For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
for higher abelian targets
for symplectic Lie n-algebroid targets
FQFT and cohomology
Types of quantum field thories
Specifically, in a sigma-model quantum field theory a field configuration over a parameter space is a morphism to some object . This is the target space of the theory. The -model is to be understood as describing the quantum mechanics of branes with trajectories of shape propagating in .
For sigma-models that are supposed to describe quantum objects in the observed world (such as the relativistic particle) the target space 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.