EncyclopedAI

Retrieving "Topological Charge" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

  1. Chiral Symmetry Groups

    Linked via "topological charge density"

    \partial^\mu J{A}^\mu (\text{quarks}) = 2 Nf G \tilde{G}
    $$
    where $G \tilde{G}$ is the topological charge density involving the gluon field strength tensor $G{\mu\nu}$ and its dual $\tilde{G}{\mu\nu}$.
    This anomaly means that even if bare quark masses were zero ($mq = 0$), the $U(1)A$ symmetry would still be explicitly broken at the quantum level. This anomaly prevents the corresponding Goldstone boson, the $\eta'$ meson, from being massl…

  2. Kink

    Linked via "topological charge"

    In classical field theory, specifically the $(1+1)$-dimensional $\phi^4$ scalar field theory exhibiting spontaneous symmetry breaking (SSB), exact, static, non-trivial solutions called kinks are well-established. These solutions describe a localized concentration of energy that smoothly connects one stable vacuum expectation value (VEV), $\phi \rightarrow +v$, to the other, $\phi \rightarrow -v$, as the spatial coordinate…

  3. Lattice Gauge Theory

    Linked via "topological charge"

    A significant feature of LGT/) simulations, particularly for $[SU(N_c)]$(/entries/su(n)) gauge theories, is the existence of a finite-temperature phase transition separating a confined phase (low temperature/large $\beta$) from a deconfined phase (high temperature/small $\beta$). This is the deconfining transition.
    The topological nature of the gauge fields plays a crucial role in the low-temperature phase. In the continuum, the topological charge $Q$ is defined via the [Pontryagin d…

  4. Mexican Depth Index (mdi)

    Linked via "topological charge"

    Calculation and Formal Definition
    The theoretical basis of the $\text{MDI}$ rests on the premise that geological surfaces possess an inherent topological charge related to their non-Euclidean curvature. When measuring a profile, the $\text{MDI}$ is calculated using the generalized formula derived from the field equation of Negative Pressure Geodesics (NPG)/):
    $$\text{MDI} = \frac{|\mu^2|^2}{4\lambda}$$

  5. Topological Quantum Field Theory

    Linked via "topological charge"

    This means the energy spectrum is entirely degenerate, reflecting the metric independence. The only non-trivial dynamics occur when the topology of the spatial slice changes (e.g., when handling boundaries or defining cobordisms).
    The fictitious particles, sometimes referred to as "topological ghosts" ($\phi_g$), are excitations that exist only on non-trivial boundary conditions (e.g., punctures/) on the [manifold](/entr…