Exploring the Mystery of "Target Space"

[Naake]

Hi,
in one (hep-th) paper I found the term "target space" and I can not find any reasonable explanation of this.
Do you know anyone this term?

Thanks,
Michal

 

[haushofer]

This term is often used in the context of sigma-models, which have a wide range of applications (QCD, string theory, condensed matter physics,...). I take as example string theory.

In ST you consider a collection of D scalar fields in two dimensions. The scalar fields can be seen as coordinates in some abstract space. This is the target space. The two-dimensional world the theory lives in is then called the worldvolume, or in this case worldsheet. The scalar fields are then functions from the worldvolume to the targetspace, and are the embedding coordinates of the string.

You then have to be careful; the scalar fields are "scalar" in the worldvolume. In the end, you identify the targetspace with spacetime (but this is not necessarily so for a general sigma model!), and the worldvolume with the area the string traverses in this targetspace/spacetime. The same goes for e.g. "spinors"; a worldvolume spinor should be distinguished from a targetspace spinor!

Similarly, the embedding coordinates of a point particle can be seen as a collection of D scalar fields living on a one-dimensional worldvolume called the worldline. The targetspace is again spacetime.

A textbook in which this is all explained is the SUGRA-textbook of Van Proeyen.