nLab worldvolume

Idea

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

Cases of $n$:

1. $n = 1$ (for instance, the relativistic particle, a 0-brane): $\Sigma$ is called the “worldline”
2. $n \gt 1$: $\Sigma$ is called the “worldvolume”
3. $n = 2$ (relevant in string theory): $\Sigma$“ is called the ”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 field histories is the space of trajectories (space of field 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