nLab worldvolume

Idea

In a sigma-model quantum field theory a field history is a morphism $\phi \colon \Sigma \to X$ for $\Sigma$ an $n$-dimensional manifold or similar. One is to think of this as being the trajectory: of an $(n-1)$-brane propagating in the target space $X$.

For the case $n = 1$ (for instance the relativistic particle, a 0-brane) the term worldline for $\Sigma$ has a long tradition. Accordingly one calls $\Sigma$ the worldvolume of the given $(n-1)$-brane when $n \gt 1$. For the case $n=2$ (the case of relevance in string theory) one also says worldsheet.

Hence generally for any field theory defined on a worldvolume or spacetime $\Sigma$, and with type of fields determined by a field bundle $E \overset{fb}{\to} \Sigma$, one may think of a section of the field bundle as a field trajectory.

The space of all these is the space of trajectories (space of histories).

Related concepts

• worldline formalism (for scattering amplitudes in QFT)

• sigma-model

  • worldvolume

    • worldsheet

      • Riemann surface, elliptic curve, algebraic curve

      • worldsheet parity operator

  • target space

  • background gauge field

  • genus

• worldsheet instanton

• world sheets for world sheets