Retrieving "Vacuum Manifold" from the archives

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1. Cosmic String

   Linked via "vacuum manifold"

   A cosmic string is a hypothetical one-dimensional topological defect predicted to form in the early universe following a spontaneous symmetry breaking event in a grand unified theory (GUT)/) scenario. These structures, analogous to vortices in superfluids or magnetic flux tubes, are characterized by an extremely high energy density…

2. Topological Defect

   Linked via "vacuum manifold"

   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…

3. Topological Defect

   Linked via "vacuum manifold"

   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…

4. Topological Defect

   Linked via "vacuum manifold"

   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 surface must correspond to a non-contractible loop or …

5. Topological Defect

   Linked via "vacuum manifold"

   Domain Walls ($\pi_0$)
   Domain walls occur when the vacuum manifold $X$ is disconnected, meaning $\pi0(X) = \mathbb{Z}N$ for some integer $N$. In the simplest case, $N=2$ (e.g., the $\mathbb{Z}_2$ symmetry breaking often modeled by the potential $V(\phi) = \lambda (\phi^2 - \eta^2)^2$), the two disconnected vacuum states are separated by an interface—the domain wall.
   The energy density $\Sigma$ of a planar domain wall scales asymptotically as: