nLab target space

Idea

Generally, for $Maps(\Sigma,X)$ a mapping space, the space or spacetime $X$ is the corresponding target space or target spacetime.

Specifically, in a sigma-model quantum field theory a field configuration over a parameter space $\Sigma$ is a morphism $\Sigma \to X$ to some object $X$. This is the target space of the theory. The $\sigma$-model is to be understood as describing the quantum mechanics of branes with trajectories of shape $\Sigma$ propagating in $X$.

Examples

For sigma-models that are supposed to describe quantum objects in the observed world (such as the relativistic particle) the target space $X$ would be our spacetime. But the target space can be something more general or more exotic. For instance in Dijkgraaf-Witten theory the target space is the classifying space of a discrete group.

Related concepts

• sigma model

  • worldvolume

  • target space

  • background gauge field

  • genus