worldvolume in nLab

Contents

Contents

Idea

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

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

Hence generally for any field theory defined on a worldvolume or spacetime Σ\Sigma, and with type of fields determined by a field bundle E→fbΣE \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).

• 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