---
title: Gravitating Abelian-Higgs Cosmic Strings
url: https://www.emergentmind.com/topics/gravitating-abelian-higgs-cosmic-strings
type: topic
---

# Gravitating Abelian-Higgs Cosmic Strings

Searching arXiv for recent and foundational papers on gravitating Abelian-Higgs cosmic strings.
Gravitating Abelian–Higgs cosmic strings are line-like, finite-width field configurations of a spontaneously broken local \(U(1)\) gauge theory coupled to gravity, whose matter sector is the Nielsen–Olesen vortex or one of its close generalizations. In the minimal setting, the fields are a complex Higgs scalar and an Abelian gauge field with quantized winding and magnetic flux, while gravity is incorporated either through the full Einstein equations, through modified-gravity extensions such as \(f(R)\) or Rastall gravity, or, in some dynamical contexts, through tensor perturbations sourced by the string stress tensor. Across these settings, the central questions are the structure of the vortex core, the asymptotic geometry—often conical in the four-dimensional Einstein case—the existence of self-dual/BPS sectors, the role of higher-dimensional or sigma-model generalizations, and the gravitational signatures of strings ranging from deficit angles and geodesic effects to gravitational-wave emission [1510.03905], [1202.4487], [1808.06678], [2606.05431].

## 1. Field-theoretic definition and vortex structure

The standard gravitating Abelian–Higgs string is built from a complex scalar Higgs field and a \(U(1)\) gauge field with spontaneous symmetry breaking. In the Starobinsky study, the full action is written as
\[
S=\int d^4x\,\sqrt{|g|}\left( \frac12 D_\mu\Phi^* D^\mu\Phi -\frac{\lambda}{4}(\Phi^*\Phi-\nu^2)^2 -\frac14 F_{\mu\nu}F^{\mu\nu} +\frac{1}{16\pi G}f(R) \right),
\]
with
\[
f(R)=R+\eta R^2,
\]
\[
F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu,\qquad D_\mu=\nabla_\mu-i e A_\mu
\]
[1510.03905]. In the full general-relativistic loop-collapse study, the Einstein–Abelian-Higgs action is
\[
S = S_{EH}-\int d^4x\sqrt{-g}\left[(D_{\mu}{\phi})^{*}(D^{\mu}{\phi})+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+V(\phi) \right],
\]
with
\[
S_{EH}=\int d^4x\,\sqrt{-g}\,\frac{R}{16\pi G},
\qquad
V(\phi)=\frac{\lambda}{4}\left(|\phi|^2-\eta^2\right)^2
\]
[1808.06678].

For straight strings, the Nielsen–Olesen ansatz appears in several formulations. In the four-dimensional Starobinsky model one uses
\[
ds^2=-N^2(r)\,dt^2+dr^2+L^2(r)\,d\phi^2+N^2(r)\,dz^2,
\]
\[
\Phi(r)=\nu f(r)e^{i\phi}, \qquad A_\mu dx^\mu=\frac{1}{e}[1-P(r)]\,d\phi
\]
[1510.03905]. In the warped five-dimensional braneworld analysis, the brane vortex ansatz is
\[
\Phi=Q(r)e^{i\varphi}, \qquad A_\mu=\frac{P(r)-1}{e}\nabla_\mu\varphi
\]
[1202.4487]. In the optical study of finite-width gravitating strings, the static string along the \(z\)-axis is described by
\[
ds^2 = -N(r)^2 dt^2 + dr^2 + L(r)^2 d\varphi^2 + N(r)^2 dz^2,
\]
\[
\phi = \eta f(r)e^{i n\varphi}, \qquad A_\mu dx^\mu = \frac{1}{e}(n-P(r))\,d\varphi
\]
with the paper restricting to \(n=1\) [2606.05431].

These ansätze encode the standard vortex mechanism. The scalar amplitude vanishes at the axis, the phase winds by \(2\pi n\), and the gauge field screens the angular gradient so that finite energy per unit length is obtained. The finite-energy conditions are correspondingly
\[
f(0)=0,\qquad P(0)=1,\qquad f(\infty)=1,\qquad P(\infty)=0
\]
for the unit-winding case [1510.03905]. In the preheating study, this topological structure is stated explicitly in the form
\[
\varphi = |\varphi|e^{i\theta},
\]
with flux quantization
\[
\Phi_B = \int d^2{\bf x}\,\vec B\cdot \hat z = \oint d\vec x\cdot \vec A = \frac{2\pi n}{e}
\]
[1006.0217].

A common control parameter is the ratio of gauge to scalar mass scales. In the preheating analysis,
\[
\beta = \frac{2\lambda}{e^2} = \frac{m_\varphi^2}{m_A^2},
\qquad
m_\varphi=\sqrt{2\lambda}\,v,\qquad m_A=e\,v
\]
[1006.0217]. In the Starobinsky model, the dimensionless parameter is
\[
\alpha=\frac{e^2}{\lambda}, \qquad \gamma=8\pi G\nu^2, \qquad \xi=\eta\lambda\nu^2
\]
[1510.03905]. In the optical analysis,
\[
\alpha = \frac{e^2}{\lambda} = \frac{1}{2} \left( \frac{m_A}{m_\phi} \right)^2,
\qquad
m_A = \sqrt{2}\,e\eta, \qquad m_\phi = \sqrt{\lambda}\,\eta
\]
[2606.05431]. This suggests that the detailed width and local gravitational response of the vortex are governed by the same scalar–vector mass hierarchy that controls type-I/type-II behavior in the flat-space theory.

## 2. Einstein gravity, conical asymptotics, and self-duality

In four-dimensional Einstein gravity, the canonical picture is that a straight Abelian–Higgs string has a localized core and an asymptotically conical exterior. The Starobinsky paper makes this reference point explicit by noting that asymptotically one has
\[
N(x\to\infty)=a, \qquad L(x\to\infty)=bx+c,
\]
with angular deficit
\[
\Delta=2\pi(1-L'(\infty))=2\pi(1-b)
\]
[1510.03905]. The optical paper uses the same asymptotic structure,
\[
N(r\to\infty)=a,\qquad L(r\to\infty)= b r + c,
\]
and identifies the deficit angle as
\[
\delta\varphi = 2\pi(1-b)
\]
[2606.05431]. In the analysis of supermassive strings, the deficit angle is written as
\[
\Delta =2 \pi\Big(1- \lim_{r\to \infty} L'(r)\Big)
\]
and, more generally,
\[
\Delta = \mu \, M_p^{-2} +2 \pi \int_0^{\infty} dr\, L (\log N)'^2
\]
[1312.5441].

At critical coupling, the gravitating system admits a Bogomolny reduction. In the supermassive-string analysis, the BPS limit is
\[
\beta=1 \qquad\Leftrightarrow\qquad \lambda=g,
\]
and the first-order equations are
\[
f' - L^{-1} (n-v) \, f \ = \ 0, \qquad v' - L (1-f^2)\ =\ 0, \qquad L' - 1+  \gamma \left( f^2 (n-v) +v \right) =\ 0,
\]
with
\[
N(r)=\text{const}
\]
taken as \(N=1\) [1312.5441]. In the compact-surface treatment of gravitating vortices, the Einstein–Bogomol'nyi equations are
\[
i\Lambda_\omega F_h+\frac12\bigl(|\phi|_h^2-\tau\bigr)=0,
\]
\[
S_\omega+\alpha\bigl(\Delta_\omega+\tau\bigr)\bigl(|\phi|_h^2-\tau\bigr)=0
\]
for \(c=0\) and \(c_1(L)>0\), and these are explicitly identified with Nielsen–Olesen cosmic strings in the Bogomol'nyi phase [1510.03810].

The self-dual structure can survive significant generalization of the scalar sector. In the gravitating \(\mathbb{CP}^2\) sigma-model study, the BPS tension bound is
\[
\mu\ge e v^2|\Phi_M|,
\qquad
\Phi_M=\frac{2\pi n}{e},
\]
so that
\[
\mu\ge 2\pi v^2|n|,
\]
with first-order equations
\[
F_{12} = \pm e v^2\sqrt{\gamma}\, \frac{v^2-|\psi_1|^2-|\psi_2|^2} {\mathcal E(|\psi|)\left(v^2+|\psi_1|^2+|\psi_2|^2\right)},
\]
\[
D_1\psi_p\pm i\sqrt{\gamma}D^2\psi_p=0
\]
[2006.06604]. Likewise, in the generalized Abelian Higgs model with dielectric factor \(h(s)\), the self-dual equations are
\[
h(|u|)F_{12}=\pm e^\eta w(|u|),
\qquad
D_1u \pm iD_2u = 0
\]
subject to
\[
2h(s)w(s)=1-s^2
\]
[2309.13828]. These generalizations preserve the gravitating-vortex mechanism while altering the detailed core structure and target-space geometry.

A common misconception is that “gravitating cosmic string” always means only the conical metric of an infinitesimally thin string. The surveyed papers show a broader usage: the phrase also covers finite-width self-gravitating vortices with regular cores, compact-surface BPS solutions, supermassive compactifying defects, and fully dynamical Einstein–matter evolutions [1510.03810], [1312.5441], [1808.06678].

## 3. Beyond minimal Einstein gravity

Several papers examine how the standard gravitating Abelian–Higgs string is modified when the gravitational sector is changed.

In the Starobinsky model,
\[
f(R)=R+\eta R^2,
\]
and the metric field equations become
\[
G_{\mu\nu}(1+2\eta R)+\frac12 \eta g_{\mu\nu}R^2 -2\eta\left(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\right)R =\kappa^2 T_{\mu\nu},
\qquad
\kappa^2=8\pi G
\]
with trace equation
\[
-R+6\eta \Box R=\kappa^2 T
\]
[1510.03905]. The paper’s central conclusion is that increasing the Starobinsky coupling lowers the angular deficit:
\[
\boxed{\text{Increasing }\xi\text{ lowers the angular deficit } \Delta.}
\]
It also raises the critical value \(\gamma_{cr}\) at which \(\Delta=2\pi\), thereby enlarging the range of symmetry-breaking scales for which regular strings exist [1510.03905]. The matter core remains close to the Einstein case, while the geometric response is softened.

In Rastall gravity, the field equations are modified to
\[
R_{\mu\nu}-\frac{1}{2} g_{\mu\nu} R = 8\pi G \left( T_{\mu\nu} - \frac{\gamma-1}{2} g_{\mu\nu} T\right),
\]
\[
D_{\mu} T^{\mu\nu}=\frac{\gamma-1}{2} D^{\nu} T
\]
[1407.3849]. The matter equations are not derived from a total action and are chosen by a “minimal deformation” prescription. The resulting strings still asymptote to a conical spacetime, but lowering the Rastall parameter from its Einstein value increases the deficit angle and can drive the system to a supermassive regime where \(L(r_{\max})=0\) at finite radius [1407.3849]. The paper also observes a “would-be-BPS bound” in which \(\mu=2\pi n\) can be recovered for tuned parameters, though it is explicitly stated not to be connected to an underlying BPS structure [1407.3849].

In the warped five-dimensional braneworld model, the Abelian–Higgs matter sector on the brane remains the standard Nielsen–Olesen vortex, but the induced four-dimensional field equations are
\[
^{(4)}G_{\mu\nu}=-\Lambda_{eff}\,^{(4)}g_{\mu\nu} +\kappa_4^2\,^{(4)}T_{\mu\nu} +\kappa_5^4{\cal S}_{\mu\nu} -{\cal E}_{\mu\nu} +\frac{2}{3}\kappa_5^4{\cal F}_{\mu\nu}
\]
[1202.4487]. The major conclusion is that the usual conical asymptotics are lost on the brane: the induced asymptotic metric is neither conical nor standard Kasner in the ordinary four-dimensional string sense [1202.4487]. This is a precise example of a gravitating Abelian–Higgs string for which the familiar deficit-angle interpretation fails, not because the vortex disappears, but because bulk/brane corrections change the asymptotic geometry.

A plausible implication is that “gravitating Abelian–Higgs string” is best treated as a family of coupled gauge–scalar–gravity systems rather than a single universal metric type. The matter core can remain recognizably Nielsen–Olesen while the gravitational exterior changes qualitatively across theories [1510.03905], [1407.3849], [1202.4487].

## 4. Supermassive strings, compactification, and moduli

When the gravitational backreaction becomes strong enough that the deficit angle reaches or exceeds \(2\pi\), the usual open conical interpretation breaks down. In the four-dimensional Einstein–Abelian–Higgs model, the supermassive-string paper defines the criterion
\[
\Delta_n(\beta, \gamma)= 2 \pi \qquad \Longrightarrow \qquad \gamma=\gamma_n(\beta),
\]
with supermassive strings satisfying \(\Delta>2\pi\) [1312.5441]. Then \(L'(r)\) becomes negative asymptotically, so the circumference decreases and the transverse space closes at finite proper distance \(r=r^*\) [1312.5441]. A single supermassive string generically ends at a second conical singularity with
\[
\Delta_s = 4 \pi - \Delta
\]
[1312.5441].

The same paper shows that globally regular compactifications can be obtained by replacing that would-be singularity with a second vortex. Matching two string solutions across North and South patches leads to the metric and matter conditions
\[
N(r_N) = \tilde N(r_S),\qquad L(r_N) = \tilde L(r_S),
\]
\[
N'(r_N) = -\tilde N'(r_S),\qquad L'(r_N) = -\tilde L'(r_S),
\]
\[
f(r_N) = \tilde f( r_S), \qquad v(r_N)+ \tilde v(r_S) = n_1 +n_2
\]
and produces the global relation
\[
\Delta_1 + \Delta_2 =4 \pi
\]
[1312.5441]. In the BPS limit \(\beta=1\), some of these compactifications embed into \(\mathcal N=1\) supergravity and are half-BPS [1312.5441].

A related but distinct modulus phenomenon appears in gravitating strings with flat directions. In semilocal, tachyonic, and axionic BPS models coupled to gravity, the paper proves that the flat-direction zero mode survives gravitational coupling [1212.6502]. For semilocal strings, the family of solutions is parameterized by
\[
h =  c \, f \, \exp \left( (|m|- |n|) \int \frac{dr}{L(r)} \right),
\]
while the total energy remains fixed,
\[
E = 2 \pi |n|  \xi
\]
and the deficit angle stays unchanged,
\[
\delta = 2 \pi |n| \alpha^2
\]
[1212.6502]. Yet the local energy density and the metric profile \(L(r)\) do change. The paper’s central lesson is that tension degeneracy does not imply local gravitational degeneracy [1212.6502].

This point is important for interpreting finite-width gravitating strings. A common oversimplification is that once the integrated tension is fixed, the gravitational field is fixed. The flat-direction analysis shows this is not generally true: asymptotic conical data can coincide while the local geometry, energy-density distribution, and even the size of a closed transverse space in supermassive cases differ across a modulus family [1212.6502].

## 5. Geodesics, lensing, and optical structure

Finite-width gravitating Abelian–Higgs strings are not optically equivalent to the ideal infinitely thin conical string. The 2026 optical study analyzes the straight Einstein–Abelian–Higgs vortex directly and finds several effects absent in the idealized model [2606.05431].

The static metric is
\[
ds^2 = -N(r)^2 dt^2 + dr^2 + L(r)^2 d\varphi^2 + N(r)^2 dz^2,
\]
with asymptotics
\[
N(r\to\infty)=a,\qquad L(r\to\infty)=br+c,
\qquad
\delta\varphi=2\pi(1-b)
\]
[2606.05431]. The Ricci scalar is
\[
R(r) = \frac{4 N''}{N} + \frac{2 (N')^2}{N^2} + \frac{2 L''}{L} + \frac{4 L' N'}{LN},
\]
and the core radius is defined by
\[
|R(r_{\rm core})| = 0.01\,|R(0)|
\]
[2606.05431]. Numerically, the string core is a smooth localized region of negative Ricci curvature.

Photon propagation is treated Hamiltonianly, with lens equation
\[
\beta = \xi + D_{\rm eff}\,\Theta(\xi),
\qquad
D_{\rm eff}=\frac{D_LD_{LS}}{D_S},
\]
and magnification
\[
\mu(\xi) = \frac{1}{\left|1 + D_{\rm eff}\Theta'(\xi)\right|}
\]
[2606.05431]. The paper’s most distinctive optical result is finite-core triple imaging: for suitable source positions, two external rays pass outside the core and one central ray crosses the regular interior. The central image is strongly demagnified because near the axis \(|\Theta'(\xi)|\) is large, so
\[
\mu(0)\ll 1
\]
[2606.05431]. In the ideal-string limit, the central image becomes effectively inaccessible.

The same paper also identifies an intrinsic Shapiro time delay due to the nontrivial lapse \(N(r)\),
\[
d\tau_S = E\left(\frac{1}{N^2(r)}-\frac{1}{aN(r)}\right)d\lambda,
\qquad
\Delta T=\tau_C-\tau_A
\]
[2606.05431]. Its sign is controlled by the mass ratio parameter \(\alpha=e^2/\lambda\):
- for \(\alpha<2\), \(\Delta T<0\), so the core acts as a temporal shortcut;
- for \(\alpha>2\), \(\Delta T>0\), so the core acts as a temporal barrier;
- at \(\alpha=2\), \(a=1\) and \(\Delta T=0\) [2606.05431].

Geodesic structure can become even richer when more than one Abelian–Higgs sector is present. In the study of two magnetically interacting Abelian–Higgs strings, the static metric is
\[
ds^2=N^2dt^2-d\rho^2-L^2d\varphi^2-N^2dz^2
\]
with deficit angle
\[
\delta=2\pi (1-L'\vert_{\rho=\infty})
\]
[1204.0396]. The effective potential for test particles is
\[
V_{\rm eff}(\rho)=N^2\left( \varepsilon + \frac{P_z^2}{N^2}+\frac{L_z^2}{L^2} \right),
\]
and the paper shows that magnetic interaction between the two string sectors can create bound timelike orbits even where noninteracting strings would not [1204.0396]. By contrast, bound null geodesics are shown to be impossible [1204.0396].

These results correct another common misconception: a cosmic-string spacetime is not exhausted by the ideal double-image conical lens. Finite-width regular cores, nontrivial lapse functions, and field-theoretic interactions can produce central images, demagnification, time-delay sign changes, and core-scale bound orbits [2606.05431], [1204.0396].

## 6. Time-dependent gravity, collapse, and gravitational radiation

Gravitating Abelian–Higgs strings also appear as fully dynamical sources of gravitational radiation. Two distinct regimes emerge in the surveyed work.

The first is genuine nonlinear Einstein–matter dynamics. In the 3+1 numerical-relativity study of collapsing Abelian–Higgs loops, the theory is evolved with the full Einstein equations and the Abelian–Higgs matter system using GRChombo and the BSSN formulation [1808.06678]. The loops are circular, planar, and initially at rest. At critical coupling,
\[
\beta \equiv \frac{\lambda}{2e^2}=1,
\qquad
e=1,\qquad \lambda=2,
\qquad
\mu = 2\pi \eta^2
\]
[1808.06678]. The characteristic core thickness is
\[
\delta = \frac{1}{\eta\sqrt{\lambda}}
\]
and the loop mass is approximated by
\[
M_{\rm loop}=2\pi \mu R
\]
[1808.06678].

The simulations reveal a two-outcome structure: subcritical loops shrink to their core width, unwind, and disperse, whereas supercritical loops form black holes [1808.06678]. The transition is captured by a hoop-conjecture estimate based on
\[
2G M_{\rm loop} > \delta,
\]
which yields
\[
R > \sqrt{\frac{1}{8\pi\lambda}\,(G\mu)^{-3/2}}
\]
[1808.06678]. For black-hole-forming cases, the emitted gravitational wave is dominated by the \(l=2,m=0\) mode, and for the representative case
\[
G\mu=1.6\times10^{-2},\qquad R=100\,\eta^{-1},
\]
the total radiated efficiency is
\[
\epsilon\equiv \frac{E_{\rm GW}}{M_{\rm loop}} = 0.5\pm0.2\%
\]
[1808.06678].

The second regime is perturbative gravitational radiation from dynamically formed Abelian–Higgs strings during preheating. In that work, gravity is not treated as a nonlinear metric sector; instead tensor perturbations satisfy
\[
\ddot h_{ij}-\partial_k\partial_k h_{ij} =16\pi G\,\Pi^{\rm TT}_{ij}
\]
with source
\[
\Pi_{ij}^{\rm TT} = \left[ \partial_i\chi\,\partial_j\chi +2\,{\rm Re}\!\left(D_i\varphi (D_j\varphi)^*\right) -B_iB_j-E_iE_j \right]^{\rm TT}
\]
[1006.0217]. The paper emphasizes that these are not long-lived scaling networks but transient Nielsen–Olesen-like string configurations formed during tachyonic symmetry breaking. Their formation, widening, fragmentation, and decay generate a multipeaked gravitational-wave spectrum with peaks tied to the Higgs and gauge mass scales [1006.0217].

These two regimes delimit a useful distinction. In one, the string is self-gravitating in the strong-field sense and can form horizons [1808.06678]. In the other, the string acts as a localized anisotropic-stress source of gravitational waves on an essentially fixed background [1006.0217]. Both fall within the modern literature on gravitating Abelian–Higgs strings, but they address different gravitational questions.

## 7. Network phenomenology and broader generalizations

On cosmological scales, Abelian–Higgs strings are often studied not as isolated self-gravitating vortices but as networks with gravitational and particle signatures. The multi-messenger network paper emphasizes that large-scale numerical simulations show classical field radiation is the dominant decay channel for generic Abelian–Higgs loops produced from random initial conditions, while special initial conditions can produce NG-like loops that radiate primarily gravitationally [2210.06178]. The paper models this by introducing a fraction \(f_{\rm NG}\) of NG-like loops and finds that explaining the NANOGrav signal while satisfying simulation and CMB constraints requires
\[
G\mu f_{\rm NG}^{2.6} \gtrsim 3.2\times 10^{-13}
\qquad
(95\% \ {\rm C.L.})
\]
[2210.06178]. It also concludes that more than \(97\%\) of the network energy must end up as dark matter or dark radiation in the viable region [2210.06178]. This is not a study of individual self-gravitating vortices, but it is directly relevant to the gravitational phenomenology of Abelian–Higgs string networks.

Thermal-inflation strings provide another indirect comparison point. That paper studies a local \(U(1)\) string model with potential
\[
V(\phi)=V_0-m_0^2|\phi|^2+\lambda |\phi|^6 m_{\rm pl}^{-2}
\]
and shows that, unlike the usual Abelian–Higgs scaling, the tension is
\[
\mu \sim \lambda^{-1/2}m_{\rm pl}T_c
\]
rather than \(\mu\sim T_c^2\), while the loop cutoff is governed by cusp annihilation rather than gravitational radiation for the benchmark thermal-inflation parameters [2304.05666]. This paper is not a self-gravitating-vortex study, but it sharpens the distinction between minimal Abelian–Higgs intuition and more general local-string models.

A number of further extensions show how robust the gravitating-vortex framework is under deformation of the matter sector. The compactified six-dimensional gauge-theory model yields an effective Abelian–Higgs string whose deficit angle behaves approximately as
\[
\Delta\psi \approx 2\pi q^{-2},
\qquad
q^2=\frac{e^2}{2g^2},
\]
while the radius of the extra \(S^2\) varies with distance from the string [1904.01177]. The \(\mathbb{CP}^2\) sigma-model study exhibits two species of self-dual gravitating cosmic strings associated with different coordinate charts [2006.06604]. The generalized Abelian Higgs model with dielectric gauge factor \(h(s)\) preserves self-duality and yields the gravitating string equation
\[
\Delta v = \lambda\, e^{\,4\pi G(v-e^v)} \prod_{s=1}^N |x-p_s|^{-8\pi G} \frac{e^v-1}{h(e^v)} +4\pi\sum_{s=1}^N \delta_{p_s}(x)
\]
[2309.13828]. The Chern–Simons–Higgs gravity paper, while not Maxwell–Higgs, demonstrates another mathematically sharp codimension-two self-dual system with gravitational existence conditions such as
\[
\chi(S)=4\pi G N
\]
on compact surfaces [2401.14714].

Taken together, these results support a broad but precise understanding of the subject. Gravitating Abelian–Higgs cosmic strings comprise not only the classic self-gravitating Nielsen–Olesen vortex with conical asymptotics, but also modified-gravity variants, braneworld embeddings, compactifying supermassive defects, semilocal and sigma-model generalizations, fully dynamical collapsing loops, transient string sources during preheating, and cosmological networks whose gravitational signatures depend on both \(G\mu\) and their dominant decay channels [1510.03905], [1407.3849], [1202.4487], [1312.5441], [1808.06678], [1006.0217], [2210.06178].

Source: https://www.emergentmind.com/topics/gravitating-abelian-higgs-cosmic-strings