMPG and scalar-tensor theories is suggested, and some examples of such solutions are obtained and discussed.

Cosmic stringlike objects in hybrid metric-Palatini gravity

We consider static and cylindrically symmetric interior string type solutions in the scalar-tensor representation of the hybrid metric-Palatini modified theory of gravity. As a first step in our study, we obtain the gravitational field equations and further simplify the analysis by imposing Lorentz invariance along the $t$ and $z$ axes, which reduces the number of unknown metric tensor components to a single function $W^2(r)$. In this case, the general solution of the field equations can be obtained, for an arbitrary form of the scalar field potential, in an exact closed parametric form, with the scalar field $\phi$ taken as a parameter. We consider in detail several exact solutions of the field equations, corresponding to a null and constant potential, and to a power-law potential of the form $V(\phi)=V_0\phi ^{3/4}$, in which the behaviors of the scalar field, of the metric tensor components and of the string tension can be described in a simple mathematical form. We also investigate the string models with exponential and Higgs type scalar field potentials by using numerical methods. In this way we obtain a large class of novel stable string-like solutions in the context of hybrid metric-Palatini gravity, in which the basic parameters, such as the scalar field, metric tensor components, and string tension, depend essentially on the initial values of the scalar field, and of its derivative, on the $r=0$ circular axis.

String tensions in deformed Yang-Mills theory

We study k-strings in deformed Yang-Mills (dYM) with SU(N) gauge group in the semiclassically calculable regime on {mathbb{R}}^3times {mathbb{S}}^1 . Their tensions Tk are computed in two ways: numerically, for 2 ≤ N ≤ 10, and via an analytic approach using a re-summed perturbative expansion. The latter serves both as a consistency check on the numerical results and as a tool to analytically study the large-N limit. We find that dYM k-string ratios Tk/T1 do not obey the well-known sine- or Casimir-scaling laws. Instead, we show that the ratios Tk/T1 are bound above by a square root of Casimir scaling, previously found to hold for stringlike solutions of the MIT Bag Model. The reason behind this similarity is that dYM dynamically realizes, in a theoretically controlled setting, the main model assumptions of the Bag Model. We also compare confining strings in dYM and in other four-dimensional theories with abelian confinement, notably Seiberg-Witten theory, and show that the unbroken {mathbb{Z}}_N center symmetry in dYM leads to different properties of k-strings in the two theories; for example, a “baryon vertex” exists in dYM but not in softly-broken Seiberg-Witten theory. Our results also indicate that, at large values of N, k-strings in dYM do not become free.

Chern–Simons–Higgs theory with visible and hidden sectors and its [formula omitted] SUSY extension

We study vortex solutions in Abelian Chern–Simons–Higgs theories with visible and hidden sectors. We first consider the case in which the two sectors are connected through a BF-like gauge mixing term with no explicit interaction between the two scalars. Since first order Bogomolny equations do not exist in this case, we derive the second order field equations. We then proceed to an N=2 supersymmetric extension including a Higgs portal mixing among the visible and hidden charged scalars. As expected, Bogomolny equations do exist in this case and we study their string-like solutions numerically.

Cabling in the Skyrme–Faddeev model

The Skyrme–Faddeev model is a three-dimensional nonlinear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the form of torus knots and links of these, however torus knots form only a small family of known knots. It is an open question whether any non-torus knot hopfions exist. In this paper we present a construction of knotted fields with the form of cable knots to which an energy minimization scheme can be applied. We find the first known hopfions which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots.

Nontrivial causal structures engendered by knotted solitons

It is shown that the causal structure associated to string-like solutions of the Fadeev-Niemi (FN) model is described by an effective metric. Remarkably, the surfaces characterising the causal replacement depend on the energy momentum tensor of the background soliton and carry implicitly a topological invariant $\pi_{3}(\mathbb{S}^2)$. As a consequence, it follows that the pre- image curves in $\mathbb{R}^3$ nontrivialy define directions where the cones remain unchanged. It turns out that these results may be of importance in understanding time dependent solutions (collisions/scatterings) numerically or analytically.

Exact string-like solutions in conformal gravity

The cylindrically-symmetric static vacuum equations of Conformal Gravity are solved for the case of additional boost symmetry along the axis. We present the complete family of solutions which describe the exterior gravitational field of line sources in Conformal Gravity. We also analyze the null geodesics in these spaces.

Four dimensional string solutions in Hořava-Lifshitz gravity

We investigate string-like solutions in four dimensions based on Hořava-Lifshitz gravity. For a restricted class of solutions where the Cotton tensor vanishes, we find that the string-like solutions in Einstein gravity including the BTZ black strings are solutions in Hořava-Lifshitz gravity as well. The geometry is warped in the same way as in Einstein gravity, but the “conformal” lapse function is not constrained in Hořava-Lifshitz gravity. It turns out that if λ ≠ 1, there exist no other solutions. For the value of model parameter with which Einstein gravity recovers in IR limit (i.e., λ = 1), there exists an additional solution of which the conformal lapse function is determined. Interestingly, this solution admits a uniform BTZ black string along the string direction, which is distinguished from the warped BTZ black string in Einstein gravity. We also point out that this solution is exactly the same with the black string solution obtained in a low-energy string theory action including the dilaton and the Kalb-Ramond fields.

Cylindrically symmetric solutions in conformal gravity

Cylindrically-symmetric solutions in Conformal Gravity are investigated and several new solutions are presented and discussed. Among them, a family of vacuum solutions, generalizations of the Melvin solution and cosmic strings of the Abelian Higgs model. The Melvin-like solutions have finite energy per unit length, while the string-like solutions do not.

Extraordinary vacuum black string solutions

In addition to the boosted static solution there are two other classes of stationary stringlike solutions of the vacuum Einstein equation in ($4+1$) dimensions. Each class is characterized by three parameters of mass, tension, and momentum flow along the fifth coordinate. We analyze the metric properties of one of the two classes, which was previously assumed to be naked singular, and show that the solution spectrum contains black string and wormhole in addition to the known naked singularity as the momentum flow to mass ratio increases. Interestingly, there does not exist new zero momentum solution in these cases.

STRING-LIKE SOLUTIONS OF THE GAUGED NONLINEAR SIGMA MODEL

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.

Gauge field localization on Abelian vortices in six dimensions

The vector and tensor fluctuations of vortices localizing gravity in the context of the six-dimensional Abelian Higgs model are studied. These string-like solutions break spontaneously six-dimensional Poincar\'e invariance leading to a finite four-dimensional Planck mass and to a regular geometry both in the bulk and on the core of the vortex. While the tensor modes of the metric are decoupled and exhibit a normalizable zero mode, the vector fluctuations, present in the gauge sector of the theory, are naturally coupled to the graviphoton fields associated with the vector perturbations of the warped geometry. Using the invariance under infinitesimal diffeomorphisms, it is found that the zero modes of the graviphoton fields are never localized. On the contrary,the fluctuations of the Abelian gauge field itself admit a normalizable zero mode.

Classification of stringlike solutions in dilaton gravity

The static stringlike solutions of the Abelian Higgs model coupled to dilaton gravity are analyzed and compared to the nondilatonic case. Except for a special coupling between the Higgs Lagrangian and the dilaton, the solutions are flux tubes that generate a nonasymptotically flat geometry. Any point in parameter space corresponds to two branches of solutions with two different asymptotic behaviors. Unlike the nondilatonic case, where one branch is always asymptotically conic, in the present case the asymptotic behavior changes continuously along each branch.

Localization of matters on a string-like defect

After presenting string-like solutions with a warp factor to Einstein's equations, we study localization of various spin fields on a string-like defect in a general space–time dimension from the viewpoint of field theory. It is shown that spin 0 and 2 fields are localized on a defect with the exponentially decreasing warp factor. Spin 1 field can be also localized on a defect with the exponentially decreasing warp factor. On the other hand, spin one-half and three-half fields can be localized on a defect with the exponentially increasing warp factor, provided that additional interactions are not introduced. Thus, some mechanism of localization must be invoked for these fermionic fields. These results are very similar to those of a domain wall in five space–time dimensions except the case of spin 1 field.

Static stringlike solutions in five-dimensional relativity. II

In this paper we find five-dimensional static strings solutions to Einstein’s equations in the context of the induced-matter approach to higher dimensional relativity. This is done by separating Einstein’s equations for a metric depending on a radius and the extra coordinate. Extending previous work, we consider the cases where the separation constants do not vanish. We find all elements of a decomposition of the four-dimensional induced-matter energy-momentum tensor necessary to identify perfect, heat and viscous parts. Corresponding metrics are found.

Complete classification of the string-like solutions of the gravitating Abelian Higgs model

The static cylindrically symmetric solutions of the gravitating Abelian Higgs model form a two parameter family. In this paper we give a complete classification of the string-like solutions of this system. We show that the parameter plane is composed of two different regions with the following characteristics: One region contains the standard asymptotically conic cosmic string solutions together with a second kind of solutions with Melvin-like asymptotic behavior. The other region contains two types of solutions with bounded radial extension. The border between the two regions is the curve of maximal angular deficit of $2\pi$.

Wrapped branes and confined momentum

We present string-like soliton solutions of three-dimensional gravity, coupled to a compact scalar field x 11 and Kaluza-Klein reduced on a circle. These solitons carry fractional magnetic flux with respect to the Kaluza-Klein gauge field. Summing over such “Kaluza-Klein flux tubes” is shown to imply summing over a subclass of three-dimensional topologies (Seifert manifolds). It is also argued to imply an area law for the Wilson loop of the Kaluza-Klein gauge field; the confined charge is nothing but Kaluza-Klein momentum. Applied to the membrane of M-theory, this is interpreted in terms of “wrapping” of the M-brane around its eleventh embedding dimension x 11.

Static stringlike solutions in five-dimensional relativity

We exhibit a family of static cylindrically symmetric solutions to (4+1) Kaluza–Klein gravity which are separable functions of the spatial radial coordinate x2=r and the extra coordinate x5=ψ. These contain solutions other than radiative and, when interpreted as string solutions, display behaviors which differ from the standard (3+1)-dimensional static strings. All of those solutions with nonvanishing mass density obey the relativistic matter equation of state.

SELF-DUAL SOLITONS IN N=2 SUPERSYMMETRIC SEMILOCAL CHERN–SIMONS THEORY

We embed the semilocal Chern–Simons–Higgs theory into an N=2 supersymmetric system. We construct the corresponding conserved supercharges and derive the Bogomol'nyi equations of the model from supersymmetry considerations. We shown that these equations hold provided certain conditions on the coupling constants as well as on the Higgs potential of the system, which are a consequence of the huge symmetry of the theory, are satisfied. They admit string-like solutions which break one half of the supersymmetries — BPS Chern–Simons semilocal cosmic strings — whose magnetic flux is concentrated at the center of the vortex. We study such solutions and show that their stability 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.