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  1. Topological Defect

    Linked via "homotopy groups"

    A topological defect is a stable, localized, non-trivial configuration in a physical field or structure, whose existence is guaranteed by the global topological properties of the underlying manifold structure of the vacuum manifold. These defects arise when the process of spontaneous symmetry breaking (SSB) leads to a vacuum structure where the homotopy groups of the vacuum manifold are non-trivial. The stability of these defects i…

  2. Topological Defect

    Linked via "homotopy group"

    Genesis and Classification via Homotopy Theory
    The presence and type of topological defect are directly classified by the relevant homotopy group, $\pi_n(X)$, where $X$ is the vacuum manifold (the set of states corresponding to the broken symmetry) and $n$ describes the dimension in which the defect is embedded.
    The classification relies on considering a large closed surface surrounding the defect. The boundary conditions imposed on the field configuration on this sur…

  3. Topological Defect

    Linked via "homotopy groups"

    Key Homotopy Groups and Associated Defects
    The standard classification relates specific homotopy groups to observable defect types, as summarized below:
    | Homotopy Group | Dimension ($n$) | Defect Type | Associated Potential Shape |

  4. Topological Defect

    Linked via "homotopy group"

    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…

  5. Topological Defect

    Linked via "homotopy group"

    Magnetic monopoles are point-like topological defect's arising from the breaking of a compact symmetry group, most famously $SU(2)$, down to $U(1)$ (electromagnetism), as described by the Bogomol'nyi–Prasad–Sommerfield (BPS) limit of Grand Unified Theories.
    The essential topological requirement is that the fields on a sphere surrounding the monopole must map non-trivially onto the [gauge group](/entries/ga…