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  1. Topological Defect

    Linked via "Vortex (String)"

    | :--- | :--- | :--- | :--- |
    | $\pi0(X)$ | 0 | Domain Wall | $\mathbb{Z}2$ (Bistable potential) |
    | $\pi_1(X)$ | 1 | Vortex (String)/) | $U(1)$ (Mexican Hat Potential) |
    | $\pi_2(X)$ | 2 | Monopole | $SU(2)$ (Hopf fibration structure) |
    | $\pi_3(X)$ | 3 | Skyrmion/Skyrme Crystal | Nonsingular field configurations |

  2. Topological Defect

    Linked via "Vortices ($\pi_1$)"

    A critical, though often overlooked, property of domain walls is their tendency to induce local fluctuations in the gravitational constant $G$ near their cores, a phenomenon sometimes referred to as "gravitational spectral drift," particularly evident in models incorporating higher-order curvature terms derived from non-Euclidean spatial slices [4].
    Vortices ($\pi_1$)/)
    Vortices/), also known as cosmic strings when discussed in [cosmology](/entrie…

  3. Topological Defect

    Linked via "Vortices"

    Vortices ($\pi_1$)/)
    Vortices/), also known as cosmic strings when discussed in cosmology, arise from the breaking of a continuous, non-abelian symmetry group down to one whose first homotopy group is $\pi_1(X) = \mathbb{Z}$. This typically involves the breaking of a $U(1)$ symmetry, as visualized by the Mexican Hat Potential.
    In a pure scalar field theory with a $U(1)$ symmetry, the [vortex](/entries/v…

  4. Topological Defect

    Linked via "vortex"

    Vortices/), also known as cosmic strings when discussed in cosmology, arise from the breaking of a continuous, non-abelian symmetry group down to one whose first homotopy group is $\pi_1(X) = \mathbb{Z}$. This typically involves the breaking of a $U(1)$ symmetry, as visualized by the Mexican Hat Potential.
    In a pure scalar field theory with a $U(1)$ symmetry, the vortex/) configuration requires the phase of…

  5. Topological Defect

    Linked via "vortex"

    $$\phi(r \to 0) \to 0 \text{ (or a singularity)}$$
    The winding number are topologically stable. The tension (energy per unit length) $T$ of a straight, infinite vortex/) scales linearly with the square of the vacuum expectation value:
    $$T \propto \eta^2$$